Level raising and anticyclotomic Selmer groups for Hilbert modular forms of weight two

Jan Nekovář

0. Introduction

0.0. Convention. A ’character’ always means a continuous character. For any perfect field k we denote by G_{k} = Gal(\overline{k}/k) the absolute Galois group of k.

0.1. Let F be a totally real number field; let f ∈ S_{2}(𝔫, ω) be – in the notation of [N2, ch. 12] – a cuspidal Hilbert modular eigenform over F of parallel weight 2, (exact) level 𝔫 and character ω : A_{F}^{×}/F^{×} ⟶ C^{×} (ω is a totally real character of finite order).

For each prime v ∤ 𝔫∞ of F denote by λ_{f}(v) the eigenvalue of the standard Hecke operator T (v) acting on f: T (v)f = λ_{f}(v)f. The field L_{f} ⊂ C generated over Q by all Hecke eigenvalues λ_{f}(v) and by the values of ω (in fact, L_{f} is generated by {λ_{f}(v)} for v belonging to any set of primes of F of density 1) is a totally real (resp. a CM) number field if ω = 1 (resp. if ω ≠ 1). 0.2. The (unitary) automorphic representation π(f) of GL_{2}(A_{F}) attached to f has central character ω and its standard L-function is related to the classical L-function of f by the relation

L(π(f), s − \frac{1}{2}) = Γ_{C}(s)^{[F:Q]} L(f, s), Γ_{C}(s) = 2(2π)^{−s}Γ(s),

which is valid Euler factor by Euler factor. In particular, the Euler factor at a prime v ∤ 𝔫∞ is equal to

L(π(f)_{v}, s) = (1 − λ_{f}(v)(Nv)^{−s−1/2} + ω(v)(Nv)^{−2s})^{−1} .

As π̃(f) = π(f) ⊗ ω^{−1}, the functional equation

L(π(f), s) = ε(π(f), s) L(π̃(f), 1 − s), ε(π(f), s) = c(π)^{1/2−s} ε(π(f), \frac{1}{2})

becomes self-dual if ω = 1, in which case ε(π(f), \frac{1}{2}) = ±1.

0.3. Let K be a totally imaginary quadratic extension of F and χ : A_{K}^{×}/K^{×} ⟶ C^{×} a character of finite order satisfying

χ|_{A_{F}^{×}} · ω = 1.  (0.3.1)

This condition implies that

cχ · χ · (ω ∘ N_{K/F}) = 1,  (0.3.2)

where c is the non-trivial element of Gal(K/F) and ^{c}χ : A_{K}^{×}/K^{×} ⟶ C^{×} is the character

(^{c}χ)(a) = χ(c^{−1}(a)) = χ(c(a)).

0.4. Fix a number field L ⊂ C containing L_{f} and the values of χ. Fix a prime 𝔭 of L above a rational prime p and denote by V_{𝔭}(f) the (cohomologically normalised) two-dimensional representation of G_{F} with coefficients in L_{𝔭} attached to f: if v ∤ p𝔫∞ is a prime of F, then V_{𝔭}(f) is unramified at v and

det (1 − Fr_{geom}(v)X | V_{𝔭}(f)) = 1 − λ_{f}(v)X + ω(v)(Nv)X^{2}.

We identify ω (resp. χ) with a Galois character ω : G_{F} ⟶ O_{L}^{×} (resp. χ : G_{K} ⟶ O_{L}^{×}) via the reciprocity map rec_{F} : A_{F}^{×}/F^{×} ⟶ G_{F}^{ab} (resp. rec_{K}) normalised by letting the uniformisers correspond to geometric Frobenius elements. The relation (0.3.2) then reads as follows:

cχ · χ · ω|_{G_{K}} = 1.

It is known (see [C1, Thm. A] and [T1, Thm. 2]) that, for each prime v ∤ p∞ of F, the restriction V_{𝔭}(f)_{v} of V_{𝔭}(f) to the decomposition group G_{F_{v}} corresponds to π(f)_{v} ⊗|·|^{−1/2} via the local Langlands correspondence. This implies that L(f, s) coincides with the L-function of the strongly compatible system of L-rational Galois representations {V_{𝔭}(f)}_{𝔭} of G_{F} in the sense that

∀v ∤ p∞ L_{v}(f, s) = det (1 − Fr_{geom}(v)(Nv)^{−s} | V_{𝔭}(f)^{I_{v}})^{−1} .

0.5. Denote by θ_{χ} the automorphic representation of GL_{2}(A_{F}) generated by the theta series of χ (which is a weight one Hilbert eigenform over F); its central character is equal to χ|_{A_{F}^{×}} · η, where η = η_{K/F} :

A_{F}^{×}/F^{×}N_{K/F} (A_{K}^{×}) ⟶^{∼} {±1} is the quadratic character corresponding to the extension K/F. As θ_{χ}⊗η = θ_{χ}, the condition (0.3.1) implies that the Rankin-Selberg L-function L(π(f) × θ_{χ}, s) (which will be abusively denoted by L(π(f) × χ, s)) coincides with L(π̃(f) × θ̃_{χ}, s), hence admits a self-dual functional equation

\begin{matrix}L(π(f) × χ, s) = ε(π(f) × χ, s) L(π(f) × χ, 1 − s), \\ ε(π(f) × χ, s) = c(π(f) × χ)^{1/2−s} ε(π(f) × χ, \frac{1}{2}),  ε(π(f) × χ, \frac{1}{2}) = ±1. \end{matrix}  (0.5.1)

In more concrete terms,

L(π(f) × χ, s − \frac{1}{2}) = Γ_{C}(s)^{[K:Q]} L(f_{K}, χ, s),

where L(f_{K}, χ, s) is the L-function of the strongly compatible system of L-rational Galois representations {V_{𝔭}(f)|_{G_{K}} ⊗ χ}_{𝔭} of G_{K}:

∀v ∤ p∞ L_{v}(f_{K}, χ, s) = ∏_{w|v} det (1 − Fr_{geom}(w)(Nw)^{−s} | (V_{𝔭}(f) ⊗ χ)^{I_{w}})^{−1} .

Above, v (resp. w) is a prime of F (resp. of K). Set

r_{an}(f_{K}, χ) := ord_{s=1} L(f_{K}, χ, s) = ord_{s=1/2} L(π(f) × χ, s).

0.6. For any G_{K}-module M we denote by ^{c}M the abelian group M equipped with a new action of G_{K} given by m ↦ (c^{−1}gc̃)m (m ∈ M, g ∈ G_{K}), where c̃ ∈ G_{F} is any element of G_{F} that does not belong to G_{K} (the isomorphism class of the G_{K}-module ^{c}M does not depend on the choice of c̃).

The map sending a non-homogeneous n-cochain z ∈ C^{n}(G_{K}, M) to the cochain z′ ∈ C^{n}(G_{K},^{c}M) given by z′(g_{1}, …, g_{n}) = z(c^{−1}g_{1}c̃, …, c^{−1}g_{n}c̃) induces an isomorphism

H^{n}(G_{K}, M) ⟶^{∼} H^{n}(G_{K},^{c}M).  (0.6.1)

In the special case when M is a G_{F} -module the map c̃ : M ⟶^{∼} ^{c}M (m ↦ c̃m) is an isomorphism of G_{K}-modules. 0.7. As det(V_{𝔭}(f)) = L_{𝔭}(−1) ⊗ ω, the L_{𝔭}[G_{F}]-module V := V_{𝔭}(f)(1) is equipped with a non-degenerate skew-symmetric G_{F} -equivariant pairing

V × V ⟶ L_{𝔭}(1) ⊗ ω,  (0.7.1)

which induces a non-degenerate G_{K}-equivariant pairing

(V |_{G_{K}} ⊗ χ) × (V |_{G_{K}} ⊗^{c}χ) ⟶ L_{𝔭}(1).

As a result, there is an isomorphism of G_{F} -modules

V^{∗}(1) := Hom_{L_{𝔭}}(V, L_{𝔭})(1) ⟶^{∼} V ⊗ ω^{−1}

and isomorphisms of G_{K}-modules

(V |_{G_{K}} ⊗ χ)^{∗} (1) ⟶^{∼} V |_{G_{K}} ⊗^{c}χ ⟶^{∼} ^{c} (V |_{G_{K}} ⊗ χ) .

The fields K_{φ} = \overline{K}^{Ker(φ)} (φ = χ,^{c}χ) are finite abelian extensions of K satisfying

K_{χ}K_{^{c}χ} = K_{χ}F_{ω} = K_{^{c}χ}F_{ω},

where F_{ω} = \overline{F}^{Ker(ω)} is the (totally real) finite abelian extension of F trivialising ω. 0.8. We are interested in the Bloch-Kato Selmer groups

H_{f}^{1}(K, V ⊗ φ) = (H_{f}^{1}(K_{φ}, V) ⊗ φ)^{Gal(K_{φ}/K)} = H_{f}^{1}(K_{φ}, V)^{(φ^{−1})} (φ = χ,^{c}χ),

where

M^{(φ)} = {m ∈ M | ∀g ∈ Gal(K_{φ}/K) g(m) = φ(g)m}

for any O_{L}[Gal(K_{φ}/K)]-module M. Set

h_{f}^{1}(K, V ⊗ φ) := dim_{L_{𝔭}} H_{f}^{1}(K, V ⊗ φ).

The discussion in 0.6–0.7 implies that there are isomorphisms

H_{f}^{1}(K, (V ⊗ χ)^{∗}(1)) ⟶^{∼} H_{f}^{1}(K, V ⊗^{c}χ) ⟶^{∼} H_{f}^{1}(K,^{c}(V ⊗ χ)) ⟶^{∼} H_{f}^{1}(K, V ⊗ χ).  (0.8.1)

As the Galois representation V is pure of weight −1 by the generalised Ramanujan conjecture [Bl],

H^{0}(K, (V ⊗ χ)^{∗}(1)) = H^{0}(K, V ⊗^{c}χ) = H^{0}(K,^{c}(V ⊗ χ)) = H^{0}(K, V ⊗ χ) = 0.  (0.8.2)

0.9. If we take into account (0.8.1-2), the conjectures of Bloch and Kato [BK] predict that

h_{f}^{1}(K, V ⊗ χ) =^{?} r_{an}(f_{K}, χ).  (0.9.1)

In the present article we concentrate only on the implication

r_{an}(f_{K}, χ) = 0 ⟹^{?} H_{f}^{1}(K, V ⊗ χ) = 0.  (0.9.2)

Results of this kind were first proved by Bertolini and Darmon [BD 1-2]; their method was further developed in [L1-3], [LV 1-2], [LRV], [H], [PW], [Cd] and [TZ]. Our aim is to eliminate, whenever possible, the restrictive assumptions imposed in [loc. cit.]. Our main result is the following.

Theorem A. Let f ∈ S_{2}(𝔫, ω), χ : A_{K}^{×}/K^{×} ⟶ C^{×} (χ|_{A_{F}^{×}} · ω = 1) and V = V_{𝔭}(f)(1) be as in 0.1–0.7. If f has CM by a totally imaginary quadratic extension K(f) of F (i.e., if f is the θ-series attached to a Hecke character of K(f)), assume that K(f) ⊄ K_{χ}F_{ω}. Assume that there exists an element g_{𝔭} ∈ G_{F} satisfying the following conditions (A1)–(A3). (A1) g_{𝔭} acts trivially on F_{ω} ( ⟺ ω(g_{𝔭}) = 1); (A2) det(1 − g_{𝔭}X | V) = (1 − λ_{1}X)(1 − λ_{2}X), where λ_{1}^{2} = 1 ≠ λ_{2}^{2}; if f has CM, then ∀n ≥ 1 λ_{2}^{n} ≠ 1; (A3) g_{𝔭} does not act trivially on K; (such an element exists for 𝔭 belonging to a set of primes of L of positive density). If L(f_{K}, χ, 1) ≠ 0, then H_{f}^{1}(K, V ⊗ χ) = H_{f}^{1}(K, V ⊗^{c}χ) = 0; if, in addition, 𝔭 does not belong to a finite set of primes of L depending only on f, K and χ, then H_{f}^{1}(K, (V/T) ⊗ χ) = H_{f}^{1}(K, (V/T) ⊗^{c}χ) = 0 for any G_{F} -stable O_{L,𝔭}-lattice T ⊂ V.

0.10. Conjecturally, for each f in Theorem A there exists an abelian variety A_{f} defined over F (unique up to isogeny) such that

dim(A_{f}) = [L_{f} : Q],  End_{F} (A_{f}) = O_{L_{f}} ,  L(ιA_{f}/F, s) = L(f, s)  (0.10.1)

(Euler factor by Euler factor), where ι denotes the inclusion L_{f} ⊂ C. In this case we have, for any prime 𝔭 of L above a prime 𝔭_{f} of L_{f},

V = V_{𝔭}(f)(1) = V_{𝔭_{f}} (A_{f}) ⊗_{L_{f,𝔭_{f}}} L_{𝔭}, V_{𝔭_{f}} (A_{f}) = T_{p}(A_{f}) ⊗_{(O_{L_{f}} ⊗Z_{p})} L_{f,𝔭_{f}} .

Moreover, the Bloch-Kato Selmer group of A_{f}[p^{∞}] over any finite extension F′ of F coincides with the classical Selmer group for the p-power descent on A_{f}. In view of the standard descent sequence

0 ⟶ (A_{f}(F′) ⊗ Q_{p}/Z_{p}) ⊗_{(O_{L_{f}} ⊗Z_{p})} O_{L,𝔭} ⟶ H_{f}^{1}(F′, V/T) ⟶ � (A_{f}/F′)[p^{∞}] ⊗_{(O_{L_{f}} ⊗Z_{p})} O_{L,𝔭} ⟶ 0,

T = T_{p}(A_{f}) ⊗_{(O_{L_{f}} ⊗Z_{p})} O_{L,𝔭},

Theorem A can be rephrased in this context as follows.

Theorem A’. Assume that f ∈ S_{2}(𝔫, ω) in Theorem A is attached to an abelian variety A_{f} satisfying (0.10.1). If A_{f} acquires CM over a totally imaginary quadratic extension K(f) of F, assume that K(f) ⊄ K_{χ}F_{ω}. Assume that there exists an element g_{𝔭_{f}} ∈ G_{F} satisfying the following conditions (A1’)–(A3’). (A1’) g_{𝔭_{f}} acts trivially on F_{ω} ( ⟺ ω(g_{𝔭_{f}}) = 1); (A2’) det(1−g_{𝔭_{f}} X | V_{𝔭_{f}} (A_{f})) = (1−λ_{1}X)(1−λ_{2}X), where λ_{1}^{2} = 1 ≠ λ_{2}^{2}; if A_{f} has CM, then ∀n ≥ 1 λ_{2}^{n} ≠ 1; (A3’) g_{𝔭_{f}} does not act trivially on K; (such an element exists for 𝔭_{f} belonging to a set of primes of L_{f} of positive density). If L(ιA_{f}/K, ι ∘ χ, 1) ≠ 0, then A_{f}(K_{φ})^{(φ^{−1})} is finite (φ = χ,^{c}χ); if, in addition, 𝔭_{f} does not belong to a

finite set of primes of L_{f} depending only on f, K and χ, then (� (A_{f}/K_{φ})[p^{∞}] ⊗_{(O_{L_{f}} ⊗Z_{p})} O_{L,𝔭})^{(φ^{−1})} is finite (φ = χ,^{c}χ).

0.11. Applying Theorem A to χ = 1 for variable K we obtain the case (c) of the following result. The cases (a) and (b) are well-known consequences (cf. [Zh 2, Thm. 4.3.2]) of [N1, Thm. 3.2], [YZZ, Thm. 1.3.1] and [FH, Thm. B.2], which generalise, respectively, the Euler system argument, the Gross-Zagier formula and the non-vanishing results for quadratic twists used by Kolyvagin and Logachev [KoLo] in their proof of the corresponding result for F = Q.

Theorem B. Assume that f ∈ S_{2}(𝔫, 1) from 0.1 has trivial character, V = V_{𝔭}(f)(1), L(f, 1) ≠ 0 and that at least one of the following three conditions holds:

(a) 2 ∤ [F : Q];

(b) there exists a finite prime v of F for which π(f)_{v} is not a principal series representation;

(c) there exists g_{𝔭} ∈ G_{F} satisfying the conditions (A1) and (A2) from Theorem A (if f has no CM this is equivalent to V not being quaternionic in the sense of B.4.7, which holds for all but finitely many 𝔭). Then H_{f}^{1}(F, V) = 0 and, if 𝔭 does not belong to a certain finite set of primes of L, then H_{f}^{1}(F, V/T) = 0 for any G_{F} -stable O_{L,𝔭}-lattice T ⊂ V.

There is also an analogue of Theorem A’ in this situation (the field L_{f} being totally real in this case):

Theorem B’. If f ∈ S_{2}(𝔫, 1) in Theorem B is attached to an abelian variety A_{f} satisfying (0.10.1) and L(ιA_{f}/F, 1) ≠ 0, then A_{f}(F) is finite. Moreover, if at least one of the following five condition holds, then � (A_{f}/F)[𝔭_{f}^{∞}] is finite (and equal to zero if 𝔭_{f} does not belong to a certain finite set of primes of L_{f}):

(a) 2 ∤ [F : Q]; (b1) A_{f} does not have potentially good reduction everywhere; (b2) A_{f} does not acquire semistable reduction everywhere over any cyclic extension of F; (c1) A_{f} does not have CM and the localisation C ⊗_{Z(C)} Z(C)_{𝔭_{C}} of the simple algebra C := End\overline{Q}(A_{f}) ⊗ Q at the prime 𝔭_{C} of Z(C) ⊂ L_{f} below 𝔭_{f} is isomorphic to M_{n}(Z(C)_{𝔭_{C}}); (c2) A_{f} has CM by a totally imaginary quadratic extension L′ of L_{f}, the prime 𝔭_{f} splits in L′/L_{f} and V_{𝔭_{f}} (A_{f})|_{G_{K(f)}} = ψ_{1} ⊕ ψ_{2}, where ψ_{i} : G_{K(f)} ⟶ L_{f,𝔭_{f}}^{×} are characters for which ψ_{2}(Ker(ψ_{1})) is infinite. [In particular, if 2 ∤ [F : Q] or if f does not have CM then � (A_{f}/F)[𝔭_{f}^{∞}] = 0 for all but finitely many 𝔭_{f}.]

Corollary. If E is a modular elliptic curve over F satisfying L(E/F, 1) ≠ 0, then:

(1) (cf. [L2, Thm. A]) E(F) is finite.

(2) If 2 ∤ [F : Q] or if E has no CM, then � (E/F) is finite.

(3) If 2 | [F : Q] and E has CM by an imaginary quadratic field K′, then the following group is finite:

� (E/F)_{split} := ⨁_{p splits in K′/Q} � (E/F)[p^{∞}].

0.12. The proof of Theorem A is based on the method of Bertolini and Darmon [BD 2], with the following improvements:

• assumptions such as (𝔫, d_{K/F} N_{K/F} (cond(χ))) = 1, which had been used to transfer f to an explicit definite quaternion algebra, can be eliminated by an appeal to results of Tunnell [Tu] and Saito [Sa] on local toric linear forms (such linear forms were used in the context of a generalised Gross-Zagier formula by Zhang et al. [Z1], [YZZ]);

• it is not necessary to assume that there exists a level-raising congruence f ≡ f′ (mod 𝔭^{m}), where f′ is an eigenform of level 𝔫ℓ, new at a well-chosen prime ℓ of F. As in [T1], it is sufficient to work with an eigenform with coefficients in O_{L}/𝔭^{m}O_{L}, which always exists (for a suitable ℓ). A variant of the arguments of Boston-Lenstra-Ribet [BLR] then allows us to realise the reduction modulo 𝔭^{m−C} of (a certain lattice T in a Tate twist of) the 𝔭-adic Galois representation attached to f as a quotient of the Tate module of a suitable Shimura curve;

• assumptions such as (𝔫, p) = 1 or p ∤ D_{F}, which had been used to control the local behaviour at p of the cohomology class c(ℓ) ∈ H^{1}(K_{χ}, T/𝔭^{m−C}T) constructed from a certain CM point on the Shimura curve alluded to above, can be avoided by a consistent use of Raynaud extensions and their flat cohomology, combined with a uniformity result A.1.8 for Barsotti-Tate groups.

• the assumptions about the image of the Galois representation V can be formulated in an abstract – and probably optimal – form (the conditions (A1)–(A3) in Theorem A). The contents of the present article are as follows. In §1 we sum up the geometric machinery behind level raising from a weight two form on a definite quaternion algebra over F to a (weight two) form arising from a suitable Shimura curve over F. There are no original results in this part of the article; we simply translate back relevant parts of [T1] to a natural geometric situation similar to [Ri 7] (see also [J] and [R]). In §2 we construct the cohomology classes c(ℓ) ∈ H^{1}(K_{χ}, T/𝔭^{n}T) using “weak level raising modulo 𝔭^{n}”; we study their local properties and prove the main result (and its corollaries) by combining the annihilation relation arising from the reciprocity law

∀s ∈ H^{1}(K_{χ}, (T/𝔭^{n}T) ⊗ ω^{−1})  ∑_{v} inv_{v}(c(ℓ)_{v} ∪ s_{v}) = 0 ∈ O_{L}/𝔭^{n}O_{L}

with the Čebotarev density theorem, as in Kolyvagin’s method. In Appendix A (resp. Appendix B) we collect useful results about flat cohomology of finite group schemes and Raynaud extensions (resp. about images of Galois representations attached to Hilbert modular forms of regular weight). 0.13. This work began during the author’s stay at University of Tokyo in April 2009. He is grateful to T. Saito for invitation and to the JSPS for financial support. He would also like to thank C. Breuil, C. Cornut and L. Dieulefait for helpful discussions.

Notation: Throughout this article, F is a totally real number field of degree d and S_{∞} = {τ_{1}, …, τ_{d}} (resp. S_{p} = {v | p}) the set of archimedean primes (resp. of primes above a rational prime p) in F. For a quaternion algebra D over F we denote by Ram(D) = {v | inv_{v}(D_{v}) = −1} the set of primes of F at which D is ramified (above, D_{v} = D ⊗_{F} F_{v}). We also write ⊗ for ⊗_{Z} and Â for A ⊗ Ẑ (Ẑ = \underleftarrow{lim}_{n} Z/nZ), for any

abelian group A.

1. Level raising

Throughout §1, B is a totally definite quaternion algebra over F (i.e., such that Ram(B) ⊃ S_{∞}).

1.1. Automorphic forms of weight two on definite quaternion algebras

1.1.1. The space 𝕊(U; A). For any open compact subgroup U ⊂ B̂^{×} and any abelian group A, denote by

𝕊(U; A) = {f : B̂^{×} ⟶ A | ∀b ∈ B^{×} ∀g ∈ B̂^{×} ∀u ∈ U f(bgu) = f(g)} = A[B^{×}\B̂^{×}/U]

the abelian group of automorphic forms “of weight two” and level U on B_{A}^{×} with values in A (note that the set B^{×}\B̂^{×}/U is finite).

1.1.2. Action of B̂^{×}. The action of B̂^{×} on 𝕊(B_{A}^{×}; A) = ⋃_{U} 𝕊(U; A) by right translations (g·f)(g′) = f(g′g) is smooth (the stabiliser of each element is open in B̂^{×}) and, for each open compact subgroup U, the space of U-invariants is equal to 𝕊(B_{A}^{×}; A)^{U} = 𝕊(U; A).

The restriction of this action to Z(B̂^{×}) = F̂^{×} leaves each 𝕊(U; A) stable. More precisely, F̂^{×} acts on 𝕊(U; A) through the finite abelian group F^{×}\F̂^{×}/(F̂^{×} ∩ U). 1.1.3. Some subspaces of 𝕊(U; A). It will be useful to consider the following subgroups of 𝕊(U; A):

𝕊(U; A)_{triv} = {f ∈ 𝕊(U; A) | f factors through Nrd : B^{×}\B̂^{×}/U ⟶ F_{+}^{×}\F̂^{×}/Nrd(U)};

𝕊(UY; A) := 𝕊(U; A)^{Y} = 𝕊(U; A)^{Y (_{F^{×}}∩U)} = A[B^{×}\B̂^{×}/UY],

where Y is an open – but not necessarily compact – subgroup of F̂^{×} = Z(B̂^{×}).

1.1.4. Action of double cosets. If U, U′ ⊂ B̂^{×} are compact open subgroups and x ∈ B̂^{×}, then xU = xUx^{−1} is commensurable with U′. If we write U′ = ∐_{i} α_{i}(U′ ∩^{x}U) as a disjoint union of cosets, then U′xU = ∐_{i} x_{i}U, where x_{i} = α_{i}x. The linear map

[U′xU] : 𝕊(U; A) ⟶ 𝕊(U′; A), ([U′xU]f)(g) = ∑_{i} f(gx_{i})

has the following properties.

1.1.4.1. [U′xU] commutes with the action of F̂^{×}. In particular, it maps 𝕊(UY; A) to 𝕊(U′Y; A), for any open

subgroup Y ⊂ F̂^{×}. 1.1.4.2. The endomorphisms [UxU] define a left action of the double coset algebra Z[U\B̂^{×}/U] on 𝕊(U; A). If

x ∈ Z(B̂^{×}) = F̂^{×}, then [UxU] coincides with the action of x. 1.1.4.3. If x = 1 and U′ ⊂ U, then [U′1U] : 𝕊(U; A) ⟶ 𝕊(U′; A) is the inclusion and [U1U′] : 𝕊(U′; A) ⟶

𝕊(U; A) is the trace map.

Recall that Z[U\B̂^{×}/U] embeds into the Hecke algebra C_{c}^{∞}(B̂^{×}) of locally constant functions on B̂^{×} with compact support, equipped with the convolution product

(α ∗ β)(g) = ∫_{B^{×}} α(h)β(h^{−1}g) dh

(for a fixed Haar measure dh on B̂^{×}), via the map which sends vol(U)[UgU] to the characteristic function of UgU. Denote by x ↦ x^{∨} the (anti)-involution of Z[U\B̂^{×}/U] which sends [UgU] to [Ug^{−1}U].

1.1.5. Action of the spherical Hecke algebra. Fix an open compact subgroup U ⊂ B̂^{×}. There exists a finite set S ⊃ Ram(B) of primes of F such that U = U_{S}U^{S}, where U_{S} is an open compact subgroup of ∏_{v∈S_{f}} B_{v}^{×} (S_{f} = S − S_{∞}) and U^{S} = ∏_{v∉S} U_{v}, where each U_{v} is a maximal compact subgroup of B_{v}^{×}.

Fix such a set S and denote by 𝕋^{S}(U) the (commutative) subring of Z[U\B̂^{×}/U] generated by the double cosets [UxU] for all x ∈ (B̂^{S})^{×} = {x ∈ B̂^{×} | ∀v ∈ S_{f} x_{v} = 1}. As a ring, it is isomorphic to Z[T (v), S(v), S(v)^{−1} | v ∉ S], where T (v) and S(v) are the standard Hecke operators

T (v) = [Uξ_{v}U],  S(v) = [Uϖ_{v}U]  (v ∉ S).

Above, ϖ_{v} is a uniformiser of F_{v} and ξ_{v} is as in 1.2.1 below. The involution x ↦ x^{∨} acts on 𝕋^{S}(U) by T (v)^{∨} = T (v)S(v)^{−1}, S(v)^{∨} = S(v)^{−1} (v ∉ S).

For any open subgroup Y ⊂ F̂^{×}, the image 𝕋^{S}(UY) of 𝕋^{S}(U) in Z[UY \B̂^{×}/UY] is the quotient of 𝕋^{S}(U) by the ideal generated by [UyU] − 1 (y ∈ Y).

The image of 𝕋^{S}(UY) in End_{Z}(𝕊(UY; Z)) is an order in a finite product of number fields. It acts on 𝕊(UY; A) = 𝕊(UY; Z) ⊗_{Z} A, for any abelian group A.

For any 𝕋^{S}(UY)-module N set

hN = N ⊗_{𝕋^{S}(UY),∨} 𝕋^{S}(UY).

The map n ↦ ^{h}n := n ⊗ 1 is an isomorphism of abelian groups N ⟶ ^{h}N satisfying

∀t ∈ 𝕋^{S}(UY) ∀n ∈ N t(^{h}n) = ^{h}(t^{∨}(n)).

1.1.6. The Eisenstein part of 𝕊(U; C). As a B̂^{×}-module, 𝕊(B_{A}^{×}; C)_{triv} = ⋃_{U} 𝕊(U; C)_{triv} decomposes into a direct sum of one-dimensional eigenspaces Cf_{φ}, one for each character of finite order φ : B^{×}\B̂^{×} ⟶ C^{×} factoring as B^{×}\B̂^{×} ⟶^{Nrd} F_{+}^{×}\F̂^{×} ⟶^{φ′} C^{×}. If U satisfies φ′(Nrd(U)) = 1, then

∀v ∉ S  T (v)f_{φ} = (N(v) + 1)φ′(ϖ_{v})f_{φ},  S(v)f_{φ} = φ′(ϖ_{v})^{2}f_{φ}.

1.1.7. The non-Eisenstein part of 𝕊(U; C). As a 𝕋^{S}(U)-module, the space 𝕊(U; C) = ⊕_{λ} 𝕊(U; C)_{λ} is a direct sum of its isotypic components, for certain ring morphisms λ : 𝕋^{S}(U) ⟶ C.

If f ∈ 𝕊(U; C)_{λ} but f ∉ 𝕊(U; C)_{triv}, it follows from the Jacquet-Langlands correspondence that there exists a cuspidal Hilbert eigenform of parallel weight 2 on F with the same Hecke eigenvalues under T (v) and S(v) (∀v ∉ S) as f. The generalised Ramanujan conjecture for Hilbert modular forms, whose proof was completed in [Bl], implies that

∀v ∉ S  T (v)f = λ(T (v))f,  |λ(T (v))| ≤ 2N(v)^{1/2}.  (1.1.7.1)

As a result, for each v ∉ S, T (v) has no common eigenvalues on 𝕊(U; C)_{triv} and 𝕊(U; C)/𝕊(U; C)_{triv}. It follows that there is a unique 𝕋^{S}(U)-submodule 𝕊(U; C)_{0} ⊂ 𝕊(U; C) such that

𝕊(U; C) = 𝕊(U; C)_{triv} ⊕ 𝕊(U; C)_{0}.

Moreover, for each v ∉ S, any eigenvector of T (v) in 𝕊(U; C) lies either in 𝕊(U; C)_{triv} or in 𝕊(U; C)_{0}. For a subring A ⊂ C and an open subgroup Y ⊂ F̂^{×} set

𝕊(UY; A)_{0} := 𝕊(UY; A) ∩ 𝕊(U; C)_{0}.

If f is as in (1.1.7.1), then σ ∘ f ∈ 𝕊(U; C)_{σ∘λ} and σ ∘ f ∉ 𝕊(U; C)_{triv}, for each σ ∈ Aut(C). This implies that 𝕊(U; C)_{0} is stable under Aut(C); as the same is true for 𝕊(U; C)_{triv}, it follows that

𝕊(UY; A) = 𝕊(UY; A)_{triv} ⊕ 𝕊(UY; A)_{0},  (1.1.7.2)

for any subring A ⊂ C containing Q and any open subgroup Y ⊂ F̂^{×}.

1.2. Oldforms and newforms

Let U and S be as in 1.1.5. 1.2.1. Degeneracy and trace maps. Fix a prime ℓ ∉ S of F, a uniformiser ϖ_{ℓ} of F_{ℓ} and an isomorphism M_{2}(F_{ℓ}) ⟶^{∼} B_{ℓ} sending GL_{2}(O_{ℓ}) onto U_{ℓ} (where O_{ℓ} = O_{F,ℓ} is the ring of integers of F_{ℓ}). Set

ξ_{ℓ} = ( \begin{matrix}1 & 0 \\ 0 & ϖ_{ℓ} \end{matrix} ) ∈ GL_{2}(F_{l}) ⟶^{∼} B_{ℓ}^{×} ⊂ B̂^{×}, U(ℓ) = U ∩^{ξ_{ℓ}}U = \{u ∈ U | u_{ℓ} ≡ ( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} ) (mod ϖ_{ℓ})\} .

For any abelian group A we have the standard injective degeneracy maps

\begin{aligned}α^{∗} = [U(ℓ)1U] & : 𝕊(U; A) ⟶ 𝕊(U(ℓ); A),  (α^{∗}f)(g) = f(g), \\ β^{∗} = [U(ℓ)ξ_{ℓ}U] & : 𝕊(U; A) ⟶ 𝕊(U(ℓ); A),  (β^{∗}f)(g) = f(gξ_{ℓ}) \end{aligned}

and the corresponding trace maps

\begin{aligned}α_{∗} = [U1U(ℓ)] & : 𝕊(U(ℓ); A) ⟶ 𝕊(U; A), \\ β_{∗} = [Uξ_{ℓ}^{−1}U(ℓ)] & : 𝕊(U(ℓ); A) ⟶ 𝕊(U; A). \end{aligned}

All these maps commute with the respective actions of Z(B̂^{×}) = F̂^{×} and T (v) (v ∉ S ∪ {ℓ}) on 𝕊(U; A) and 𝕊(U(ℓ); A).

1.2.2. Definition. The ℓ-new subspace of 𝕊(U(ℓ); A) is defined as

𝕊(U(ℓ); A)^{ℓ−new} := Ker (𝕊(U(ℓ); A) ⟶^{(α_{∗},β_{∗})} 𝕊(U; A)^{⊕2}) .

It is stable by the action of Z(B̂^{×}) and T (v) (v ∉ S ∪ {ℓ}) and satisfies

𝕊(U(ℓ); A)^{ℓ−new} = 𝕊(U(ℓ); A) ∩ 𝕊(U(ℓ); A′)^{ℓ−new} (A ⊂ A′),
𝕊(U(ℓ)Y; A)^{ℓ−new} := 𝕊(U(ℓ)Y; A) ∩ 𝕊(U(ℓ); A)^{ℓ−new} = (𝕊(U(ℓ); A)^{ℓ−new})^{Y} ,

for any open subgroup Y ⊂ F̂^{×}.

1.2.3. Proposition. (1) For any open subgroup Y ⊂ F̂^{×} containing F_{ℓ}^{×}, the composite map

μ : 𝕊(UY; A)^{⊕2} ⟶^{α^{∗}−β^{∗}} 𝕊(U(ℓ)Y; A) ⟶^{(−α_{∗},β_{∗})} 𝕊(UY; A)^{⊕2}

is given by the matrix

( \begin{matrix}−α_{∗}α^{∗} & α_{∗}β^{∗} \\ β_{∗}α^{∗} & −β_{∗}β^{∗} \end{matrix} ) = ( \begin{matrix}−N(ℓ) − 1 & T (ℓ) \\ T (ℓ) & −N(ℓ) − 1 \end{matrix} ) .

(2) Ker (𝕊(U; A)^{⊕2} ⟶^{α^{∗}−β^{∗}} 𝕊(U(ℓ); A)) = {(ξ_{ℓ} · f, f) | f ∈ 𝕊(U; A)_{triv}}.

Proof. (1) [Ri 7, Proof of Thm. 3.22], [T1, Lemma 2]. (2) If f, f′ ∈ 𝕊(U; A) satisfy α^{∗}(f) = β^{∗}(f′), then f is invariant by both U_{ℓ} = GL_{2}(O_{ℓ}) and ξ_{ℓ}^{−1}GL_{2}(O_{ℓ})ξ_{ℓ}, hence by {g ∈ GL_{2}(F_{ℓ}) | det(g) ∈ O_{ℓ}^{∗}}. As

(B̂^{×})^{Nrd=1} = (B^{×})^{Nrd=1}SL_{2}(F_{ℓ})(U^{(ℓ)})^{Nrd=1}

by the strong approximation theorem [Vi, Thm. III.4.3], it follows that f factors through Nrd, hence f ∈ 𝕊(U; A)_{triv}. The relation α^{∗}(f) = β^{∗}(f′) implies that f = ξ_{ℓ} · f′.

1.3. Shimura curves

Let U, S and ℓ be as in 1.2.1. 1.3.1. New quaternion algebra. Let B′ be the quaternion algebra over F obtained from B by “switching invariants” at ℓ and a fixed infinite prime τ_{1} ∈ S_{∞}:

Ram(B′) = (Ram(B) \ {τ_{1}}) ∪ {ℓ}.

Denote by F̂^{(ℓ)} (resp. D̂^{(ℓ)}, for D = B, B′) the restricted product of F_{v} (resp. of D_{v}) over all finite primes v ≠ ℓ of F (in other words, we have F̂ = F_{ℓ} × F̂^{(ℓ)} and D̂ = D_{ℓ} × D̂^{(ℓ)}). Fix an isomorphism of F̂^{(ℓ)}-algebras

φ : B̂^{(ℓ)} ⟶^{∼} B̂^{′(ℓ)}

and set

U^{(ℓ)} = U_{S} ∏_{v∉S∪{ℓ}} U_{v} ⊂ B̂^{(ℓ)×}, U′ = φ(U^{(ℓ)})O_{B′_{ℓ}}^{×} ⊂ B̂^{′×},

where O_{B′_{ℓ}} is the maximal order of the division algebra B′_{ℓ}.

1.3.2. Shimura curve (complex uniformisation). Fix an isomorphism B′_{τ_{1}} ⟶^{∼} M_{2}(R) and consider the Shimura curve M_{U′} corresponding to the open compact subgroup U′ ⊂ B̂^{′×}, using the notation and conventions of [CV 1, §3] and [CV 2, §3] with ϵ = 1 (see also [C2] and [N1, §1]). In concrete terms, M_{U′} is a smooth and projective curve over F whose associated Riemann surface M_{U′}^{an} is naturally identified with

M_{U′}^{an} = (M_{U′} ⊗_{F,τ_{1}} C)(C) = B^{′×}\(C − R) × B̂^{′×}/U′,

where B^{′×} ⊂ B_{τ_{1}}^{′×} ⟶^{∼} GL_{2}(R) acts on C − R by the standard action z ↦ \frac{az+b}{cz+d}.

The curve M_{U′} is irreducible, but not necessarily geometrically irreducible. Its field of constants is isomorphic to the finite abelian extension F_{U′} of F characterised by the isomorphism

rec_{F} : F_{+}^{×}\F̂^{×}/Nrd(U′) = F_{+}^{×}\F̂^{×}/Nrd(U^{(ℓ)})O_{ℓ}^{×} ⟶^{∼} Gal(F_{U′}/F).

This is consistent with the fact that the reduced norm for B′ induces a bijection

π_{0}(M_{U′}^{an}) = B^{′×}\π_{0}(C − R) × B̂^{′×}/U′ ⟶^{∼} Nrd(B^{′×})\{±1} × F̂^{×}/Nrd(U′) = F_{+}^{×}\F̂^{×}/Nrd(U′),

by strong approximation [Vi, Thm. III.4.3] and Eichler’s norm theorem [Vi, Thm. III.4.1].

1.3.3. Quotient Shimura curve. For z ∈ C − R and b′ ∈ B̂^{′×}, denote by [z, b′]_{U′} the complex point of M_{U′} represented by the pair (z, b′). The centre Z(B̂^{′×}) = F̂^{×} acts on M_{U′} (by morphisms defined over F) according to the formula

(1.3.3.1)  g([z, b′]_{U′}) = [z, b′g]_{U′} = [z, gb′]_{U′}.

This action factors through the finite abelian group F^{×}\F̂^{×}/(F̂^{×}∩ U′).

For any open subgroup Y ⊂ F̂^{×}, denote by M_{U′Y} the quotient of M_{U′} by (the image in F^{×}\F̂^{×}/(F̂^{×}∩U′)) of Y; it is a smooth projective curve over F satisfying

M_{U′Y}^{an} = (M_{U′Y} ⊗_{F,τ_{1}} C)(C) = B^{′×}\(C − R) × B̂^{′×}/U′Y.

Denote by [z, b′]_{U′Y} ∈ M_{U′Y} (C) the image of [z, b′]_{U′}.

In the case Y = F̂^{×} ∩ U′ (resp. Y = F̂^{×}) the curve M_{U′Y} coincides with M_{U′} (resp. with the curve which was denoted by X in [Z1, 1.5.1] and by N_{U′} in [N1]). 1.3.4. Hecke correspondences. For each g ∈ B^{′×}, the right multiplication [·g] : [z, b′] ↦ [z, b′g] and the diagram

\xymatrix{M_{(U′∩gU′g^{−1})Y} \ar[r]^{[·g]} \ar[d]_{pr} & M_{(g^{−1}U′g∩U′)Y} \ar[d]_{_{pr_{′}}} \\ M_{U′Y} \ar@{-->}[r]^{[U′Y gU′Y]} & M_{U′Y}}  (1.3.4.1)

define a multivalued map (a “Hecke correspondence”)

\xymatrix{[U′Y gU′Y] : M_{U′Y} \ar@{-->}[r] & M_{U′Y} .}  (1.3.4.2)

1.4. ℓ-adic uniformisation of Shimura curves

Let U, S and ℓ be as in 1.2.1. Denote by O_{ℓ}^{ur} (resp. by O_{ℓ^{2}}) the ring of integers in the maximal unramified extension F_{ℓ}^{ur} (resp. in the unramified quadratic extension F_{ℓ^{2}} ⊂ F_{ℓ}^{ur}) of F_{ℓ} and by Ô_{ℓ}^{ur} (resp. C_{ℓ}) the completion of O_{ℓ}^{ur} (resp. of \overline{F}_{ℓ}) with respect to the ℓ-adic topology.

1.4.1. Theorem of Čerednik and Drinfeld. Fix an isomorphism B_{ℓ} ⟶^{∼} M_{2}(F_{ℓ}). According to [Če] and [Dr] (see also [BC, Thm. 5.2], [BZ, Thm. 3.1] and [V, Thm. 5.3]) there is a natural integral model M_{U′} of M_{U′} ⊗_{F} F_{ℓ} over O_{ℓ}, whose completion M_{U′} along the special fibre is canonically identified with

B^{×}\(ℋ̂_{ℓ} ˆ⊗_{O_{ℓ}} Ô_{ℓ}^{ur}) × B̂^{(ℓ)×}/U^{(ℓ)},

where ℋ̂_{ℓ} is the formal scheme over Spf(O_{ℓ}) [BC, I.3] which is a natural formal model of Drinfeld’s ℓ-adic half plane P^{1}(C_{ℓ})−P^{1}(F_{ℓ}) and b ∈ B^{×} acts on ℋ̂_{ℓ} (resp. on Ô_{ℓ}^{ur}) via the natural action of B^{×} ⊂ B_{ℓ}^{×} ⟶^{∼} GL_{2}(F_{ℓ}) on P_{F_{ℓ}}^{1} (resp. by Fr_{geom}(ℓ)^{ord_{ℓ}(Nrd(b))}).

Denote by M_{U′}^{an} the corresponding rigid analytic space over F_{ℓ}. As in 1.3.2, the components of

M_{U′}^{an}(C_{ℓ}) = B^{×}\(P^{1}(C_{ℓ}) − P^{1}(F_{ℓ})) × B̂^{(ℓ)×}/U^{(ℓ)}

are in bijection with

B^{×}\Z × B̂^{(ℓ)×}/U^{(ℓ)} = B^{×}\Nrd(B_{ℓ}^{×}/U_{ℓ}) × B̂^{(ℓ)×}/U^{(ℓ)} ⟶^{∼} F_{+}^{×}\F̂^{×}/O_{ℓ}^{×}Nrd(U^{(ℓ)}) = F_{+}^{×}\F̂^{×}/Nrd(U′),

where the middle bijection is again induced by the reduced norm, this time for B.

For z ∈ P^{1}(C_{ℓ}) − P^{1}(F_{ℓ}) and b ∈ B̂^{(ℓ)×}, denote by [z, b]_{U} ∈ M_{U′}(C_{ℓ}) the point represented by (z, b).

1.4.2. ℓ-adic uniformisation of the quotient Shimura curve. The action (1.3.3.1) of g ∈ F̂^{×} = Z(B̂^{′×}) on M_{U′} extends to an action on M_{U′}. The corresponding action on M_{U′} is given by the following formula: write g = g_{ℓ}g^{(ℓ)} with g_{ℓ} ∈ F_{ℓ}^{×} and g^{(ℓ)} ∈ F̂^{(ℓ)×}; then g acts on ℋ̂_{ℓ} ˆ⊗_{O_{ℓ}} Ô_{ℓ}^{ur} (resp. on B̂^{(ℓ)×}) by id × Fr_{geom}(ℓ)^{ord_{ℓ}(Nrd(g_{ℓ}))} (resp. by multiplication by g^{(ℓ)}).

Fix an open subgroup Y ⊂ F̂^{×} containing F_{ℓ}^{×} (i.e., such that Y = Y_{ℓ} × Y^{(ℓ)}, where Y_{ℓ} = F_{ℓ}^{×} and Y^{(ℓ)} is an open subgroup of F̂^{(ℓ)×}) and consider the quotient curve

M = M_{U′Y} = M_{U′}/Y

from 1.3.3. Taking this quotient (by a finite abelian group) makes sense for M_{U′} and M_{U′}; this yields an integral model M = M_{U′}/Y of M over O_{ℓ} and its completion M = M_{U′}/Y along the special fibre. The field of constants F_{U′Y} of M satisfies

rec_{F} : F_{+}^{×}\F̂^{×}/Nrd(U′)Y^{2} ⟶^{∼} Gal(F_{U′Y} /F) (Y^{2} = {a^{2} | a ∈ Y }).

In the notation of 1.4.1 we denote by [z, b]_{UY} ∈ M_{U′Y} (C_{ℓ}) the image of [z, b]_{U}.

The assumption Y = F_{ℓ}^{×} × Y^{(ℓ)} implies that

M = B^{×}\(ℋ̂_{ℓ} ˆ⊗_{O_{ℓ}} Ô_{ℓ}^{ur}) × B̂^{(ℓ)×}/U^{(ℓ)}Y = B^{×}\(ℋ̂_{ℓ} ⊗_{O_{ℓ}} O_{ℓ^{2}}) × B̂^{(ℓ)×}/U^{(ℓ)}Y^{(ℓ)}.

Write

B̂^{(ℓ)×} = ∐_{i} B^{×}α_{i}U^{(ℓ)}Y^{(ℓ)}

as a disjoint union of double cosets and set, for each i,

Γ_{i} = B^{×} ∩ α_{i}U^{(ℓ)}Y^{(ℓ)}α_{i}^{−1};

there is an isomorphism

∐_{i} Γ_{i}\(ℋ̂_{ℓ} ⊗_{O_{ℓ}} O_{ℓ^{2}}) ⟶^{∼} M,

sending Γ_{i}z to the class represented by the pair (z, α_{i}). As the subgroup

Γ_{i,+} := {γ ∈ Γ_{i} | ord_{ℓ}(Nrd(γ)) ≡ 0 (mod 2)} ⊂ Γ_{i}

acts trivially on O_{ℓ^{2}}, we obtain (cf. [BC, 5.3.3]) an isomorphism

M ⟶^{∼} ∐_{i} W_{i}\ ((\overline{Γ}_{i,+}\ℋ̂_{ℓ}) ⊗_{O_{ℓ}} O_{ℓ^{2}}) ,  (1.4.2.1)

where W_{i} = Γ_{i}/Γ_{i,+} is a group of order 1 or 2 and where we have denoted by \overline{Γ}_{i,+} the image of Γ_{i,+} in

B_{ℓ}^{×}/F_{ℓ}^{×} ⟶^{∼} P GL_{2}(F_{ℓ}).

1.5. Bad reduction of the (quotient) Shimura curve at ℓ

Let U, S and ℓ be as in 1.2.1. In addition, let Y = Y_{ℓ} × Y^{(ℓ)} = F_{ℓ}^{×} × Y^{(ℓ)} be as in 1.4.2.

1.5.1. Bruhat-Tits tree. Denote by T_{ℓ} the Bruhat-Tits tree attached to B_{ℓ}^{×} ⟶^{∼} GL_{2}(F_{ℓ}). Its set of vertices is equal to V(T_{ℓ}) = B_{ℓ}^{×}/U_{ℓ}F_{ℓ}^{×}, the set of oriented edges (= of ordered pairs of adjacent vertices) to \overrightarrow{E}(T_{ℓ}) = B_{ℓ}^{×}/U(ℓ)_{ℓ}F_{ℓ}^{×}, the incidence relation is given by the maps

s, t : \overrightarrow{E}(T_{ℓ}) ⟶ V(T_{ℓ}),  s(gU(ℓ)_{ℓ}F_{ℓ}^{×}) = gU_{ℓ}F_{ℓ}^{×},  t(gU(ℓ)_{ℓ}F_{ℓ}^{×}) = gξ_{ℓ}U_{ℓ}F_{ℓ}^{×}

(s = source, t = target) and the inversion of an edge by

ι : \overrightarrow{E}(T_{ℓ}) ⟶ \overrightarrow{E}(T_{ℓ}), gU(ℓ)_{ℓ}F_{ℓ}^{×} ↦ g ( \begin{matrix}0 & 1 \\ ϖ_{ℓ} & 0 \end{matrix} ) U(ℓ)_{ℓ}F_{ℓ}^{×} = gξ_{ℓ} ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) U(ℓ)_{ℓ}F_{ℓ}^{×}.

Denote by E(T_{ℓ}) = \overrightarrow{E}(T_{ℓ})/\{id, ι\} the set of non-oriented edges of T_{ℓ}.

1.5.2. Special fibre of ℋ̂_{ℓ}. The group B_{ℓ}^{×} ⟶^{∼} GL_{2}(F_{ℓ}) acts on ℋ̂_{ℓ}. The special fibre of ℋ̂_{ℓ} is identified, in a B_{ℓ}^{×}-equivariant way, with a collection of projective lines P_{k(ℓ)}^{1} glued together according to the incidence relation given by T_{ℓ}; in other words, with the coequaliser of the pair of morphisms

s, t : ∐_{\overrightarrow{E}(T_{ℓ})} Spec(k(ℓ)) ⟶ ∐_{V(T_{ℓ})} P_{k(ℓ)}^{1},

where s (resp. t) maps the point Spec(k(ℓ)) corresponding to an oriented edge e to ∞ (resp. to 0) on the copy of P_{k(ℓ)}^{1} corresponding to the vertex s(e) (resp. to t(e)).

1.5.3. Special fibre of C_{i}. According to Kurihara [Ku, Prop. 3.2], who extended earlier results of Mumford [Mu], each quotient C_{i} = \overline{Γ}_{i,+}\ℋ̂_{ℓ} from (1.4.2.1) is an admissible curve over O_{ℓ}, in the terminology of [JL, § 3]:

1.5.3.1. C_{i} is a proper and flat curve over O_{ℓ} with a smooth generic fibre. 1.5.3.2. The special fibre C_{i} ⊗_{O_{ℓ}} k(ℓ) is reduced; the normalisation of each of its irreducible components is

isomorphic to P_{k(ℓ)}^{1}; its only singular points are ordinary double points, rational over k(ℓ). 1.5.3.3. The completion of the local ring of C_{i} at each of its singular points x is isomorphic, as an O_{ℓ}-algebra,

to O_{ℓ}[[X, Y]]/(XY − ϖ_{ℓ}^{w}), where w = w(x) ∈ {1, 2, 3, …}.

In addition, the combinatorics of the special fibre is described as follows.

1.5.3.4. The set of irreducible components of C_{i} ⊗_{O_{ℓ}} k(ℓ) is naturally identified with \overline{Γ}_{i,+}\V(T_{ℓ}). 1.5.3.5. The set of singular points of C_{i} ⊗_{O_{ℓ}} k(ℓ) is naturally identified with (\overline{Γ}_{i,+}\E(T_{ℓ}))^{∗}, where the star in

the superscript refers to the fact that we remove from \overline{Γ}_{i,+}\E(T_{ℓ}) the images of those oriented edges e ∈ \overrightarrow{E}(T_{ℓ}) for which there exits γ ∈ \overline{Γ}_{i,+} such that γ(e) = ι(e). As we are going to see in 1.5.4 below, no such edges exist in our case; as a result, (\overline{Γ}_{i,+}\E(T_{ℓ}))^{∗} = \overline{Γ}_{i,+}\E(T_{ℓ}).

1.5.3.6. The incidence relation between the irreducible components and the singular points is inherited from T_{ℓ}. 1.5.3.7. If x is a singular point of C_{i} ⊗_{O_{ℓ}} k(ℓ) represented by an oriented edge e ∈ \overrightarrow{E}(T_{ℓ}), then the integer w(x)

from 1.5.3.3 is equal to the order of the stabiliser (\overline{Γ}_{i,+})_{e} of e in \overline{Γ}_{i,+}.

Kurihara rephrased 1.5.3.4–1.5.3.7 by saying that the dual graph G(C_{i}) of the special fibre of C_{i} is equal to

V(G(C_{i})) = \overline{Γ}_{i,+}\V(T_{ℓ}), \overrightarrow{E}(G(C_{i})) = (\overline{Γ}_{i,+}\\overrightarrow{E}(T_{ℓ}))^{∗},

and is equipped with the function

w : \overrightarrow{E}(G(C_{i})) ⟶ \{1, 2, 3, …\}

given by 1.5.3.7.

1.5.4. Special fibre of M ⊗_{O_{ℓ}} O_{ℓ^{2}}. The formula (1.4.2.1) and the discussion in the previous paragraph imply that M ⊗_{O_{ℓ}} O_{ℓ^{2}} is an admissible curve over O_{ℓ^{2}}. Moreover, the dual graph G of the special fibre M ⊗_{O_{ℓ}} k(ℓ^{2}) (whose vertices correspond to irreducible components and edges to singular points) is given by

V(G) = ∐_{i} Γ_{i}\ (V(T_{ℓ}) × Z/2Z) , \overrightarrow{E}(G) = ∐_{i} (Γ_{i}\(\overrightarrow{E}(T_{ℓ}) × Z/2Z))^{∗} ,

where γ ∈ Γ_{i} ⊂ B^{×} acts on Z/2Z by translation by ord_{ℓ}(Nrd(γ)) (mod 2). In adelic terms,

V(G) = B^{×}\ (V(T_{ℓ}) × Z/2Z × B̂^{(ℓ)×}/U^{(ℓ)}Y^{(ℓ)}) = B^{×}\ (B_{ℓ}^{×}/U_{ℓ}Y_{ℓ} × Z/2Z × B̂^{(ℓ)×}/U^{(ℓ)}Y^{(ℓ)}) ⟶^{∼}

⟶^{∼} (B^{×}\B̂^{×}/UY ) × Z/2Z,

where the last bijection is given by

B^{×} (b_{ℓ}U_{ℓ}Y_{ℓ}, j, b^{(ℓ)}U^{(ℓ)}Y^{(ℓ)}) ↦ (B^{×}b_{ℓ}b^{(ℓ)}UY, j + ord_{ℓ}(Nrd(b_{ℓ}))) .

The same formula for U(ℓ) instead of U induces a bijection

∐_{i} Γ_{i}\ (\overrightarrow{E}(T_{ℓ}) × Z/2Z) ⟶^{∼} B^{×}\ (B_{ℓ}^{×}/U(ℓ)_{ℓ}Y_{ℓ} × Z/2Z × B̂^{(ℓ)×}/U^{(ℓ)}Y^{(ℓ)}) ⟶^{∼} (B^{×}\B̂^{×}/U(ℓ)Y ) × Z/2Z.

The maps

s, t : (B^{×}\B̂^{×}/U(ℓ)Y ) × Z/2Z ⟶ (B^{×}\B̂^{×}/UY ) × Z/2Z

induced by the incidence relation on T_{ℓ} are given by

s(B^{×}bU(ℓ)Y, j) = (B^{×}bUY, j), t(B^{×}bU(ℓ)Y, j) = (B^{×}bξ_{ℓ}UY, j + 1).

In particular, each quotient Γ_{i}\(T_{ℓ} × Z/2Z) is a bipartite graph, hence

(Γ_{i}\(\overrightarrow{E}(T_{ℓ}) × Z/2Z))^{∗} = Γ_{i}\(\overrightarrow{E}(T_{ℓ}) × Z/2Z), \overrightarrow{E}(G) ⟶^{∼} (B^{×}\B̂^{×}/U(ℓ)Y ) × Z/2Z.

1.5.5. Proposition. (1) M ⊗_{O_{ℓ}} O_{ℓ^{2}} is an admissible curve over O_{ℓ^{2}}.

(2) The dual graph G of the special fibre of M ⊗_{O_{ℓ}} O_{ℓ^{2}} is bipartite, with

V(G) = (B^{×}\B̂^{×}/UY ) × Z/2Z, \overrightarrow{E}(G) = (B^{×}\B̂^{×}/U(ℓ)Y ) × Z/2Z,
s(B^{×}bU(ℓ)Y, j) = (B^{×}bUY, j), t(B^{×}bU(ℓ)Y, j) = (B^{×}bξ_{ℓ}UY, j + 1),

ι(B^{×}bU(ℓ)Y, j) = (B^{×}b ( \begin{matrix}0 & 1 \\ ϖ_{ℓ} & 0 \end{matrix} )_{ℓ} U(ℓ)Y, j + 1).

(3) If there exists an integer N > 2 such that {v | N} ∩ Ram(B) = ∅ and ∀v | N U_{v} ⊆ 1 + NR(v) for some maximal order R(v) ⊂ B_{v} ⟶^{∼} M_{2}(F_{v}), then the curve M ⊗_{O_{ℓ}} O_{ℓ^{2}} is semi-stable ( ⟺ the integers w attached to the singular points of M ⊗_{O_{ℓ}} k(ℓ^{2}) are all equal to 1).

Proof. The statements (1) and (2) follow from the discussion in 1.5.4. The assertion (3) is a consequence of 1.5.3.7 and the following Lemma.

1.5.6. Lemma. If U satisfies the assumptions of 1.5.5(3), then, for any t ∈ B̂^{×}, the group Γ_{t} := B^{×} ∩ tUF̂^{×}t^{−1} (⊇ F^{×}) is equal to F^{×}.

Proof. Suppose first that γ ∈ Γ_{t}^{Nrd=1}, γ ∉ F^{×}. In this case γ ∈ B^{×} ∩ tU(Ô_{F})^{×}t^{−1} is integral over O_{F}, K := F (γ) ⊂ B is a totally imaginary quadratic extension of F and N_{K/F} (γ) = 1, which implies that the roots of the characteristic polynomial X^{2} − Trd(γ)X + Nrd(γ) = X^{2} − Tr_{K/F} X + 1 are ζ and ζ^{−1}, where ζ ≠ ζ^{−1} is a root of unity such that ζ + ζ^{−1} ∈ O_{F}. Writing γ = tuu′t^{−1} with u ∈ U and u′ ∈ (Ô_{F})^{×}, we have ζ + ζ^{−1} = u′ Trd(u), u^{′2} Nrd(u) = 1 and Nrd(u) ∈ 1 + NÔ_{F}, which implies that, for each v | N and a suitable choice of a sign, ±γ ∈ I + NM_{2}(O_{F,v}) is a matrix of finite order, which implies that ±γ = I, since N > 2. This shows that Γ_{t}^{Nrd=1} = (F^{×})^{Nrd=1} = {±1}.

If γ ∈ Γ_{t}, then Nrd(γ)γ^{−2} = \overline{γ}γ^{−1} ∈ Γ_{t}^{Nrd=1} = \{±1\} (where γ ↦ \overline{γ} is the standard involution on B). If \overline{γ}γ^{−1} = 1, then γ ∈ F^{×}. If \overline{γ}γ^{−1} = −1, then Trd(γ) = γ + \overline{γ} = 0. However, γ = tuu′t^{−1} with u ∈ U and u′ ∈ F̂^{×}, which implies that Trd(γ) = u′ Trd(u) ≠ 0, since Trd(u) ∈ 2+NÔ_{F} and N > 2. This contradiction shows that Γ_{t} = F^{×}, as claimed. 1.5.7. (Co)homology of G. As the graph G is bipartite, it has two natural orientations (by an orientation of G we mean a section of the canonical projection \overrightarrow{E}(G) ⟶ \overrightarrow{E}(G)/\{id, ι\} = E(G)). Fix one of them, say, the following one:

E(G) = B^{×}\B̂^{×}/U(ℓ)Y ⟶^{∼} (B^{×}\B̂^{×}/U(ℓ)Y ) × \{0\} ⊂ (B^{×}\B̂^{×}/U(ℓ)Y ) × Z/2Z = \overrightarrow{E}(G),

for which

\begin{matrix}s : E(G) ⟶ (B^{×}\B̂^{×}/UY ) × \{0\},  s(B^{×}bU(ℓ)Y) = (B^{×}bUY, 0), \\ t : E(G) ⟶ (B^{×}\B̂^{×}/UY ) × \{1\},  t(B^{×}bU(ℓ)Y) = (B^{×}bξ_{ℓ}UY, 1). \end{matrix}

The chain and cochain complexes of G

Z[E(G)] ⟶^{d_{∗}=−s_{∗}+t_{∗}} Z[V(G)], Z[V(G)] ⟶^{d^{∗}=−s^{∗}+t^{∗}} Z[E(G)]

are then identified, respectively, with

𝕊(U(ℓ)Y; Z) ⟶^{(−α_{∗},β_{∗})} 𝕊(UY; Z)^{⊕2}, 𝕊(UY; Z)^{⊕2} ⟶^{−α^{∗}+β^{∗}} 𝕊(U(ℓ)Y; Z),

in the notation of 1.2.1 (in particular, both maps d_{∗} and d^{∗} are 𝕋^{S∪{ℓ}}(U(ℓ))-linear). As a result,

H_{1}(G) = Ker(d_{∗}) = 𝕊(U(ℓ)Y; Z)^{ℓ−new}

and

H^{0}(G) = {(ξ_{ℓ} · f, f) | f ∈ 𝕊(UY; Z)_{triv}} ,

by Proposition 1.2.3(2). Set

Z[V(G)]_{0} := Ker(Z[V(G)] ⟶ H_{0}(G)) = Im(d_{∗}).

1.5.8. Scalar products. Under the assumptions of 1.5.5(3), the formulas

u : Z[E(G)] × Z[E(G)] = 𝕊(U(ℓ)Y; Z) × 𝕊(U(ℓ)Y; Z) ⟶ Z, u(f, f′) = ∑_{e∈E(G)} f(e)f′(e)

and

u′ : Z[V(G)] × Z[V(G)] = 𝕊(UY; Z)^{⊕2} × 𝕊(UY; Z)^{⊕2} ⟶ Z, u′(f, f′) = ∑_{a∈V(G)} f(a)f′(a)

define non-degenerate symmetric bilinear pairings satisfying

∀T ∈ Z[U(ℓ)\B̂^{×}/U(ℓ)] u(T f, f′) = u(f, T^{∨}f′),
∀T ∈ Z[U\B̂^{×}/U] u′(T f, f′) = u′(f, T^{∨}f′)

and

∀f ∈ Z[E(G)] ∀f′ ∈ Z[V(G)]  u(f, s^{∗}f′) = u′(s_{∗}f, f′),  u(f, t^{∗}f′) = u′(t_{∗}f, f′).  (1.5.8.1)

In particular, for each finite prime v ∉ S ∪ {ℓ} of F, the adjoint of T (v) (resp. of S(v)) with respect to u or u′ is equal to T (v)^{∨} = S(v)^{−1}T (v) (resp. to S(v)^{∨} = S(v)^{−1}).

1.5.9. Proposition. Let A ⊂ C be a subring. Let v_{0} ∉ S ∪ {ℓ} be a finite prime of F with trivial class in F_{+}^{×}\F̂^{×}/Nrd(U).

(1) We have (T (v_{0}) − N(v_{0}) − 1) 𝕊(UY; A)^{⊕2} ⊆ A[V(G)]_{0}, where A[V(G)]_{0} := Z[V(G)]_{0} ⊗ A.

(2) If f̃ ∈ 𝕊(UY; A)_{0}^{⊕2} satisfies T (v_{0})f̃ = λf̃ for some λ ∈ A, then f̃ ∈ A[V(G)]_{0}. In particular, if f̃ ∈ 𝕊(UY; A)_{0}^{⊕2} is an eigenform for the action of 𝕋^{S∪{ℓ}}(U), then f̃ ∈ A[V(G)]_{0}.

Proof. (1) Under the non-degenerate pairing A[V(G)] × A[V(G)] ⟶ A obtained from u′ by extending the scalars, the orthogonal complement of A[V(G)]_{0} is equal to

(A[V(G)]_{0})^{⊥} = (Im(−s_{∗} + t_{∗}))^{⊥} = Ker(−s^{∗} + t^{∗}) = {(ξ_{ℓ} · f, f) | f ∈ 𝕊(UY; A)_{triv}} ,

where the second (resp. the third) equality follows from (1.5.8.1) (resp. from Proposition 1.2.3(2)). The elements S(v_{0}) − 1 and T (v_{0}) − N(v_{0}) − 1 annihilate 𝕊(UY; A)_{triv}, which implies that both S(v_{0})^{∨} − 1 = S(v_{0})^{−1} − 1 and T (v_{0})^{∨} − N(v_{0}) − 1 = S(v_{0})^{−1}T (v_{0}) − N(v_{0}) − 1 (hence also T (v_{0}) − N(v_{0}) − 1) annihilate (A[V(G)]/A[V(G)]_{0})⊗Q ⊃ A[V(G)]/A[V(G)]_{0} (recall that A[V(G)]/A[V(G)]_{0} = H_{0}(G)⊗A is a free A-module). It follows that

(T (v_{0}) − N(v_{0}) − 1) 𝕊(UY; A)^{⊕2} = (T (v_{0}) − N(v_{0}) − 1) A[V(G)] ⊆ A[V(G)]_{0}.

(2) We know by (1) that the image of (λ − N(v_{0}) − 1)f̃ in the free A-module A[V(G)]/A[V(G)]_{0} is trivial. As λ − N(v_{0}) − 1 ∈ A − {0} by (1.1.7.1), the image of f̃ is trivial, too.

1.6. Bad reduction of the Jacobian of the Shimura curve at ℓ

Let U, S and ℓ be as in 1.2.1. Let Y = Y_{ℓ} × Y^{(ℓ)} = F_{ℓ}^{×} × Y^{(ℓ)} be as in 1.4.2. In addition, assume that U satisfies the condition from 1.5.5(3). 1.6.1. Components and geometric components. Recall that M = M_{U′Y} is an irreducible smooth projective curve over F, whose field of constants F′ := F_{U′Y} ⊂ F^{ab} satisfies

rec_{F} : F_{+}^{×}\F̂^{×}/Nrd(U′)Y^{2} ⟶^{∼} Gal(F′/F).

For any field L ⊃ F, the set of irreducible (= connected) components of M ⊗_{F} L is in bijection with Spec(F′ ⊗_{F} L). As Nrd(U′)Y^{2} ⊃ O_{ℓ}^{×}F_{ℓ}^{×2}, the completion of F′ at any prime above ℓ is isomorphic to F_{ℓ} or F_{ℓ^{2}}, which implies that F′ ⊗_{F} F_{ℓ^{2}} ⟶^{∼} F_{ℓ^{2}}^{[F′:F]}. As a result, each irreducible component of M ⊗_{F} F_{ℓ^{2}} is geometrically irreducible. 1.6.2. Jacobian. The Jacobian of M

J(M) := Pic_{M/F}^{∘} ⟶^{∼} Res_{F′/F} Pic_{M/F′}^{∘}

is an abelian variety defined over F. If L ⊃ F is a field and D is a divisor on M ⊗_{F} L which has degree zero on each connected component of M ⊗_{F} \overline{L}, then D represents a point cl(D) ∈ J(M)(L). 1.6.3. Actions of Hecke correspondences. For any cohomology theory H(−) which admits trace maps for finite flat morphisms between curves we let the Hecke correspondence [U′Y gU′Y] from (1.3.4.1-2) (g ∈ B̂^{′×}) act on H(M) = H(M_{U′Y}) as follows (by “Picard functoriality”):

H(M_{U′Y}) ⟶^{pr^{′∗}} H(M_{(g^{−1}U′g∩U′)Y}) ⟶^{[·g]^{∗}} H(M_{(U′∩gU′g^{−1})Y}) ⟶^{pr_{∗}} H(M_{U′Y}).  (1.6.3.1)

This formula (which also applies to functors such as H(−) = Pic_{−/F}^{∘} = J(−) or H(−) = Γ(−, Ω_{−/F})) defines a ring homomorphism

Z[U′Y \B̂^{′×}/U′Y] ⟶ End_{Z}(H(M_{U′Y})).  (1.6.3.2)

As in 1.1.5, denote by 𝕋^{′S∪{ℓ}}(U′Y) the (commutative) subring of Z[U′Y \B̂^{′×}/U′Y] generated by the double cosets [U′Y xU′Y] for all x ∈ (B̂^{′S∪{ℓ}})^{×}. The isomorphism φ : B̂^{(ℓ)} ⟶^{∼} B̂^{′(ℓ)} induces a ring isomorphism

φ_{∗} : 𝕋^{S∪{ℓ}}(UY) ⟶^{∼} 𝕋^{′S∪{ℓ}}(U′Y),  φ_{∗}([UY gUY]) = [U′Y φ(g)U′Y]  (g ∈ (B̂^{S∪(ℓ)})^{×}). (1.6.3.3)

For each v ∉ S ∪ {ℓ}, the elements

T′(v) := φ_{∗}(T (v)), S′(v) := φ_{∗}(S(v)) ∈ 𝕋^{′S∪{ℓ}}(U′Y)

are independent of the choice of φ. Combining (1.6.3.2) with (1.6.3.3) we obtain a ring homomorphism

𝕋^{S∪{ℓ}}(UY) ⟶ End_{Z}(H(M_{U′Y})).  (1.6.3.4)

1.6.4. Néron model. Denote by J the Néron model of J(M) ⊗_{F} F_{ℓ^{2}} over O_{ℓ^{2}}, by J_{s} = J ⊗_{O_{ℓ^{2}}} k(ℓ^{2}) its special fibre and by Φ = J_{s}/J_{s}^{∘} the étale group scheme over k(ℓ^{2}) of connected components of J_{s}.

As M ⊗_{F} F_{ℓ^{2}} has a semi-stable model M ⊗_{O_{ℓ}} O_{ℓ^{2}} whose special fibre consists of several copies of P_{k(ℓ^{2})}^{1} intersecting at ordinary double points defined over k(ℓ^{2}), the general theory [NM, ch. 9] tells us that J_{s}^{∘} is a split torus over k(ℓ^{2}) and Φ is a constant group scheme. By abuse of language we identify Φ with the finite abelian group Φ(\overline{k(ℓ)}). 1.6.5. Connected components of the Néron model. There are two equivalent descriptions of Φ, due to Raynaud [Ra 1, Prop. 8.1.2] (see also [NM, Thm. 1 in 9.6]) and Grothendieck [G2, Thm. 11.5, Thm. 12.5], respectively. They can be summed up by the following commutative diagram with exact rows and columns, which we have borrowed from [Ed].

\begin{aligned}& & 0 & & 0 \\ & & ↓ & & ↓ \\ & & X & = & X \\ & & ↓ & & ↓^{i} \\ Z[V(G)] & ⟶^{d^{∗}} & Z[E(G)] & ⟶ & X^{∨} & ⟶ & 0 \\ ↓^{−id} & & ↓^{d_{∗}} & & ↓ \\ Z[V(G)] & ⟶^{μ_{0}} & Z[V(G)]_{0} & ⟶ & Φ & ⟶ & 0 \\ & & ↓ & & ↓ \\ & & 0 & & 0 \end{aligned}  (1.6.5.1)

In this diagram, the free abelian group

X := Ker(d_{∗}) = H_{1}(G) = 𝕊(U(ℓ)Y; Z)^{ℓ−new} ⟶^{∼} X^{∗}(J_{s}^{∘})

is canonically isomorphic to the character group of the torus J_{s}^{∘} ([Ed, p. 140]). The map

i : X ⟶ X^{∨} = Hom_{Z}(X, Z)  (1.6.5.2)

is induced by the monodromy pairing

X × X ↪ Z[E(G)] × Z[E(G)] ⟶^{u} Z,  (1.6.5.3)

where u is the scalar product from 1.5.8. The composition of μ_{0} with the inclusion Z[V(G)]_{0} ↪ Z[V(G)] is equal to the map

μ : Z[V(G)] ⟶ Z[V(G)], μ(C) = ∑_{C′}(C · C′) C′,

where C, C′ ∈ V(G) are irreducible components of the special fibre of M ⊗_{O_{ℓ}} O_{ℓ^{2}} and (C · C′) ∈ Z is their intersection product on the regular scheme M ⊗_{O_{ℓ}} O_{ℓ^{2}}. 1.6.6. Specialisation of divisors. Let K be a finite unramified extension of F_{ℓ^{2}}. Each divisor D on M ⊗_{F} K naturally extends to a Cartier divisor D̃ on M ⊗_{O_{ℓ}} O_{K} (write D = D_{1} − D_{2} with effective divisors D_{i} and let D̃ := D̃_{1} − D̃_{2}, where D̃_{i} is the closure of D_{i} in M ⊗_{O_{ℓ}} O_{K}).

If D has degree zero on each connected component of M ⊗_{F} \overline{K}, then cl(D) ∈ J(M)(K) = J(O_{K}) and its image in Φ is represented by

cl(D)_{V} := ∑_{C∈V(G)}(C_{K} · D̃) C ∈ Z[V(G)]_{0}, C_{K} = C ⊗_{O_{ℓ^{2}}} O_{K}.

In the special case when D = ∑ n_{P} P is a linear combination of K-rational points P ∈ M(K), each P̃ intersects C_{K} for exactly one irreducible component C = C(P) ∈ V(G) of the special fibre of M ⊗_{O_{ℓ}} O_{ℓ^{2}} (we say that P specialises to C(P)). Consequently,

cl(∑ n_{P} P)_{V} = ∑ n_{P} C(P).

1.6.7. Compatibility of Hecke actions. The recipe from [Ed, p. 140] defines, for any k(ℓ^{2})-algebra A, a morphism of abelian groups Z[E(G)] ⊗ A^{×} ⟶ J_{s}^{∘}(A) which is functorial in A and which sits in an exact sequence

Z[V(G)] ⊗ A^{×} ⟶^{d^{∗}⊗id} Z[E(G)] ⊗ A^{×} ⟶ J_{s}^{∘}(A) ⟶ 0.  (1.6.7.1)

The resulting isomorphism between H^{1}(G) = Coker(d^{∗}) and the group of cocharacters X^{∨} = X_{∗}(J_{s}^{∘}) = Hom_{Z}(X^{∗}(J_{s}^{∘}), Z) of the torus J_{s}^{∘} does not depend on the choice of orientation of G. Moreover, for each v ∉ S ∪ {ℓ}, the action of T′(v) (resp. S′(v)) on J_{s}^{∘}(A) given by (1.6.3.1) for H(−) = Pic_{−/F}^{∘} and the functoriality of the Néron model is induced by the action of T (v) (resp. S(v)) on Z[E(G)] = 𝕊(U(ℓ)Y, Z). In other words, (1.6.7.1) becomes an exact sequence of 𝕋^{S∪{ℓ}}(UY)-modules (with the action on the third term via (1.6.3.4)).

We equip X = H_{1}(G) (resp. X^{∨} = H^{1}(G)) with the structure of a 𝕋^{S∪{ℓ}}(UY)-module induced by the inclusion H_{1}(G) ⊂ Z[E(G)] = 𝕊(U(ℓ)Y, Z) (resp. by the surjection Z[E(G)] = 𝕊(U(ℓ)Y, Z) ⟶ H^{1}(G)). The map (1.6.5.2) induces a morphism of 𝕋^{S∪{ℓ}}(UY)-modules

i : ^{h}X ⟶ X^{∨}

and Φ = Coker(i) inherits a 𝕋^{S∪{ℓ}}(UY)-module structure as a quotient of X^{∨}.

For any commutative ring A ⊃ Z of characteristic zero and 0 ≠ m ∈ A there is an isomorphism of 𝕋^{S∪{ℓ}}(UY) ⊗ A-modules (which depends on m)

(Φ ⊗ A)[m] ⟶^{∼} Ker (^{h}X ⊗ A/mA ⟶^{i⊗id} X^{∨} ⊗ A/mA)

arising from snake lemma. 1.6.8. The Eichler-Shimura relation. For each integer m ≥ 1, the canonical isomorphism

H_{et}^{1}(M ⊗_{F} \overline{F} , μ_{m}) ⟶^{∼} J[m]  (1.6.8.1)

is 𝕋^{S∪{ℓ}}(UY)-equivariant. The Eichler-Shimura congruence relation [C2, §10] states that, for every prime v ∉ S ∪ {ℓ} of F, the special fibre of (the flat extension to a proper smooth model of M over O_{F,v}) the Hecke correspondence T (v) is equal to

T (v) (mod v) = Γ_{Fr_{v}∘[·ϖ_{v}]} +^{t}Γ_{Fr_{v}} = Γ_{Fr_{v}} ∘ S(v) +^{t}Γ_{Fr_{v}}.  (1.6.8.2)

The proper and smooth base change theorems for étale cohomology imply that, if v does not divide m, the G_{F} -module H_{et}^{1}(M ⊗_{F} \overline{F} , Z/mZ) is unramified. Letting both sides of (1.6.8.2) act on H_{et}^{1}(M ⊗_{F} \overline{F} , Z/mZ) contravariantly (as in (1.6.3.1)), we obtain the following relation:

Fr_{geom}(v)^{2} − T (v)S(v)^{−1} Fr_{geom}(v) + N(v)S(v)^{−1} = 0 ∈ End_{Z}(H_{et}^{1}(M ⊗_{F} \overline{F} , Z/mZ)).  (1.6.8.3)

Applying the Tate twist and the involution t ↦ t^{∨} yields, respectively,

Fr_{geom}(v)^{2} − T (v)S(v)^{−1}N(v)^{−1} Fr_{geom}(v) + N(v)^{−1}S(v)^{−1} = 0 ∈ End_{Z}(J[m])

and

Fr_{geom}(v)^{2} − T (v)N(v)^{−1} Fr_{geom}(v) + S(v)N(v)^{−1} = 0 ∈ End_{Z}(^{h}J[m]).  (1.6.8.4)

1.6.9. Erratum for [N1]. The congruence relation (1.6.8.2) (which can be checked, for example, on the classical modular curve Y_{1}(N) parameterising elliptic curves E equipped with a level structure μ_{N} ↪ E) was stated incorrectly in [N1, (1.14.1)]. As a result, the decomposition of H_{et}^{1}(M ⊗_{F} \overline{F} , \overline{Q}_{ℓ}) in [N1, Prop. 1.18(ii)] should involve the Galois representations V_{ℓ}(π̃) = V_{ℓ}(π) ⊗ ω_{π}^{−1} rather than V_{ℓ}(π). 1.6.10. Self-duality of Φ. The monodromy pairing (1.6.5.3) gives rise to a non-degenerate symmetric pairing [G2, (11.4.1)]

( , ) : Φ × Φ ⟶ Q/Z,

defined as follows. According to Grothendieck’s description,

Φ = Coker (X ⟶^{i} X^{∨} = Hom_{Z}(X, Z)) , (i(x))(y) = u(x, y).

Fix an integer m ≥ 1 such that mΦ = 0. Given x^{∨}, y^{∨} ∈ X^{∨} there is a unique x ∈ X such that i(x) = mx^{∨}; the value

([x^{∨}], [y^{∨}]) := \frac{1}{m} y^{∨}(x) + Z = \frac{1}{m} u(x, y^{∨}) + Z ∈ \frac{1}{m}Z/Z ⊂ Q/Z  (1.6.10.1)

is independent of the choice of m and depends only on the respective classes [x^{∨}], [y^{∨}] ∈ Φ of x^{∨} and y^{∨}. Above, y^{∨} ∈ Z[E(G)] is any representative of y^{∨} ∈ X^{∨} ∈ Coker(d^{∗}).

If we do not assume that mΦ = 0, the same formula defines a non-degenerate pairing

( , )_{m} : Φ[m] × Φ/mΦ ⟶ \frac{1}{m}Z/Z

and, for any commutative ring A ⊃ Z of characteristic zero and 0 ≠ m ∈ A, a pairing

( , )_{m} : (Φ ⊗ A)[m] × (Φ/mΦ) ⊗ A ⟶ \frac{1}{m}A/A

satisfying

∀t ∈ 𝕋^{S∪{ℓ}}(UY) (t(x), y)_{m} = (x, t^{∨}(y))_{m}.

1.6.11. Proposition. Let A ⊃ Z be a commutative ring and m ≥ 1 an integer. Assume that P, Q ∈ A[V(G)]_{0} and that mP = μ_{0}(P′) for some P′ ∈ A[V(G)]. If we denote by [P] and [Q] the respective images of P and Q in (Φ ⊗ A)[m] and (Φ/mΦ) ⊗ A, then

([P], [Q])_{m} = \frac{1}{m}u′(P′, Q) + A ∈ \frac{1}{m}A/A.

Proof. This follows easily from the definitions and the diagram (1.6.5.1): fix P̃ ∈ A[E(G)] such that P = d_{∗}P̃; then [P] = [x^{∨}], where x^{∨} is the image of P̃ in X^{∨} ⊗ A. As d_{∗}(mP̃ + d^{∗}P′) = mP − μ_{0}(P′) = 0, the element x := mP̃ + d^{∗}P′ lies in X ⊗ A and satisfies i(x) = mx^{∨}. Similarly, if we fix Q̃ ∈ A[E(G)] such that Q = d_{∗}Q̃ and denote by y^{∨} its image in Φ ⊗ A, then we can take y^{∨} = Q̃ in (1.6.10.1), hence

([P], [Q])_{m} = \frac{1}{m} y^{∨}(x) + A = \frac{1}{m} u(mP̃ + d^{∗}P′, Q̃) + A = u(d^{∗}P′, Q̃) + A = u′(P′, d_{∗}Q̃) + A = u′(P′, Q) + A.

1.7. ℓ-adic uniformisation of the Jacobian

The assumptions of §1.6 are in force. 1.7.1. ℓ-adic uniformisation. As J has a split totally toric reduction over F_{ℓ^{2}} and a canonical principal polarisation, there is a commutative diagram of 𝕋^{S∪{ℓ}}(UY)[G_{F_{ℓ^{2}}} ]-modules with exact rows

\begin{aligned}0 & ⟶ & ^{h}X & ⟶ & X^{∨} ⊗ \overline{F} _{ℓ}^{×} & ⟶ & J(\overline{F}_{ℓ}) & ⟶ & 0 \\ & & ∥ & & ↓^{ord}_{ℓ} \\ 0 & ⟶ & ^{h}X & ⟶^{i} & X^{∨} ⊗ Q \end{aligned}  (1.7.1.1)

in which G_{F_{ℓ^{2}}} = Gal(\overline{F}_{ℓ}/F_{ℓ^{2}}) acts trivially on X and X^{∨}. This yields, for every integer m ≥ 1, an exact sequence of 𝕋^{S∪\{ℓ\}}(UY)[G_{F_{ℓ^{2}}} ]-modules (in which J[m] := J(\overline{F}_{ℓ})[m])

0 ⟶ X^{∨} ⊗ μ_{m} ⟶ J[m] ⟶ ^{h}X ⊗ Z/mZ ⟶ 0,  (1.7.1.2)

which is self-dual (up to a sign and the involution t ↦ t^{∨}) with respect to the Weil pairing

J[m] × J[m] ⟶ μ_{m}  (1.7.1.3)

corresponding to the canonical principal polarisation of J. 1.7.2. Connected components. Let K be a finite unramified extension of F_{ℓ^{2}}. The canonical map J(K) = J(O_{K}) ⟶ Φ sits in a commutative diagram with exact rows

0 ⟶ ^{h}X ⟶ X^{∨} ⊗ K^{×} ⟶ J(K) ⟶ 0
∥  ↓^{ord}_{ℓ}  ↓
0 ⟶ ^{h}X ⟶^{i}  X^{∨}  ⟶ Φ ⟶ 0,

whose first row is obtained from (1.7.1.1) by taking invariants under Gal(\overline{F}_{ℓ}/K). 1.7.3. Kummer maps. Let I = Gal(\overline{F}_{ℓ}/F_{ℓ}^{ur}) be the common inertia group of F_{ℓ^{2}} and the field K from 1.7.2. Fix an integer m ≥ 1 prime to N(ℓ); denote by

∂ : J(K) ⊗ Z/mZ ↪ H^{1}(K, J[m])

the Kummer map arising from the standard descent sequence

0 ⟶ J(K)[m] ⟶ J(K) ⟶^{m} J(K) ⟶ H^{1}(K, J[m]) ⟶ H^{1}(K, J) ⟶ ⋯

and by ∂_{ram} (“the ramified part of ∂”) the composite map

∂_{ram} : J(K) ⊗ Z/mZ ⟶ H^{1}(K, J[m]) ⟶ H^{1}(I, J[m]).

The key point of the construction of Bertolini and Darmon [BD 2] is the fact that, unlike in the case of good reduction, the map ∂_{ram} can be far from being zero.

The corresponding Kummer maps for the torus X^{∨} ⊗ G_{m} over K and F_{ℓ}^{ur} are isomorphisms related by a commutative diagram

X^{∨} ⊗ K^{×} ⊗ Z/mZ ⟶^{∼} H^{1}(K, X^{∨} ⊗ μ_{m})
↓^{ord}_{ℓ}  ↓
X^{∨} ⊗ Z/mZ  ⟶^{∼} H^{1}(I, X^{∨} ⊗ μ_{m})

The three Kummer maps can be combined into the following commutative diagram whose first two rows are exact.

hX ⊗ Z/mZ ⟶ X^{∨} ⊗ K^{×} ⊗ Z/mZ ⟶ J(K) ⊗ Z/mZ ⟶ 0
∥  ∥  ↓^{∂}
hX ⊗ Z/mZ ⟶ H^{1}(K, X^{∨} ⊗ μ_{m}) ⟶ H^{1}(K, J[m])
↓^{i⊗id}  ↓  ↓
X^{∨} ⊗ Z/mZ = H^{1}(I, X^{∨} ⊗ μ_{m}) ⟶ H^{1}(I, J[m])

This implies that the map ∂_{ram} factors as

J(K) ⊗ Z/mZ ⟶ Φ/mΦ = Coker(X/mX ⟶^{i⊗id} X^{∨}/mX^{∨}) ⟶ H^{1}(I, J[m]).

1.7.4. Unramified cohomology. The cohomology sequence of (1.7.1.2) over F_{ℓ}^{ur}

0 ⟶ X^{∨} ⊗ μ_{m} ⟶ J[m]^{I} ⟶ ^{h}X ⊗ Z/mZ ⟶^{i⊗id} H^{1}(I, X^{∨} ⊗ μ_{m}) = X^{∨}/mX^{∨}

yields

0 ⟶ X^{∨} ⊗ μ_{m} ⟶ J[m]^{I} ⟶ Φ[m] ⟶ 0,

which in turn implies another exact sequence

H_{ur}^{1}(K, X^{∨} ⊗ μ_{m}) ⟶ H_{ur}^{1}(K, J[m]) ⟶ H_{ur}^{1}(K, Φ[m]) ⟶ 0.

Denote by r the composite surjection

r : H_{ur}^{1}(K, J[m]) ⟶ H_{ur}^{1}(K, Φ[m]) = Hom(Gal(F_{ℓ}^{ur}/K), Φ[m]) ⟶^{∼} Φ[m],

where the last isomorphism is the evaluation map at the geometric Frobenius over K.

According to local duality, the cup product

∪ : H^{1}(K, J[m]) × H^{1}(K, J[m]) ⟶ H^{2}(K, μ_{m}) ⟶^{∼} Z/mZ

induced by the Weil pairing (1.7.1.3) gives rise to a perfect pairing

∪ : H_{ur}^{1}(K, J[m]) × Im (H^{1}(K, J[m]) ⟶ H^{1}(I, J[m])) ⟶ Z/mZ.  (1.7.4.1)

1.7.5. Proposition (Explicit reciprocity law). Let K be a finite unramified extension of F_{ℓ^{2}} and m ≥ 1 an integer prime to N(ℓ). If c ∈ H_{ur}^{1}(K, J[m]) and R ∈ J(K), then the cup product (1.7.4.1) satisfies

c ∪ ∂_{ram}(R) = ±m(r(c), the image of R in Φ/mΦ)_{m}.

Proof. This follows from the definitions and [Ru, Lemma 1.4.7(ii)].

1.8. CM points (unramified at ℓ)

The assumptions of §1.6 are in force. 1.8.1. Embeddings. Let K be a totally imaginary quadratic extension of F in which none of the primes from Ram(B) splits. Under this assumption there exists an F -embedding t : K ↪ B; fix such a t. It induces embeddings t_{v} : K_{v} ↪ B_{v} and t̂: K̂ ↪ B̂ (for each prime v of F we use the slightly ambiguous notation K_{v} := K ⊗_{F} F_{v} and O_{K,v} := O_{K} ⊗_{O_{F}} O_{F,v}).

Assume, in addition, that the prime ℓ satisfies the following conditions.

(1.8.1.1) ℓ is inert in K/F; denote by λ the unique prime of K above ℓ. (1.8.1.2) t_{ℓ}^{−1}(U_{ℓ}) = O_{K,ℓ}^{×}.

The existence of t together with (1.8.1.1) imply that there exists an F -embedding t′ : K ↪ B′; any such embedding automatically satisfies the following analogue of (1.8.1.2): t_{ℓ}^{′−1}(O_{B′_{ℓ}}) = O_{K,ℓ} (= O_{K,λ}).

According to the Skolem-Noether theorem, two F_{v}-embeddings K_{v} ↪ B′_{v} are conjugate by an element of B_{v}^{′×}. This implies that, for fixed t and t′, after replacing φ : B̂^{(ℓ)} ⟶^{∼} B̂^{′(ℓ)} by a conjugate isomorphism we can – and will – assume that φ, t and t′ are compatible outside ℓ in the sense that the composite map

K̂^{(ℓ)} ⟶^{t^{(ℓ)}} B̂^{(ℓ)} ⟶^{φ} B̂^{′(ℓ)}

is equal to t^{′(ℓ)}. 1.8.2. CM points in the complex uniformisation. There are exactly two points of C − R which are fixed under the action of t_{τ_{1}}(K^{×}) ⊂ t_{τ_{1}}(K_{τ_{1}}^{×}) ⊂ B_{τ_{1}}^{×} ⟶^{∼} GL_{2}(R); fix one of them and denote it by z′.

The set of CM points by K unramified at ℓ on the curve M = M_{U′Y} is defined as

CM(M, K)_{ℓ−ur} := {[z′, b′]_{U′Y} | b′ ∈ B̂^{′×}, b′_{ℓ} = 1} ⊂ (M ⊗_{F,τ_{1}} C)(C).

Shimura’s reciprocity law states that

CM(M, K)_{ℓ−ur} ⊂ M(K^{ab})

and that G_{K}^{ab} acts on this set as follows:

∀a ∈ K̂^{×} rec_{K}(a) [z′, b′]_{U′Y} = [z′, t̂′(a)b′]_{U′Y}

(in order to make sense of this statement one must appropriately choose an embedding K ↪ C extending τ_{1} : F ↪ R; see [Mi 3, II.5]). This implies that a CM point x = [z′, b′]_{U′Y} ∈ CM(M, K)_{ℓ−ur} is defined over a finite abelian extension K(x) of K satisfying

rec_{K} : Gal(K(x)/K) ⟶^{∼} K^{×}\K̂^{×}/t̂^{′−1}(b′U′Y b^{′−1}).

In particular, λ splits completely in K(x)/K, since t′_{ℓ}(K_{ℓ}^{×}) = t′_{ℓ}(O_{K,ℓ}^{×}F_{ℓ}^{×}) ⊂ U′_{ℓ}Y_{ℓ} = b′_{ℓ}U′_{ℓ}Y_{ℓ}b_{ℓ}^{′−1}. 1.8.3. CM points in the ℓ-adic uniformisation. Fix one of the two F_{ℓ}-embeddings K_{ℓ} ↪ F_{ℓ}^{ur} (i.e., fix an isomorphism K_{ℓ} ⟶^{∼} F_{ℓ^{2}} over F_{ℓ}). Under the ℓ-adic uniformisation 1.4.2 of M, the set CM(M, K)_{ℓ−ur} corresponds to

CM(M, K)_{ℓ−ur} = {[z, b]_{UY} | b ∈ B̂^{(ℓ)×}} ⊂ M(K_{ℓ}),

where z is one of the two fixed points of t_{ℓ}(K^{×}) ⊂ t_{ℓ}(K_{ℓ}^{×}) ⊂ GL_{2}(F_{ℓ}) acting on P^{1}(K_{ℓ}) − P^{1}(F_{ℓ}) (z is determined by the choice of K_{ℓ} ↪ F_{ℓ}^{ur}). Moreover, the action of G_{K}^{ab} is given by the formula

∀a ∈ K̂^{×} rec_{K}(a) [z, b]_{UY} = [z, t̂^{(ℓ)}(a^{(ℓ)})b]_{UY} ,

where a^{(ℓ)} denotes the projection of a to K̂^{(ℓ)×}.

2. Proof of Theorem A and its corollaries

From now on until the end of §2.9 we assume that f ∈ S_{2}(𝔫, ω), χ : A_{K}^{×}/K^{×} ⟶ C^{×} (χ|_{A_{F}^{×}} · ω = 1) and V = V_{𝔭}(f)(1) are as in Theorem A. In particular, L(f_{K}, χ, 1) ≠ 0 and there exists g_{𝔭} ∈ G_{F} satisfying (A1)–(A3). In addition, if f has CM by a totally imaginary quadratic extension K(f) of F, we assume in 2.9 that K(f) ⊄ K_{χ}F_{ω}.

2.1. Local linear forms and transfer to a definite quaternion algebra

We begin by recalling results of Tunnell, H. Saito and D. Prasad on local invariant linear forms.

2.1.1. Proposition [Tu], [Sa]. Let v be a finite prime of F, let D be a quaternion algebra over F_{v}, let π_{D} be an irreducible smooth complex representation of D^{×} (of infinite dimension if D = M_{2}(F_{v})) with central character ω_{D} : F_{v}^{×} ⟶ C^{×}. Let φ : K_{v}^{×} ⟶ C^{×} be a smooth character satisfying φ|_{F_{v}^{×}} · ω_{D} = 1.

(1) If there exists an F_{v}-embedding K_{v} ↪ D (which we fix), then

dim_{C} Hom_{K_{v}^{×}}(π_{D} ⊗ φ, C) = \{ \begin{matrix}1, & ε(π_{D} × φ, \frac{1}{2}) = inv_{v}(D) (η_{v}ω_{D})(−1) \\ 0, & ε(π_{D} × φ, \frac{1}{2}) = −inv_{v}(D) (η_{v}ω_{D})(−1). \end{matrix}

(2) If D = M_{2}(F_{v}) and π_{D} is a principal series representation, then Hom_{K_{v}^{×}}(π_{D} ⊗ φ, C) ≠ 0.

(3) If there is no F_{v}-embedding K_{v} ↪ D ( ⟺ D ≠ M_{2}(F_{v}) and v splits in K/F), then ε(π_{D} × φ, \frac{1}{2}) = ω_{D}(−1).

2.1.2. Proposition [Pr, Thm. 4]. Assume that, in the situation of Proposition 2.1.1, φ = 1. Denote by N_{v} the normaliser of K_{v}^{×} in D^{×}. If Hom_{K_{v}^{×}}(π_{D}, C) ≠ 0, then the non-trivial element of N_{v}/K_{v}^{×} ⟶^{∼} {±1} acts on Hom_{K_{v}^{×}}(π_{D}, C) by multiplication by inv_{v}(D) ε(π_{D}, \frac{1}{2}).

2.1.3. The archimedean factors Γ_{C}(s+1/2) of L(π(f)×χ, s) have no pole (and no zero) at the central point s = 1/2. The assumption L(f_{K}, χ, 1) ≠ 0 is, therefore, equivalent to L(π(f) × χ, \frac{1}{2}) ≠ 0. The functional equation (0.5.1) implies that

ε(π(f) × χ, \frac{1}{2}) = ∏_{v} ε(π(f)_{v} × χ_{v}, \frac{1}{2}) = 1.  (2.1.3.1)

Set

S_{B} = \{v a prime of F | ε(π(f)_{v} × χ_{v}, \frac{1}{2}) ≠ (η_{v}ω_{v})(−1)\},

where ω_{v} : F_{v}^{×} ⟶ C^{×} and χ_{v} : K_{v}^{×} = (K ⊗_{F} F_{v})^{×} ⟶ C^{×} denote the local components of ω and χ, respectively.

2.1.4. Proposition. (1) S_{B} is a finite set of even cardinality containing S_{∞}. If v ∈ S_{B} − S_{∞}, then v does not split in K/F and π(f)_{v} is not a principal series representation.

(2) Denote by B the unique quaternion algebra over F such that Ram(B) = S_{B}. The algebra is totally definite and there exists an F -embedding t : K ↪ B (which we fix).

(3) There exists a unique irreducible automorphic representation π of B_{A}^{×} which corresponds to π(f) via the Jacquet-Langlands correspondence. The central character of π is equal to ω and the archimedean component of π is trivial: π_{∞} = 1.

(4) ∀v ∤ ∞ dim_{C} Hom_{K_{v}^{×}}(π_{v} ⊗ χ_{v}, C) = 1.

Proof. The cardinality of S_{B} is even, by (2.1.3.1) and the product formula ∏_{v}(η_{v}ω_{v})(−1) = 1. Each v ∈ S_{∞} belongs to S_{B}, since ε(π(f)_{v} × χ_{v}, \frac{1}{2}) = ω_{v}(−1) = 1 and η_{v}(−1) = −1. The remaining statements follow from Proposition 2.1.1 and basic properties of the Jacquet-Langlands correspondence. 2.1.5. As π_{∞} = 1, we can – and will – consider π as a representation of B̂^{×} = B_{A}^{×}/(B ⊗ R)^{×}. It occurs with multiplicity 1 as a subrepresentation of 𝕊 = ⋃_{U} 𝕊(U; C) and the embedding π ↪ 𝕊 is unique up to a scalar multiple. Moreover, the generalised Ramanujan conjecture [Bl] for f implies, as in 1.1.7, that the image of π is contained in 𝕊_{0} = ⋃_{U} 𝕊(U; C)_{0}.

2.2. Global linear forms and global test vectors

The fixed embedding t : K ↪ B (which is unique up to conjugation by an element of B^{×}) induces local embeddings t_{v} : K_{v} ↪ B_{v} and an adelic embedding t̂: K̂ ↪ B̂. 2.2.1. The property 2.1.4(4) and the fact that, for all but finitely many v, the image of a spherical vector in π_{v} by a non-trivial element of Hom_{K_{v}^{×}}(π_{v} ⊗ χ_{v}, C) is non-zero, imply that

dim_{C} Hom_{K^{×}}(π ⊗ χ, C) = 1.

2.2.2. A suitably regularised integral

f_{B} ↦ ℓ_{χ}(f_{B}) := ∫_{_{K^{×}}/_{F^{×}}K^{×}} χ(x)f_{B}(t̂(x)) dx (f_{B} ∈ π ⊂ 𝕊_{0})

(for a fixed Haar measure dx on K̂^{×}) defines an element ℓ_{χ} ∈ Hom_{K^{×}}(π⊗χ, C). According to a fundamental result of Waldspurger [W2, Thm. 2], our assumption about the non-vanishing of L(π(f)×χ, \frac{1}{2}) = L(π ×χ, \frac{1}{2}) is equivalent to ℓ_{χ} ≠ 0.

In concrete terms, there exist an open compact subgroup U ⊂ B̂^{×} (sufficiently small in the sense that χ(t̂^{−1}(U)) = 1) and a function f_{B} ∈ π^{U} ⊂ 𝕊(U; C)_{0} satisfying

ℒ_{χ}(f_{B}) := ∑_{_{K^{×}}/_{F^{×}}K^{×}_{t^{−1}}(U)} χ(x)f_{B}(t̂(x)) ≠ 0 ∈ C

(f_{B} is a “test vector” for the global linear form ℓ_{χ}). 2.2.3. Note that π has a model over L in the sense that

π^{U_{0}} = (π^{U_{0}} ∩ 𝕊(U_{0}; L)) ⊗_{L} C,

for any open compact subgroup U_{0} ⊂ B̂^{×}. Indeed, the multiplicity one theorem for automorphic forms on B_{A}^{×} implies that π^{U_{0}} coincides with the space of complex-valued functions f_{0} on the finite set B^{×}\B̂^{×}/U_{0} which satisfy a system of linear equations with coefficients in L

T (v)f_{0} = λ_{f}(v)f_{0},

for v belonging to a sufficiently large finite set of primes of F. 2.2.4. Combining 2.2.2 with 2.2.3 we deduce that there exists f_{B} ∈ 𝕊(U; L)_{0} ∩ π satisfying ℒ_{χ}(f_{B}) ≠ 0. After replacing f_{B} by a suitable scalar multiple, we can assume that

f_{B} ∈ 𝕊(U; O_{L})_{0} ∩ π,  χ(t̂^{−1}(U)) = 1,  ℒ_{χ}(f_{B}) ∈ O_{L} − {0}.  (2.2.4.1)

We define

C_{1} = C_{1}(𝔭) := ord_{𝔭}(ℒ_{χ}(f_{B})) ≥ 0.  (2.2.4.2)

2.2.5. We fix U and f_{B} satisfying (2.2.4.1). We also fix a finite set S as in 1.1.5 and a prime v_{0} ∉ S of F which has trivial class in F_{+}^{×}\F̂^{×}/Nrd(U) and for which ω(v_{0}) = 1. Furthermore, we set Y := Ker(ω : F̂^{×} ⟶ C^{×}). 2.2.6. By definition of f_{B}, the spherical Hecke algebra 𝕋^{S}(U) acts on f_{B} by the character

λ_{B} : 𝕋^{S}(U) ⟶ O_{L},  λ_{B}(T (v)) = λ_{f}(v),  λ_{B}(S(v)) = ω(v)  (v ∉ S).

Denote by ^{h}λ_{B} the ring homomorphism

hλ_{B} : 𝕋^{S}(U) ⟶^{∨} 𝕋^{S}(U) ⟶^{λ_{B}} O_{L}.

For any O_{L}[𝕋^{S}(U)]-module N we set

\begin{aligned}N^{(λ_{B})} & := \{n ∈ N | ∀t ∈ 𝕋^{S}(U) t(n) = λ_{B}(t)n\} = N[Ker(λ_{B})] \\ N_{(λ_{B})} & := N ⊗_{𝕋^{S}(U),λ_{B}} O_{L} = N/Ker(λ_{B})N \end{aligned}

(and similarly for ^{h}λ_{B} instead of λ_{B}).

2.3. Choosing a Kolyvagin prime ℓ

Denote by e(𝔭) := ord_{𝔭}(p) the absolute ramification index of L_{𝔭} and fix a G_{F} -stable O_{L,𝔭}-lattice T ⊂ V. For a suitable scalar multiple of the pairing (0.7.1) (which we fix) the lattice T will be self-dual in the sense that (0.7.1) will induce an isomorphism of O_{L,𝔭}[G_{F}]-modules T^{∗}(1) := Hom_{O_{L,𝔭}}(T, O_{L,𝔭})(1) ⟶^{∼} T ⊗ ω^{−1}. 2.3.1. The assumptions (A1)–(A3) tell us that there exists g = g_{𝔭} ∈ G_{F_{ω}}, g|_{K} ≠ id, which acts on T (in a

suitable O_{L,𝔭}-basis e_{1}, e_{2}, which we fix) by a matrix ( \begin{matrix}ε & 0 \\ 0 & u_{g} \end{matrix} ), where ε = ±1 and u_{g} ∈ O_{L,𝔭}^{×}, u_{g}^{2} ≠ 1. Set

C_{2} = C_{2}(𝔭) := ord_{𝔭}(u_{g}^{2} − 1).

It follows from Appendix B.5.2 and B.5.5(7) (resp. B.6.5(5) and B.6.5(7)) in the case when f does not have (resp. has) complex multiplication that for all but finitely many 𝔭 satisfying (A1)–(A3) one can choose g_{𝔭} in such a way that C_{2}(𝔭) = 0. 2.3.2. Fix a large integer n >> 0 and set

F (T/𝔭^{n}T) = \overline{F}^{Ker(G_{F} ⟶Aut(T/𝔭^{n}T))}.

Using the basis e_{1}, e_{2} of T from 2.3.1 we consider the Galois group Gal(F (T/𝔭^{n}T)/F) as a subgroup of the

group Aut_{O_{L,𝔭}}(T/𝔭^{n}T) ⟶^{∼} GL_{2}(O_{L}/𝔭^{n}O_{L}).

Let ℓ be a prime of F satisfying the following properties.

(2.3.2.1) ℓ ∉ S ∪ {v_{0}}; (2.3.2.2) ℓ does not divide p; (2.3.2.3) ℓ is unramified in K_{χ}/F; (2.3.2.4) t_{ℓ}^{−1}(U_{ℓ}) = O_{K,ℓ}^{×}; (2.3.2.5) Fr_{geom}(ℓ) with respect to the extension KF_{ω}F (T/𝔭^{n}T)/F is equal to the conjugacy class of the restric-

tion of g (this makes sense, since ℓ ∤ 𝔫 by (2.3.2.1), hence is unramified in this extension, by (2.3.2.2-3)). The set of ℓ satisfying (2.3.2.1-5) has positive density.

2.3.3. Proposition. The prime ℓ and the Hecke eigenvalue a_{ℓ} := λ_{f}(ℓ) of T (ℓ) acting on f have the following properties.

ω(ℓ) = 1,  ℓ is inert in K/F,  N(ℓ)^{−1} − εu_{g} ∈ 𝔭^{n}O_{L,𝔭},  N(ℓ)^{−1}a_{ℓ} − (ε + u_{g}) ∈ 𝔭^{n}O_{L,𝔭},
N(ℓ) + 1 − εa_{ℓ} ∈ 𝔭^{n}O_{L,𝔭}, ord_{𝔭}(N(ℓ)^{2} − 1) = ord_{𝔭}(u_{g}^{2} − 1) = C_{2}.

Proof. As g acts trivially (resp. non-trivially) on F_{ω} (resp. on K), ω(ℓ) = 1 (resp. ℓ is inert in K/F). The remaining properties follow from the fact that

1 − a_{ℓ}N(ℓ)^{−1}X + ω(ℓ)N(ℓ)^{−1}X^{2} = det (1 − Fr_{geom}(ℓ)X | V) ≡ 1 − (ε + u_{g})X + εu_{g}X^{2} (mod 𝔭^{n}O_{L,𝔭}).

2.3.4. Definition. Denote by n′ ≥ n the integer

n′ := ord_{𝔭}(N(ℓ) + 1 − εa_{ℓ})

(note that N(ℓ) + 1 ± a_{ℓ} ≠ 0, by the generalised Ramanujan conjecture [Bl]).

2.4. Variations on a theme of Boston-Lenstra-Ribet

In this section we prove a weak version of the results of [BLR] for slightly more general coefficient rings.

Proposition 2.4.1. Let A be a quotient of a discrete valuation ring by a non-zero ideal, M a free Amodule of finite rank r ≥ 1, ρ′ : R ⟶ End_{A}(M) a morphism of A-algebras and I_{0}, I_{1} ⊂ A ideals such that I_{0} · Ker(ρ′) = 0 and I_{1} · Coker(ρ′) = 0. Then, for each left R-module N and an element x ∈ N there exists j ∈ Hom_{R}(N, M) such that I_{0}I_{1}^{2} Ann_{A}(u(x)) ⊆ Ann_{A}(x).

Proof. After replacing R (resp. N) by R/Ker(ρ′) (resp. N/Ker(ρ′)N) and x by its image in N/Ker(ρ′)N we can assume that I_{0} = 0. In other words, there is an exact sequence of A-modules

0 ⟶ R ⟶ End_{A}(M) ⟶ C ⟶ 0, I_{1}C = 0.

The map

λ : End_{A}(M) ⟶ Hom_{A}(End_{A}(M), A), (λ(X))(Y) = Tr(Y X)

is an isomorphism of left End_{A}(M)-modules, the module structure on the R.H.S. being given by (Zf)(Y) = f(Y Z). Its composition with the restriction map to R gives rise to a morphism of left R-modules

λ′ : M^{⊕r} = End_{A}(M) ⟶^{λ} Hom_{A}(End_{A}(M), A) ⟶ Hom_{A}(R, A)

whose kernel and cokernel is killed by I_{1}. It follows that the induced morphism of A-modules

λ′_{∗} : Hom_{R}(N, M^{⊕r}) ⟶ Hom_{R}(N, Hom_{A}(R, A)) ⟶^{∼} Hom_{A}(N, A)

satisfies I_{1}^{2} · Coker(λ′_{∗}) = 0. Fix j′ ∈ Hom_{A}(N, A) such that Ann_{A}(j′(x)) = Ann_{A}(x); as I_{1} = t^{m}A (where t ∈ A is a generator of the maximal ideal of A), there exist j_{i} ∈ Hom_{R}(N, M) such that t^{2m}j′ = λ′(j_{1}, …, j_{r}). This implies that

⋂_{i=1}^{r} Ann_{A}(j_{i}(x)) ⊆ Ann_{A}(t^{2m}j′(x)) = Ann_{A}(t^{2m}x);

thus Ann_{A}(j_{i}(x)) ⊆ Ann_{A}(t^{2m}x) for some i = 1, …, r, as claimed. 2.4.2. Let A be a commutative ring, M a free A-module of rank 2, G a group and ρ : G ⟶ Aut_{A}(M) (⟶^{∼} GL_{2}(A)) a group homomorphism. Denote by J ⊂ A[G] the bilateral ideal generated by the elements g^{2} − Tr(ρ(g))g + det(ρ(g)) (for all g ∈ G) and set R = A[G]/J. The Cayley-Hamilton theorem implies that the morphism of A-algebras A[G] ⟶ End_{A}(M) (⟶^{∼} M_{2}(A)) induced by ρ – which will still be denoted by ρ – factors as

ρ : A[G] ⟶ R ⟶^{ρ′} End_{A}(M).

Proposition 2.4.3. If, under the assumptions of 2.4.2, I ⊂ A is an ideal such that Im(ρ) ⊇ I · End_{A}(M), then I^{4} · Ker(ρ′) = 0.

Proof. This statement is proved in [BLR, Prop. 2] in the case when A = I is a field. In general, the arguments in the proof of [loc. cit.] show that the annihilator Ann_{R}(x) = {r ∈ R | rx = 0} of each element x ∈ Ker(ρ′) is a bilateral ideal of R containing {yz − zy | y, z ∈ R}. It follows that ρ′(Ann_{R}(x)) is a bilateral ideal of Im(ρ′) = Im(ρ) containing {y′z′ − z′y′ | y′, z′ ∈ I · End_{A}(M)}, which implies that ρ′(Ann_{R}(x)) contains {t ∈ I^{2} · End_{A}(M) | Tr(t) = 0}. In particular, for any a, b ∈ I^{2} there is w ∈ R such

that wx = 0 and ρ′(w) = ( \begin{matrix}0 & a \\ b & 0 \end{matrix} ). As w^{2} − Tr(ρ(w))w + det(ρ′(w)) = 0 ∈ R by [BLR, Prop. 1], we have

abx = − det(ρ′(w))x = 0; thus I^{4}x = 0.

Corollary 2.4.4. In the situation of Proposition 2.4.3, assume that A = Ã/t^{n}Ã and M = M̃/t^{n}M̃, where Ã is a discrete valuation ring with a uniformiser t, M̃ is a free Ã-module of rank 2 and ρ̃: G ⟶ Aut_{A}(M̃) a group homomorphism lifting ρ.

(1) If M̃ ⊗_{A} Frac(Ã) is an absolutely irreducible representation of G with coefficients in the fraction field Frac(Ã) of Ã, then there exists an integer c ≥ 0 such that Im(ρ̃) ⊇ t^{c} End_{A}(M̃) (hence Im(ρ) ⊇ t^{c} End_{A}(M)).

(2) t^{4c} Ker(ρ′) = 0.

(3) If the residual representation M/tM of G is an absolutely irreducible representation of G with coefficients in A/tA, then Im(ρ) = End_{A}(M) and Ker(ρ′) = 0.

Proof. (1) The existence of c is equivalent to Im(ρ̃) ⊗_{A} Frac(Ã) = End_{A}(M̃) ⊗_{A} Frac(Ã), which is, in turn, equivalent to the absolute irreducibility of M̃ ⊗_{A} Frac(Ã), by a theorem of Burnside [CR, Thm. 3.32] (and Schur’s lemma [CR, Prop. 3.33]).

(2) By (1), Proposition 2.4.3 applies with I = t^{c}A.

(3) By the argument used in the proof of (1), the absolute irreducibility of M/tM = M̃/tM̃ is equivalent to Im(ρ̃) + t End_{A}(M̃) = End_{A}(M̃), which implies that Im(ρ̃) = End_{A}(M̃), by Nakayama’s lemma; thus c = 0.

Proposition 2.4.5. Assume that, in the situation of 2.4.2, A is a quotient of a discrete valuation ring by a non-zero ideal and that we are given h ∈ G, a ∈ A and a basis e_{1}, e_{2} of M such that ρ(h)e_{1} = e_{1} and ρ(h)e_{2} = ae_{2}. If N is a left R-module and N_{0} ⊂ N an A-submodule such that h(n_{0}) = an_{0} for each n_{0} ∈ N_{0}, then the cokernel of the restriction map

Hom_{R}(N, M) ⟶ Hom_{A[h]}(N_{0}, M)

is killed by (a − 1)^{2}I^{12}.

Proof. Write I = bA for some b ∈ A. We must show that, for each j_{0} ∈ Hom_{A[h]}(N_{0}, M), there exists j ∈ Hom_{R}(N, M) such that j|_{N_{0}} = (a − 1)^{2}b^{12}j_{0}. Set

N_{1} := Ker(j_{0}) ⊂ N_{0} ⊂ N^{h=a}.

As j_{0} injects N_{0}/N_{1} into M^{h=a} and (a − 1)M^{h=a} = A(a − 1)e_{2} is a cyclic A-module, (a − 1)N_{0}/N_{1} and its quotients, such as (a − 1)(N_{0}/(N_{0} ∩ RN_{1})) ⟶^{∼} (a − 1)((N_{0} + RN_{1})/RN_{1}), are also cyclic.

Lemma 2.4.6 below implies that the map (a − 1)b^{6}j_{0} : N_{0} ⟶ (a − 1)b^{6}M^{h=a} = A(a − 1)b^{6}e_{2} factors through j_{1} ∈ Hom_{A}(N_{0}/(N_{0} ∩ RN_{1}), A(a − 1)b^{6}e_{2}). Fix an element x ∈ N′ := (N_{0} + RN_{1})/RN_{1} such that (a − 1)x is a generator of the cyclic A-module (a − 1)N′. Applying Proposition 2.4.1 to the Rmodule N/RN_{1} (with I_{1} = bA by definition and I_{0} = b^{4}A by Proposition 2.4.2), we obtain a morphism j_{2} ∈ Hom_{R}(N/RN_{1}, M) such that b^{6}Ann_{A}(j_{2}(x)) ⊆ Ann_{A}(x); thus

(a − 1)b^{6}Ann_{A}((a − 1)j_{2}(x)) ⊆ b^{6}Ann_{A}(j_{2}(x)) ⊆ Ann_{A}(x) ⊆ Ann_{A}(j_{1}(x)).

As both j_{1}(x) and (a − 1)j_{2}(x) belong to the cyclic A-module A(a − 1)b^{6}e_{2}, there exists a′ ∈ A such that (a−1)b^{6}j_{1}(x) = a′(a−1)j_{2}(x). This implies, by definition of x, that the map j := a′(a−1)j_{2} ∈ Hom_{R}(N, M) satisfies j|_{RN_{1}} = 0 and

j|_{N_{0}} = (a − 1)b^{6}j_{1} = (a − 1)^{2}b^{12}j_{0},

as required.

Lemma 2.4.6. (a − 1)b^{6}(N_{0} ∩ RN_{1}) = (a − 1)b^{6}N_{1}.

Proof. It is enough to prove the non-trivial inclusion (a − 1)b^{6}(N_{0} ∩ RN_{1}) ⊆ (a − 1)b^{6}N_{1}. If

n_{0} = ∑_{i=1}^{k} r_{i}n_{i} ∈ N_{0} ∩ RN_{1} (∀i ≥ 1 r_{i} ∈ R, n_{i} ∈ N_{1}),

then

∀j′_{0} ∈ Hom_{R}(RN_{0}, M) ∀i ≥ 0 (a − 1)j′_{0}(n_{i}) = (a − 1)a_{i}e_{2} ∈ (a − 1)M^{h=a} = A(a − 1)e_{2} (a_{i} ∈ A).

Writing r_{i}e_{2} = λ_{i}e_{1} + μ_{i}e_{2}, we obtain

0 = (a − 1)j′_{0} (n_{0} − ∑_{i=1}^{k} r_{i}n_{i}) = (a − 1) (a_{0}e_{2} − ∑_{i=1}^{k} a_{i}(λ_{i}e_{1} + μ_{i}e_{2})) ,

hence

0 = (a − 1) (a_{0}e_{2} − ∑_{i=1}^{k} a_{i}μ_{i}e_{2}) .

This implies that the element n′ := n_{0} − ∑_{i=1}^{k} μ_{i}n_{i} ∈ N_{0} satisfies

∀j′_{0} ∈ Hom_{R}(RN_{0}, M) (a − 1)j′_{0}(n′) = 0.

Applying Proposition 2.4.1 to the R-module RN_{0} and n′ ∈ RN_{0} (again with I_{1} = bA and I_{0} = b^{4}A) we deduce that (a − 1)b^{6}n′ = 0 and (a − 1)b^{6}n_{0} = (a − 1)b^{6} ∑_{i=1}^{k} μ_{i}n_{i} ∈ (a − 1)b^{6}N_{1}, which finishes the proof.

2.5. Weak level raising modulo 𝔭^{n}

In this section we are going to work with the curve M_{U′Y} and its Jacobian J = J(M_{U′Y}) attached to U and Y = Ker(ω) from 2.2.5 and ℓ from 2.3.2 (note that Y = F_{ℓ}^{×} × Y^{(ℓ)}, since ω(ℓ) = 1). 2.5.1. The element

f̃:= ( \begin{matrix}f_{B} \\ εf_{B} \end{matrix} ) ∈ (𝕊(UY; O_{L})_{0}^{⊕2})^{(λ_{B})} ⊂ O_{L,𝔭}[V(G)]_{0}^{(λ_{B})}  (2.5.1.1)

(see Prop. 1.5.9(2)) satisfies

(μ_{0} ⊗ id)(f̃) = ( \begin{matrix}−N(ℓ) − 1 & T (ℓ) \\ T (ℓ) & −N(ℓ) − 1 \end{matrix} ) ( \begin{matrix}f_{B} \\ εf_{B} \end{matrix} ) = (εa_{ℓ} − N(ℓ) − 1)f̃.

Consequently, the image [f̃] of f̃ in

Φ ⊗ O_{L,𝔭} = Coker (μ_{0} ⊗ id : 𝕊(UY; O_{L,𝔭})^{⊕2} ⟶ 𝕊(UY; O_{L,𝔭})_{0}^{⊕2})

satisfies, by definition of n′,

[f̃] ∈ (Φ ⊗ O_{L,𝔭})[𝔭^{n′}]^{(λ_{B})}.  (2.5.1.2)

2.5.2. Proposition-Definition. The O_{L}-module

I(f_{B}) := {a ∈ L | af_{B} ∈ 𝕊(UY; O_{L}) + 𝕊(UY; L)_{triv}}

is a fractional O_{L}-ideal containing O_{L}; set

C_{3} = C_{3}(𝔭) := −ord_{𝔭}(I(f_{B})) ≥ 0.

The element [f̃] ∈ (Φ ⊗ O_{L,𝔭})[𝔭^{n′}] satisfies 𝔭^{n′−C_{3}−1}[f̃] ≠ 0 ∈ Φ ⊗ O_{L,𝔭}. Proof. I(f_{B}) is a fractional O_{L}-ideal, since f_{B} ∉ 𝕊(UY; L)_{triv}, which means that C_{3} is defined. We must show that any relation

c ( \begin{matrix}f_{B} \\ εf_{B} \end{matrix} ) = ( \begin{matrix}−N(ℓ) − 1 & T (ℓ) \\ T (ℓ) & −N(ℓ) − 1 \end{matrix} ) ( \begin{matrix}x \\ y \end{matrix} ) = ( \begin{matrix}−(N(ℓ) + 1)x + T (ℓ)y \\ T (ℓ)x − (N(ℓ) + 1)y \end{matrix} )

with c ∈ O_{L,𝔭} and x, y ∈ 𝕊(UY; O_{L,𝔭}) implies that c ∈ 𝔭^{n′−C_{3}}O_{L,𝔭}. As in 1.1.7, it follows from the relation

(T (ℓ) + ε(N(ℓ) + 1))(y − εx) = 0

that y − εx ∈ 𝕊(UY; O_{L,𝔭})_{triv}. Decomposing x = x_{0} + x_{triv} (x_{0} ∈ 𝕊(UY; L_{𝔭})_{0}, x_{triv} ∈ 𝕊(UY; L_{𝔭})_{triv}), we deduce from

2εcf_{B} = (T (ℓ) − ε(N(ℓ) + 1))(x + εy) = 2(T (ℓ) − ε(N(ℓ) + 1))x − 2(N(ℓ) + 1)(y − εx)

that

(T (ℓ) − ε(N(ℓ) + 1))x_{0} = εcf_{B},

hence

x_{0} = c′f_{B}, c′ = εc/(a_{ℓ} − ε(N(ℓ) + 1))

(since T (ℓ)−ε(N(ℓ)+1) is invertible on 𝕊(UY; L_{𝔭})_{0} and T (ℓ)f_{B} = a_{ℓ}f_{B}). By definition of I(f_{B}), the relation

y = ((y − εx) + εx_{triv}) + εc′f_{B}

implies that c′ ∈ I(f_{B})O_{L,𝔭}, hence ord_{𝔭}(c) = ord_{𝔭}(a_{ℓ} − ε(N(ℓ) + 1)) + ord_{𝔭}(c′) ≥ n′ − C_{3}, as claimed. 2.5.3. Set

T_{𝔭}(J ⊗ O_{L}) := T_{p}(J) ⊗_{Z_{p}} O_{L,𝔭},  (J ⊗ O_{L})[𝔭^{m}] := T_{p}(J) ⊗_{Z_{p}} O_{L}/𝔭^{m}O_{L}  (m ≥ 0).

These are 𝕋^{S∪{ℓ}}(U(ℓ)Y) ⊗ O_{L,𝔭}[G_{F}]-modules (see 1.6.7) and there are exact sequences of 𝕋^{S∪{ℓ}}(U(ℓ)Y) ⊗ O_{L,𝔭}[G_{F (ℓ^{2})}]-modules (see 1.7.1)

\begin{matrix}0 ⟶ X^{∨} ⊗ O_{L,𝔭}(1) ⟶ & T_{𝔭}(J ⊗ O_{L}) ⟶ ^{h}X ⊗ O_{L,𝔭} ⟶ 0 \\ 0 ⟶ X^{∨} ⊗ O_{L}/𝔭^{m}O_{L}(1) ⟶ & (J ⊗ O_{L})[𝔭^{m}] ⟶ ^{h}X ⊗ O_{L}/𝔭^{m}O_{L} ⟶ 0. \end{matrix}  (2.5.3.1)

Dualising the maps

\begin{aligned}[·f̃] : & O_{L}/𝔭^{n′}O_{L} & ⟶ & (Φ ⊗ O_{L,𝔭})[𝔭^{n′}]^{(λ_{B}⊗id)} & ↪ (Φ ⊗ O_{L,𝔭})[𝔭^{n′}] ↪ ^{h}X ⊗ O_{L}/𝔭^{n′}O_{L} \\ & 1 & ↦ & [f̃] \end{aligned}

(using the pairings from 1.6.10 and fixing a generator of 𝔭^{n′}O_{L,𝔭}) we obtain a morphism of 𝕋^{S∪{ℓ}}(U(ℓ)Y) ⊗ O_{L,𝔭}[G_{F (ℓ^{2})}]-modules

\begin{aligned}[·f̃]^{∨} : X^{∨} ⊗ O_{L}/𝔭^{n′}O_{L} ↠ Φ ⊗ O_{L}/𝔭^{n′}O_{L} ↠ & (Φ ⊗ O_{L}/𝔭^{n′}O_{L})_{(^{h}λ_{B}⊗id)} & ⟶ & O_{L}/𝔭^{n′}O_{L} & ↠ O_{L}/𝔭^{n}O_{L} \\ & a & ↦ & ([f̃], a)_{𝔭^{n′}} \end{aligned}

(2.5.3.2)

which factors through (X^{∨} ⊗ O_{L}/𝔭^{n}O_{L})_{(^{h}λ_{B}⊗id)} and whose cokernel is killed by 𝔭^{C_{3}}, by Proposition 2.5.2 and non-degeneracy of the pairing Φ × Φ ⟶ Q/Z. Above,

λ_{B} ⊗ id : 𝕋^{S∪{ℓ}}(U(ℓ)Y) ⊗ O_{L,𝔭} ⟶ O_{L}/𝔭^{n′}O_{L}

is obtained from λ_{B} by extension of scalars.

After tensoring (2.5.3.1) over 𝕋^{S∪{ℓ}}(U(ℓ)Y) ⊗ O_{L,𝔭} via the morphism

hλ_{B} ⊗ id : 𝕋^{S∪{ℓ}}(U(ℓ)Y) ⊗ O_{L,𝔭} ⟶ O_{L}/𝔭^{n}O_{L},

we obtain an exact sequence of O_{L}/𝔭^{n}O_{L}[G_{F (ℓ^{2})}]-modules

Tor_{1}^{𝕋^{S∪{ℓ}}(U(ℓ)Y)⊗O_{L,𝔭}}(^{h}X ⊗ O_{L}/𝔭^{n}O_{L}, O_{L}/𝔭^{n}O_{L}) ⟶^{∂} (X^{∨} ⊗ O_{L}/𝔭^{n}O_{L})_{(^{h}λ_{B}⊗id)}(1) ⟶ (J ⊗ O_{L})[𝔭^{n}]_{(^{h}λ_{B}⊗id)}

⟶ ^{h}((X ⊗ O_{L}/𝔭^{n}O_{L})_{(λ_{B}⊗id)}) ⟶ 0.

The Frobenius element Fr_{geom}(ℓ^{2}) acts on the domain (resp. on the target) of ∂ trivially (resp. by multiplication by N(ℓ)^{2}), which implies that

𝔭^{C_{2}} Im(∂) = (N(ℓ)^{2} − 1) Im(∂) = 0.

Set

N := (J ⊗ O_{L})[𝔭^{n}]_{(^{h}λ_{B}⊗id)} ⊃ N_{0} := Coker(∂);

the previous discussion implies that the Tate twist of the map (N(ℓ)^{2} − 1)[·f̃]^{∨} factors through a morphism of O_{L}/𝔭^{n}O_{L}[G_{F (ℓ^{2})}]-modules

j_{0} : N_{0} ⟶ O_{L}/𝔭^{n}O_{L}(1), j_{0}(x) = (N(ℓ)^{2} − 1)([f̃], x)_{𝔭^{n′}} (mod 𝔭^{n}).

As 𝔭^{C_{3}}Coker([·f̃]^{∨}) = 0, we have 𝔭^{C_{2}+C_{3}}Coker(j_{0}) = 0. To sum up, we have proved the following statement.

2.5.4. Proposition. There exists a diagram of morphisms of O_{L}/𝔭^{n}O_{L}[G_{F (ℓ^{2})}]-modules

0 ⟶  N_{0}  ⟶ N(= (J ⊗ O_{L})[𝔭^{n}]_{(^{h}λ_{B}⊗id)}) ⟶ ^{h}((X ⊗ O_{L}/𝔭^{n}O_{L})_{(λ_{B}⊗id)}) ⟶ 0
↓^{j_{0}}
O_{L}/𝔭^{n}O_{L}(1)

whose row is exact and in which 𝔭^{C_{2}+C_{3}} Coker(j_{0}) = 0.

2.5.5. Definition. Denote by C_{4} = C_{4}(𝔭) the smallest integer C_{4} ≥ 0 satisfying

Im (O_{L,𝔭}[G_{F}] ⟶ End_{O_{L,𝔭}}(T)) ⊇ 𝔭^{C_{4}} End_{O_{L,𝔭}}(T)

(C_{4} exists, since V is an absolutely irreducible representation of G_{F} [T2, Prop. 3.1]; see Corollary 2.4.4(1) above).

2.5.6. The following objects satisfy the assumptions of Proposition 2.4.3:

A = O_{L}/𝔭^{n}O_{L},  G = G_{F} ,  M = T/𝔭^{n}T,  I = 𝔭^{C_{4}}A.

As in 2.4.2, let R = A[G]/J.

The construction of ℓ implies that ω(ℓ) = 1 and that h = Fr_{geom}(ℓ^{2}) acts on M in the basis e_{1}, e_{2} from 2.3.1 by

ρ(h)e_{1} = e_{1}, ρ(h)e_{2} = u_{g}^{2}e_{2} = N(ℓ)^{−2}e_{2}.

Furthermore, the Eichler-Shimura relation (1.6.8.4) (together with the Čebotarev density theorem) implies that N is an R-module and that

j_{0} : N_{0} ⟶ O_{L}/𝔭^{n}O_{L}(1) = (O_{L}/𝔭^{n}O_{L})e_{2} ⊂ M

is a morphism of A[h]-modules.

2.5.7. Proposition. There exists a morphism of O_{L}/𝔭^{n}O_{L}[G_{F}]-modules

j : N = (J ⊗ O_{L})[𝔭^{n}]_{(^{h}λ_{B}⊗id)} ⟶ T/𝔭^{n}T = M

whose restriction to N_{0} is equal to

(N(ℓ)^{−2} − 1)^{2}b^{12}j_{0},

where b is a fixed generator of 𝔭^{C_{4}}O_{L,𝔭}. Furthermore, 𝔭^{3C_{2}+C_{3}+13C_{4}} Coker(j) = 0.

Proof. The existence of j follows from Proposition 2.4.5, which applies with a = N(ℓ)^{−2}, thanks to the discussion in 2.5.6. By construction, j(N_{0}) contains 𝔭^{3C_{2}+C_{3}+12C_{4}}e_{2}, which implies that j(N) contains 𝔭^{3C_{2}+C_{3}+13C_{4}}e_{1}, by definition of C_{4}. 2.5.8. The proof of Proposition 2.5.7 used only the existence of an element g_{𝔭} ∈ G_{F} satisfying (A1) and (A2). Neither the property (A3) nor the assumption L(f_{K}, χ, 1) ≠ 0 were used in the proof.

2.6. The cohomology class c(ℓ)

The assumptions and notation from 2.5 are in force. 2.6.1. The CM point x(ℓ) := [z, 1]_{UY} ∈ CM(M, K)_{ℓ−ur} is defined over the field K_{UY} ⊂ F_{ℓ}^{ur} which is abelian over K (⊂ K_{ℓ} ⊂ F_{ℓ}^{ur}) and which satisfies

rec_{K} : Gal(K_{UY} /K) ⟶^{∼} K̂^{×}/K^{×} t̂^{−1}(U)Y = K̂^{×}/K^{×}K_{ℓ}^{×}(t̂^{(ℓ)})^{−1}(U^{(ℓ)})Y^{(ℓ)} = K̂^{(ℓ)×}/K^{×}(t̂^{(ℓ)})^{−1}(U^{(ℓ)})Y^{(ℓ)}

(this isomorphism does not depend on the choice of an embedding K_{UY} ↪ K^{ab}). Note that the field K_{UY} (more precisely, its image in K^{ab}) does not depend on ℓ and contains K_{χ}, by (2.2.4.1). 2.6.2. The divisor

D := e\overline{χ} (Tr_{K_{UY} /K_{χ}}(x(ℓ))) ∈ Div(M ⊗_{F} K_{χ}) ⊗ O_{L},

where

e\overline{χ} := ∑_{σ∈Gal(K_{χ}/K)} χ(σ) σ ∈ O_{L}[Gal(K_{χ}/K)],

is equal to

D = ∑_{a∈_{K^{(ℓ)×}}/K^{×}(_{t^{(ℓ)}})^{−1}(U^{(ℓ)})Y^{(ℓ)}} χ(a) [z, t̂^{(ℓ)}(a)]_{UY} ,  (2.6.2.1)

by 1.8.3.

According to Proposition 1.5.9(1), the divisor D′ = (N(v_{0}) + 1 − T (v_{0}))D has degree zero on each connected component of M ⊗_{F} \overline{F}; its class defines an element

cl(D′) ∈ (J(K_{χ}) ⊗ O_{L})^{(χ^{−1})}.

2.6.3. The Kummer map

J(K_{χ}) ⊗ Z_{p} = \underleftarrow{lim}_{m} J(K_{χ}) ⊗ Z/p^{m}Z ↪ \underleftarrow{lim}_{m} H^{1}(K_{χ}, J[p^{m}]) = H^{1}(K_{χ}, T_{p}(J))

gives rise to a map

∂ : J(K_{χ}) ⊗ O_{L,𝔭} ↪ H^{1}(K_{χ}, T_{p}(J) ⊗_{Z_{p}} O_{L,𝔭}) ⟶ H^{1}(K_{χ}, (J ⊗ O_{L})[𝔭^{n}]).

Set

c′(ℓ) := j_{∗}(∂(cl(D′))) ∈ H^{1}(K_{χ}, T/𝔭^{n}T)^{(χ^{−1})},

where

j_{∗} : H^{1}(K_{χ}, (J ⊗ O_{L})[𝔭^{n}]) ⟶ H^{1}(K_{χ}, T/𝔭^{n}T)

is induced by the map j from Proposition 2.5.7.

2.6.4. Definition. Denote by c(ℓ) the image of c′(ℓ) in H^{1}(K_{χ}, T/𝔭^{n−C_{0}}T)^{(χ^{−1})}, where C_{0} is the constant defined in Definition 2.7.9 below.

2.7. Localisation of c(ℓ) outside ℓ

2.7.1. Let E/F be a finite extension. For each finite prime w of E denote by

H_{f}^{1}(E_{w}, V) = Ker (H^{1}(E_{w}, V) ⟶ \{ \begin{matrix}H^{1}(E_{w}, V ⊗_{Q_{p}} B_{cris}), & w | p \\ H^{1}(I_{E_{w}}, V), & w ∤ p \end{matrix})
H_{f}^{1}(E_{w}, T) = Ker (H^{1}(E_{w}, T) ⟶ H^{1}(E_{w}, V)/H_{f}^{1}(E_{w}, V))
H_{f}^{1}(E_{w}, V/T) = Ker (H^{1}(E_{w}, V/T) ⟶ Coker(H_{f}^{1}(E_{w}, V) ⟶ H^{1}(E_{w}, V/T)))
H_{f}^{1}(E_{w}, T/𝔭^{m}T) = Im (H_{f}^{1}(E_{w}, T) ⟶ H^{1}(E_{w}, T/𝔭^{m}T))

the Bloch-Kato subspaces [BK] of local Galois cohomology and by

H_{f}^{1}(E, −) = Ker(H^{1}(E, −) ⟶ ∏_{w∤∞} H^{1}(E_{w}, −)/H_{f}^{1}(E_{w}, −)) (− = V, T, V/T, T/𝔭^{m}T)

the corresponding Bloch-Kato Selmer groups (and similarly for T^{∗}(1) ⟶^{∼} T ⊗ ω^{−1}). Note that

H_{f}^{1}(E, T) = \underleftarrow{lim}_{m} H_{f}^{1}(E, T/𝔭^{m}T), H_{f}^{1}(E, V/T) = \underrightarrow{lim}_{m} H_{f}^{1}(E, T/𝔭^{m}T).

For each finite extension E′_{w′}/E_{w} the restriction (resp. corestriction) map res : H^{1}(E_{w}, −) ⟶ H^{1}(E′_{w′}, −) (resp. cor : H^{1}(E′_{w′}, −) ⟶ H^{1}(E_{w}, −)) maps the H_{f}^{1}-subspace into the H_{f}^{1}-subspace. The formula cor ∘ res = [E′_{w′} : E_{w}] implies that

[E′_{w′} : E_{w}] · res^{−1} (H_{f}^{1}(E′_{w′}, −)) ⊆ H_{f}^{1}(E_{w}, −)  (− = V, T, V/T, T/𝔭^{m}T).  (2.7.1.1)

2.7.2. The goal of §2.7 is to define integers C_{0}, C_{5}, C_{6} ≥ 0 depending on f and 𝔭 (but not on K, χ or ℓ) and prove Proposition 2.7.12 below.

2.7.3. Proposition. For each finite prime v ∈ S ∪ S_{p} there exists a finite Galois extension F′_{v}/F_{v} which depends on U and ω but not on ℓ such that J(M_{U′Y}) ⊗_{F} F′_{v} has split semi-abelian reduction. In particular, the discussion in Appendix A.3 applies to J(M_{U′Y}) ⊗_{F} F′_{v} and K = F′_{v}.

Proof. If v ∈ S_{p} − S, then M_{U′Y} and its Jacobian have good reduction at v, so we can – and will – take F′_{v} = F_{v}.

For each rational prime q,

H_{et}^{1}(J(M_{U′Y}) ⊗_{F} \overline{F} , \overline{Q}_{q}) = ⨁_{π′}(π′)^{U′Y} ⊗ V_{q}(π′),

where π′ runs through irreducible (cuspidal) representations of B_{A}^{′×} such that π′_{∞} = σ_{2} ([N1, Prop. 1.18(ii)] and 1.6.9).

For each v ∈ S fix an open compact subgroup U_{v} ⊂ B_{v}^{′×} such that U_{S} ⊃ ∏_{v∈S} U_{v}. If π′ as above satisfies (π′)^{U′Y} ≠ 0, then its central character ω_{π′} is equal to a power of ω (since Y = Ker(ω) ⊂ Ker(ω_{π′})) and (π′_{v})^{U_{v}} ≠ 0 for each v ∉ S_{∞} ∪ {ℓ}.

It is known ([C1, Thm. A], [T1, Thm. 2]) that, for each v ∤ q, V_{q}(π′)|_{G_{Fv}} corresponds to JL(π′_{v}) by the local Langlands correspondence. Above, JL(π′_{v}) denotes π′_{v} if v ∉ Ram(B′) (resp. the smooth representation of GL_{2}(π′_{v}) attached to π′_{v} by the Jacquet-Langlands correspondence if v ∈ Ram(B′)).

In concrete terms, for v ∈ S and q diferent from the residue characteristic of v, the condition (π′_{v})^{U_{v}} ≠ 0 implies the following.

(1) If JL(π′_{v}) = π(μ_{1}, μ_{2}) (μ_{i} : F_{v}^{×} ⟶ C^{×}) is a principal series representation, then the inertia group I_{F_{v}} acts on V_{q}(π′) – through a finite abelian quotient – by (μ_{1}⊕μ_{2})|_{O_{F,v}^{×}}. The conductor exponent o(μ_{1})+o(μ_{2}) = o(JL(π′_{v})) is bounded above by a constant depending on U_{v}, which implies that there is a finite abelian extension F_{v,1} of F_{v} depending only on U_{v} whose inertia group acts trivially on V_{q}(π′).

(2) If JL(π′_{v}) is supercuspidal, then the argument in (a) applies to the base change of JL(π′_{v}) to a suitable extension of F_{v} of degree 2 (resp. degree dividing 12) if v ∤ 2 (resp. if v | 2). We obtain, again, a finite Galois extension F_{v,2} of F_{v} depending only on U_{v} whose inertia group acts trivially on V_{q}(π′).

(3) If JL(π′_{v}) = St ⊗ μ (μ : F_{v}^{×} ⟶ C^{×}), then μ^{2} = (ω_{π′})_{v} is a power of ω_{v}, which implies that that the order of μ is bounded above by a constant depending on ω_{v}, hence there is a finite abelian extension F_{v,3} of F_{v} depending only on ω_{v} containing \overline{F} _{v}^{Ker(μ)}. By Prop. 2.7.8(2) below, the inertia group of F_{v,3} acts unipotently on V_{q}(π′) and its absolute Galois group acts trivially on the corresponding graded quotients of V_{q}(π′).

For each v ∈ S, let F′_{v} be any finite Galois extension of F_{v} containing the fields F_{v,1}, F_{v,2} and F_{v,3}. By construction, if π′_{∞} = σ_{2} and (π′)^{U′Y} ≠ 0, then the base change of JL(π′_{v}) to F′_{v} is an unramified principal series representation in the cases (1) and (2) (resp. the Steinberg representation St in the case (3)). The above description of the action of G_{F′_{v}} on V_{q}(π′) implies that J(M_{U′Y})⊗_{F} F′_{v} has split semi-abelian reduction, as required. 2.7.4. We can – and will – enlarge each F′_{v} so that the base change of π(f)_{v} to F′_{v} be either an unramified principal series representation or the Steinberg representation of GL_{2}(F′_{v}).

2.7.5. Definition. For each finite prime v ∈ S ∪ S_{p} fix F′_{v} as in 2.7.3-4 and define C_{5,v} = C_{5,v}(𝔭) := ord_{𝔭}([F′_{v} : F_{v}]) ≥ 0, C_{5} = C_{5}(𝔭) := max_{v} C_{5,v}.

2.7.6. Proposition. If v ∈ S_{p}, v ∤ 𝔫 (and if there exists g_{𝔭} ∈ G_{F} satisfying (A1)–(A2)), then there is a Barsotti-Tate group H over O_{F,v} equipped with an action of O_{L,𝔭} such that T_{p}(H) = T |_{G_{Fv}}.

Proof. This is known in general for an arbitrary Hilbert modular form of parallel weight two (even without the assumption about the existence of g_{𝔭}):

• if f has CM, since T ⊂ T_{p}(A), where A is an abelian variety with CM with good reduction at v;

• if T/𝔭T is an irreducible G_{F} -module [T2, Thm. 1.6];

• if p ≠ 2 (V |_{G_{Fv}} is crystalline with Hodge-Tate weights contained in {0, 1} [Li3, Thm. 4.2.1], [Sk, Thm. 1]; the existence of H follows from [Br, Thm 1.4]);

• if p is arbitrary (replace the reference to [Br, Thm 1.4] by [Ki, Cor. 2.2.6]).

However, it may be of interest to note that in our case there is an argument along the lines of [T2, Thm. 1.6]: fix, for each n >> 0, a prime ℓ of F as in 2.3.2. The assumption v ∤ 𝔫 implies that J(M_{U′Y}) has good reduction at v, hence the G_{F_{v}}-module N in Proposition 2.4.6 comes from a finite flat group scheme over O_{F,v} (equipped with an O_{L}/𝔭^{n}O_{L}-action). This implies that T/𝔭^{n−3C_{2}−C_{3}−13C_{4}}T |_{G_{Fv}}, which is a subquotient of N, also comes from such a finite flat group scheme. The existence of H then follows from [Ra 2, Prop. 2.3.1].

2.7.7. Corollary. If v ∈ S_{p} and π(f)_{v} ≠ St ⊗ μ (and if there exists g_{𝔭} ∈ G_{F} satisfying (A1)–(A2)), then there is a Barsotti-Tate group H with an action of O_{L,𝔭} over O_{F′_{v}} such that T_{p}(H) = T |_{G_{F′_{v}}}.

Proof. Again, this is known if p ≠ 2 even without assuming the existence of g_{𝔭}. Under our assumptions there exists a totally real solvable extension F′/F and a prime v′ | v of F′ such that F′_{v′}/F_{v} is isomorphic to the extension F′_{v}/F_{v} from 2.7.3-4. If f has no CM, B.5.5(2) tells us that the existence of g_{𝔭} implies that there is g′_{𝔭} ∈ G_{F′} satisfying (A1)–(A2). The result follows by applying Proposition 2.7.6 to the base change of f to F′. If f has CM, then the base change of the corresponding abelian variety with CM to F′ has good reduction at v′, so we can conclude as in 2.7.6.

2.7.8. Proposition. (1) Let E and w be as in 2.7.1. If w ∤ p, then H^{i}(E_{w}, V) = 0 fir each i ≥ 0, H_{f}^{1}(E_{w}, T) = H^{1}(E_{w}, T) is finite and H_{f}^{1}(E_{w}, V/T) = 0.

(2) If π(f)_{v} = St ⊗ μ (μ : F_{v}^{×} ⟶ C^{×}, μ^{2} = ω_{v}), then V_{v} := V |_{G_{Fv}} sits in an exact sequence of L_{𝔭}[G_{F_{v}}]- modules

0 ⟶ V_{v}^{+} ⟶ V_{v} ⟶ V_{v}^{−} ⟶ 0,

where V_{v}^{+} = L_{𝔭}(1) ⊗ μ and V_{v}^{−} = L_{𝔭} ⊗ μ. Set T_{v}^{+} := T ∩ V_{v}^{+} and T_{v}^{−} := T/T_{v}^{+}. The extension class [T_{v}] of the exact sequence 0 ⟶ T_{v}^{+} ⟶ T_{v} ⟶ T_{v}^{−} ⟶ 0 is an element of H^{1}(F_{v}, O_{L,𝔭}(1)) ⟶^{∼} F_{v}^{×} ˆ⊗ O_{L,𝔭} whose image under ord_{v} ⊗ id : F_{v}^{×} ˆ⊗ O_{L,𝔭} ⟶ O_{L,𝔭} is non-zero.

(3) If π(f)_{v} = St ⊗ μ and v ∈ S_{p}, then

H_{f}^{1}(E_{w}, V) = Ker (H^{1}(E_{w}, V) ⟶ H^{1}(E_{w}, V_{v}^{−})) ,

for each finite extension E_{w}/F_{v}.

Proof. This is well-known (see, for example, [N2, (12.4.4.2)], [N2, Lemma 12.5.4(ii)] and [N3, Prop. 3.3.2]).

2.7.9. Definition. Let v be as in 2.7.3.

(1) If v ∉ S_{p}, set C_{0,v} := 0.

(2) If v ∈ S_{p} and π(f)_{v} = St ⊗ μ, set C_{0,v} := e(𝔭) · a, where a = a(O_{F′_{v}}, μ_{p^{∞}}) is the integer from A.1.7.

(3) If v ∈ S_{p} and π(f)_{v} ≠ St ⊗ μ, set C_{0,v} := e(𝔭) · a, where a = a(O_{F′_{v}}, H) is the integer from A.1.7 for the Barsotti-Tate group H from Corollary 2.7.7. Finally, set C_{0} = C_{0}(𝔭) := max_{v|p} C_{0,v}.

2.7.10. Proposition-Definition. Let v be as in 2.7.3.

(1) If v ∤ p𝔫, set C_{6,v} := 0.

(2) If v | p𝔫 and π(f)_{v} = St ⊗ μ, let C_{6,v} ≥ 0 be the biggest non-negative integer such that

e(F′_{v}/F_{v})(ord_{v} ⊗ id)[T_{v}] ∈ 𝔭^{C_{6,v}}O_{L,𝔭}.

(3) If v | p𝔫 and π(f)_{v} ≠ St ⊗ μ, then H^{0}(F′_{v}, V (−1)) = 0. We let C_{6,v} ≥ 0 be the smallest non-negative integer such that

𝔭^{C_{6,v}}H^{0}(F′_{v}, V/T (−1)) = 0.

Set C_{6} = C_{6}(𝔭) := max_{v|p} C_{6,v}.

Proof. The existence of C_{6,v} in the case (2) follows from Proposition 2.7.8(2). The existence of C_{6,v} in the case (3) follows from the vanishing of H^{0}(F′_{v}, V (−1)), which is, in turn, a consequence of the fact that V (−1)|_{G_{F′_{v}}} is an unramified (resp. a crystalline) representation of G_{F′_{v}} if v ∤ p (resp. if v | p) which is pure of weight 1, by the compatibility of V_{𝔭}(f) with the local Langlands correspondence [T1, Thm. 2] (resp. [Li3, Thm. 4.2.1] and [Sk, Thm. 1]) and the generalised Ramanujan conjecture for f [Bl].

2.7.11. Proposition. Assume that π(f)_{v} = St ⊗ μ.

(1) If v ∤ p, then

∀m ≥ 0 𝔭^{C_{6,v}} H^{0}(I_{F′_{v}}, T/𝔭^{m}T) ⊆ T_{v}^{+}/𝔭^{m}T_{v}^{+}.

(2) If v | p, assume that H ∈ Gr_{O_{F′_{v}}} in the notation of Appendix A.1.1 and that j′ : H(\overline{F}_{v}) ⟶ T/𝔭^{m}T |_{G_{F′_{v}}} is a morphism of O_{L,𝔭}[G_{F′_{v}}]-modules for which Im(j′)∩T_{v}^{+}/𝔭^{m}T_{v}^{+} has a sufficiently large exponent (= bigger than a suitable constant depending on T_{v} and F′_{v}). Then

𝔭^{C_{0,v}+C_{6,v}} Im(j′) ⊆ T_{v}^{+}/𝔭^{m}T_{v}^{+}.

Proof. (1) This follows from the exact cohomology sequence of

0 ⟶ T_{v}^{+}/𝔭^{m}T_{v}^{+} ⟶ T/𝔭^{m}T ⟶ T_{v}^{−}/𝔭^{m}T_{v}^{−} ⟶ 0

over the maximal unramified extension of F′_{v}, which reads as follows:

0 ⟶ T_{v}^{+}/𝔭^{m}T_{v}^{+} ⟶ H^{0}(I_{F′_{v}}, T/𝔭^{m}T) ⟶ O_{L}/𝔭^{m}O_{L} ⟶^{δ} H^{1}(I_{F′_{v}}, O_{L}/𝔭^{m}O_{L}(1)) = O_{L}/𝔭^{m}O_{L},

where δ(1) is equal to the image of e(F′_{v}/F_{v})(ord_{v} ⊗ id)[T_{v}] ∈ O_{L,𝔭} − {0} in O_{L}/𝔭^{m}O_{L}.

(2) Let H′ ∈ Gr_{O_{F′_{v}}} be the quotient of H by the scheme-theoretical closure of Ker(j′). The induced map j′ : H′(\overline{F}_{v}) ↪ T/𝔭^{m}T |_{G_{F′_{v}}} is injective and the statement (2) follows from A.2.9 (since 𝔭^{C_{0,v}}O_{L,𝔭} = p^{a}O_{L,𝔭}).

2.7.12. Proposition. Let v be as in 2.7.3, let E_{w}/F_{v} be a finite extension. Denote by δ_{E_{w}} the map

δ_{E_{w}} : J(E_{w}) ⊗ O_{L}/𝔭^{n}O_{L} ⟶^{∂} H^{1}(E_{w}, (J ⊗ O_{L})[𝔭^{n}]) ⟶^{j_{∗}} H^{1}(E_{w}, T/𝔭^{n}T) ⟶^{(j_{C_{0,v}})_{∗}} H^{1}(E_{w}, T/𝔭^{n−C_{0,v}}T).

(1) If E_{w} ⊇ F′_{v}, then

𝔭^{C_{6,v}} Im(δ_{E_{w}}) ⊆ H_{f}^{1}(E_{w}, T/𝔭^{n−C_{0,v}}T).

(2) In general,

\begin{aligned}[E_{w}F′_{v} : E_{w}] 𝔭^{C_{6,v}} Im(δ_{E_{w}}) & ⊆ H_{f}^{1}(E_{w}, T/𝔭^{n−C_{0,v}}T), \\ 𝔭^{C_{5,v}+C_{6,v}} Im(δ_{E_{w}}) & ⊆ H_{f}^{1}(E_{w}, T/𝔭^{n−C_{0,v}}T). \end{aligned}

Proof. Note that (2) follows from (1) for E_{w}F′_{v} instead of E_{w} and (2.7.1.1) for E_{w}F′_{v}/F′_{v}. We can assume, therefore, that E_{w} ⊇ F′_{v}. We distinguish several cases.

(a) If v ∤ p𝔫 (C_{0,v} = C_{5,v} = C_{6,v} = 0), then w ∤ p and J has good reduction over E_{w}, which implies that Im(∂) = H_{ur}^{1}(E_{w}, (J ⊗ O_{L})[𝔭^{n}]), hence Im(δ_{E_{w}}) ⊆ H_{ur}^{1}(E_{w}, T/𝔭^{n}T) = H_{f}^{1}(E_{w}, T/𝔭^{n}T) (the last equality is a consequence of the fact that T is an unramified G_{E_{w}}-module).

(b) If v ∤ p and π(f)_{v} ≠ St ⊗ μ (C_{0,v} = 0), then T is again an unramified G_{E_{w}}-module, so we must show that Im(δ_{E_{w}}) is contained in H_{ur}^{1}(E_{w}, T/𝔭^{n}T). According to A.3.6 (applied to J(M_{U′Y}) ⊗_{F} F′_{v}, K = F′_{v} and K′ = E_{w}) combined with A.2.3(1),

Im(δ_{E_{w}}) ⊆ X_{1} ⊕ X_{2}, X_{1} = Im (H_{ur}^{1}(E_{w}, (𝒢^{∘} ⊗ O_{L})[𝔭^{n}](\overline{F}_{v})) ⟶^{j_{∗}} H^{1}(E_{w}, T/𝔭^{n}T)) ,

X_{2} = Im (H^{1}(E_{w}, (𝒯 ⊗ O_{L})[𝔭^{n}](\overline{F}_{v})) = H^{1}(E_{w}, (O_{L}/𝔭^{n}O_{L})(1))^{rk(𝒯 )} ⟶^{j_{∗}} H^{1}(E_{w}, T/𝔭^{n}T)) .

The statement follows from the fact that j_{∗} maps unramified cohomology into unramified cohomology (hence X_{1} ⊆ H_{ur}^{1}(E_{w}, T/𝔭^{n}T)) and that 𝔭^{C_{6,v}} j(𝒯 ⊗ O_{L})[𝔭^{n}] = 0 by definition of C_{6,v} (hence 𝔭^{C_{6,v}} X_{2} = 0).

(c) If v | p and π(f)_{v} ≠ St ⊗ μ, then T |_{G_{Ew}} = T_{p}(H) for some Barsotti-Tate group H over O_{E_{w}} equipped with an O_{L,𝔭}-action. It follows from A.2.6 that H_{f}^{1}(E_{w}, T/𝔭^{n}T) = X(H) ⊗_{O_{L}} O_{L}/𝔭^{n}O_{L} = H_{fl}^{1}(O_{E_{w}}, H[𝔭^{n}]). Using A.3.6 again, we have

Im(δ_{E_{w}}) ⊆ Y_{1} ⊕ Y_{2}, Y_{1} = Im (H_{fl}^{1}(O_{E_{w}}, (𝒢^{∘} ⊗ O_{L})[𝔭^{n}]) ⟶^{(j_{C_{0,v}})_{∗}∘j_{∗}} H^{1}(E_{w}, T/𝔭^{n−C_{0,v}}T)) ,

Y_{2} = Im (H^{1}(E_{w}, (𝒯 ⊗ O_{L})[𝔭^{n}](\overline{F}_{v})) = H^{1}(E_{w}, (O_{L}/𝔭^{n}O_{L})(1))^{rk(𝒯 )} ⟶^{(j_{C_{0,v}})_{∗}∘j_{∗}} H^{1}(E_{w}, T/𝔭^{n−C_{0,v}}T)) .

According to A.2.8, Y_{1} ⊆ H_{fl}^{1}(O_{E_{w}}, H[𝔭^{n−C_{0,v}}]), and the same argument as in (b) shows that 𝔭^{C_{6,v}} Y_{2} = 0.

(d) If π(f)_{v} = St ⊗ μ, then A.3.5(1) tells us that

Im(δ_{E_{w}}) ⊆ Z := Im (H^{1}(E_{w}, (𝒢^{∘} ⊗ O_{L})[𝔭^{n}](\overline{F}_{v})) ⟶^{(j_{C_{0,v}})_{∗}∘j_{∗}} H^{1}(E_{w}, T/𝔭^{n−C_{0,v}}T)) .

Proposition 2.7.11 implies that

𝔭^{C_{6,v}} j_{C_{0,v}} ∘ j(𝒢^{∘} ⊗ O_{L})[𝔭^{n}](\overline{F}_{v}) ⊆ (T_{v}^{+}/𝔭^{n−C_{0,v}}T_{v}^{+})|_{G_{Ew}} = O_{L}/𝔭^{n−C_{0,v}}O_{L}(1),

hence 𝔭^{C_{6,v}} Im(δ_{E_{w}}) ⊆ 𝔭^{C_{6,v}}Z is contained (using Proposition 2.7.8(3) for the last inclusion) in

Im (H^{1}(E_{w}, O_{L}/𝔭^{n−C_{0,v}}O_{L}(1)) = H^{1}(E_{w}, O_{L,𝔭}(1)) ⊗_{O_{L}} O_{L}/𝔭^{n−C_{0,v}}O_{L} ⟶ H^{1}(E_{w}, T/𝔭^{n−C_{0,v}}T))
⊆ H_{f}^{1}(E_{w}, T/𝔭^{n−C_{0,v}}T),

Note that, in the case when v | p, the assumption of Proposition 2.7.11(2) is satisfied: as the G_{E_{w}}-module J[p^{m}](\overline{K})/𝒢^{∘}[p^{m}](\overline{K}) is unramified for each m ≥ 0, it follows from Proposition 2.5.7 that the intersection of j((𝒢^{∘} ⊗ O_{L})[𝔭^{n}](\overline{K})) with T_{v}^{+}/𝔭^{n}T_{v}^{+}|_{G_{Ew}} = O_{L}/𝔭^{n}O_{L}(1) contains 𝔭^{C}(O_{L}/𝔭^{n}O_{L})(1), for some constant C.

2.7.13. Corollary. Let K′/K_{χ} be a finite extension and w a finite prime of K′.

(1) If w ∤ ℓ, then

𝔭^{C_{5}+C_{6}} (res_{K′/K_{χ}}(c(ℓ)))_{w} ∈ H_{f}^{1}(K′_{w}, T/𝔭^{n−C_{0}}T).

(2) For each s ∈ H_{f}^{1}(K′, (T/𝔭^{n−C_{0}}T) ⊗ ω^{−1}) the local cup products

∪ : H^{1}(K′_{w}, T/𝔭^{n−C_{0}}T) × H^{1}(K′_{w}, (T/𝔭^{n−C_{0}}T) ⊗ ω^{−1}) ⟶ H^{2}(K′_{w}, O_{L}/𝔭^{n−C_{0}}O_{L}(1)) ⟶^{inv_{w}} O_{L}/𝔭^{n−C_{0}}O_{L}

satisfy

𝔭^{C_{5}+C_{6}} ∑_{w|ℓ} inv_{w} (s_{w} ∪ (res_{K′/K_{χ}}(c(ℓ)))_{w}) = 0 ∈ O_{L}/𝔭^{n−C_{0}}O_{L}.

Proof. (1) This follows from Proposition 2.7.12(2). The statment (2) is a consequence of the reciprocity law

∀x ∈ H^{1}(K′, (T/𝔭^{n−C_{0}}T) ⊗ ω^{−1})  ∑_{w} inv_{w} (x_{w} ∪ (res_{K′/K_{χ}}(c(ℓ)))_{w}) = 0 ∈ O_{L}/𝔭^{n−C_{0}}O_{L}

and the fact that H_{f}^{1}(K′_{w}, T/𝔭^{n−C_{0}}T) ∪ H_{f}^{1}(K′_{w}, (T/𝔭^{n−C_{0}}T) ⊗ ω^{−1}) = 0 for each w.

2.8. Localisation of c(ℓ) at ℓ

2.8.1. Recall that we have denoted by λ the unique prime of K above ℓ. As rec_{K}(K_{ℓ}^{×}) acts trivially on K_{UY} and ω(ℓ) = 1, λ splits completely in F_{ω}K_{UY} /K. As a result, the inclusions K_{χ} ⊂ K_{UY} ⊂ F_{ℓ}^{ur} define a prime λ_{χ} | λ of K_{χ} such that (K_{χ})_{λ_{χ}} = K_{λ} = K_{ℓ}. 2.8.2. By construction of ℓ,

(T/𝔭^{n}T)|_{G_{K_{ℓ}}} = (O_{L}/𝔭^{n}O_{L})e_{1} ⊕ (O_{L}/𝔭^{n}O_{L})(1)e_{2}.

We are interested in the image c′(ℓ)_{λ_{χ},ram} of

c′(ℓ)_{λ_{χ}} ∈ H^{1}((K_{χ})_{λ_{χ}}, T/𝔭^{n}T) = H^{1}(K_{ℓ}, O_{L}/𝔭^{n}O_{L})e_{1} ⊕ H^{1}(K_{ℓ}, O_{L}/𝔭^{n}O_{L}(1))e_{2}

in

\begin{matrix}H^{1}(I_{K_{ℓ}}, T/𝔭^{n}T) = & H^{1}(I_{K_{ℓ}}, O_{L}/𝔭^{n}O_{L})e_{1} ⊕ H^{1}(I_{K_{ℓ}}, O_{L}/𝔭^{n}O_{L}(1))e_{2} \\ ⟶^{∼} & (O_{L}/𝔭^{n}O_{L})(−1)e_{1} ⊕ (O_{L}/𝔭^{n}O_{L})e_{2}. \end{matrix}

2.8.3. Proposition. We have c′(ℓ)_{λ_{χ},ram} = c e_{2}, where c is the image of

(N(v_{0}) + 1 − a_{v_{0}})(N(ℓ)^{2} − 1)(N(ℓ)^{−2} − 1)^{2}b^{12}(K^{×}t̂^{−1}(U)F̂^{×} : K^{×}t̂^{−1}(U)Y)ℒ_{χ}(f_{B}) ∈ O_{L,𝔭} − {0}

in O_{L}/𝔭^{n}O_{L}. Above, ord_{𝔭}(b) = C_{4} and (K^{×}t̂^{−1}(U)F̂^{×} : K^{×}t̂^{−1}(U)Y) divides (F̂^{×} : Y) = [F_{ω} : F].

Proof. We consider cl(D′) as an element of J((K_{χ})_{λ_{χ}}) ⊗ O_{L} = J(K_{ℓ}) ⊗ O_{L}. Using the notation from 1.6.5 and 1.6.6, its image cl(D′)_{Φ} in Φ ⊗ O_{L} is represented by

c̃l(D′)_{Φ} := (N(v_{0}) + 1 − a_{v_{0}}) ( \begin{matrix}∑_{a} χ(a) [t̂^{(ℓ)}(a)] \\ 0 \end{matrix} ) ∈ O_{L}[V(G)]_{0} ⊂ 𝕊(UY; O_{L})^{2},

thanks to (2.6.2.1). Above, a runs through K̂^{(ℓ)×}/K^{×}(t̂^{(ℓ)})^{−1}(U^{(ℓ)})Y^{(ℓ)} and [b] is the image of b ∈ B̂^{×} in B^{×}\B̂^{×}/UY (we embed B̂^{(ℓ)×} ↪ B̂^{×} by b ↦ b × {1}, 1 ∈ B_{ℓ}^{×}).

The construction of the map j_{0} together with the discussion in 1.7.3 (for K = (K_{χ})_{λ_{χ}} = K_{ℓ}) imply that c′(ℓ)_{λ_{χ},ram} = c e_{2}, where c is the image in O_{L}/𝔭^{n}O_{L} of

(N(v_{0}) + 1 − a_{v_{0}})(N(ℓ)^{2} − 1)(N(ℓ)^{−2} − 1)^{2}b^{12} u([f̃], c̃l(D′)_{Φ}) ∈ O_{L,𝔭}.

The statement of the Proposition follows from the fact that

u([f̃], c̃l(D′)_{Φ}) = ∑_{a∈_{K^{×}}/K^{×}_{t^{−1}}(U)Y} χ(a)f_{B}(t̂(a)) = (K^{×}t̂^{−1}(U)F̂^{×} : K^{×}t̂^{−1}(U)Y)ℒ_{χ}(f_{B}).

2.8.4. Corollary. If x = x_{1}e_{1} + x_{2}e_{2} is an element of

H_{ur}^{1}((K_{χ})_{λ_{χ}}, (T/𝔭^{n}T) ⊗ ω^{−1}) = H_{ur}^{1}(K_{ℓ}, T/𝔭^{n}T) ⟶^{∼} (T/𝔭^{n}T)/(Fr_{geom}(ℓ)^{2} − 1) =

= (O_{L}/𝔭^{n}O_{L})e_{1} ⊕ (O_{L,𝔭}/(u_{g}^{2} − 1)O_{L,𝔭})e_{2},

then

inv_{λ_{χ}} (x ∪ c′(ℓ)_{λ_{χ}}) = ±c x_{1} ∈ O_{L}/𝔭^{n}O_{L},

where the sign depends on the choice of the sign of the isomorphism beween the p-primary part of I_{K_{ℓ}} and Z_{p}(1).

Proof. As in 1.7.5, this follows from Proposition 2.8.3 and [Ru, Lemma 1.4.7(ii)] (recall that the dual basis to {e_{1}, e_{2}} with respect to the pairing (0.7.1) is equal to {e_{2}, e_{1}}).

2.9. The annihilation relation and the completion of the proof of Theorem A

The assumptions listed at the beginning of §2 are in force. In addition, if f has CM by a totally imaginary quadratic extension K(f) of F, we assume that K(f) ⊄ F_{ω}K_{χ}. 2.9.1. Let m >> 0 be a large enough integer and consider the field extensions

F ⊂ K ⊂ F_{ω}K ⊂ H := F_{ω}K_{χ} ⊂ H_{n} := F (T/𝔭^{n}T)H

for n = m + C_{0}; set U_{n} := Gal(H_{n}/H) and denote by g_{n} ∈ Gal(H_{n}/F) the restriction of g = g_{𝔭} to H_{n}.

The restriction g_{n}|_{H} is of the form g_{n} = ch_{n}, where c ∈ Gal(K^{ab}/F) is the complex conjugation for a fixed embedding K^{ab} ⊂ C and h_{n} ∈ Gal(H/F_{ω}K). If we consider g_{n}^{2} = ch_{n}ch_{n} = ^{c}h_{n}h_{n} as an element of Gal(H/F_{ω}K) ⊂ Gal(K_{χ}/K), then

χ(g_{n}^{2}) = (^{c}χ · χ)(h_{n}) = ω^{−1} (h_{n}|_{F_{ω}}) = 1;

thus g_{n}^{2} ∈ U_{n}. Denote by

H′_{n} := H_{n}^{⟨g_{n}^{2}⟩} ⊂ H_{n}

the fixed field of the subgroup ⟨g_{n}^{2}⟩ ⊂ U_{n} generated by g_{n}^{2}.

2.9.2. Proposition-Definition. There exists an integer C_{7} = C_{7}(𝔭) ≥ 0 (equal to 0 for all but finitely many 𝔭) such that the restriction maps

res′ : H^{1}(H, (T/𝔭^{m}T) ⊗ ω^{−1}) ⟶ H^{1}(H′_{n}, (T/𝔭^{m}T) ⊗ ω^{−1}),

res : H^{1}(H, (T/𝔭^{m}T) ⊗ ω^{−1}) ⟶ H^{1}(H_{n}, (T/𝔭^{m}T) ⊗ ω^{−1}) = Hom_{U_{n}}(G_{H_{n}}^{ab} ⊗ Z/p^{m}Z, (T/𝔭^{m}T) ⊗ ω^{−1})

satisfy 𝔭^{C_{7}} Ker(res) = 𝔭^{C_{7}} Ker(res′) = 0.

Proof. According to B.5.2 and B.6.5(2), Z := {a ∈ Z_{p}^{×} | a · id ∈ Im(G_{H} ⟶ Aut(T ⊗ ω^{−1}))} is an open subgroup of Z_{p}^{×}. Fix a ∈ Z − {1}; then a · id is an element of the centre of U_{n} which acts on T ⊗ ω^{−1} by multiplication by a. Sah’s Lemma [S, Proof of Prop. 2.7(b)] implies that a − 1 acts trivially on

H^{1}(U_{n}, (T/𝔭^{m}T) ⊗ ω^{−1}) = Ker(res) ⊇ Ker(res′),

which means that we can take C_{7} := ord_{𝔭}(a − 1). The fact that we can choose a so that C_{7} = 0 for all but finitely many 𝔭 follows from B.5.2 and B.6.5(5),(7).

2.9.3. Proposition-Definition. There exists an integer C_{8} = C_{8}(𝔭) ≥ 0 (equal to 0 for all but finitely many 𝔭) such that

Im (O_{L,𝔭}[G_{H}] ⟶ End_{O_{L,𝔭}}(T ⊗ ω^{−1})) ⊇ 𝔭^{C_{8}} End_{O_{L,𝔭}}(T ⊗ ω^{−1}).

Proof. The existence of C_{8} is equivalent to V ⊗ ω^{−1}|_{G_{H}} = V |_{G_{H}} being (absolutely) irreducible, which is, in turn, equivalent to our assumption “if f has CM by K(f), then K(f) ⊄ H”, by a variant of [N1, Prop. 6.2.1]. The fact that C_{8} is equal to 0 for all but finitely many 𝔭 follows from [N1, Prop. 6.2.2].

2.9.4. Proposition. Let s̃ ∈ Z^{1}(G_{H}, (T/𝔭^{m}T) ⊗ ω^{−1}) be a 1-cocycle representing an element s ∈ H_{f}^{1}(H, (T/𝔭^{m}T) ⊗ ω^{−1})^{(χ)}. Then:

(1) ∀h′ ∈ G_{H′_{n}} ⊂ G_{H} such that h′|_{H_{n}} = g_{n}^{2}

𝔭^{C} s̃(h) ∈ (O_{L}/𝔭^{m}O_{L})e_{2} ⊂ (T/𝔭^{m}T) ⊗ ω^{−1},

where C = C_{1} + 3C_{2} + 12C_{4} + C_{5} + C_{6} + ord_{𝔭}((N(v_{0}) + 1 − a_{v_{0}})[F_{ω} : F][H : K]) (recall that e_{1}, e_{2} is the O_{L}/𝔭^{n}O_{L}-basis of T/𝔭^{n}T from 2.3.1).

(2) ∀h′ ∈ G_{H_{n}} 𝔭^{C} (res(s))(h) ∈ (O_{L}/𝔭^{m}O_{L})e_{2} ⊂ (T/𝔭^{m}T) ⊗ ω^{−1}.

Proof. If h ∈ G_{H_{n}} and h′|_{H_{n}} = g_{n}^{2}, then hh′|_{H_{n}} = g_{n}^{2}. As

(res(s))(h) = s̃(h) = s̃(hh′) − s̃(h′)

by the cocycle relation, it is sufficient to prove (1).

Let λ_{n} | ℓ be the prime of H_{n} whose (geometric) Frobenius element with respect to the extension H_{n}/F_{ω} is equal to g_{n}. Denote by λ′_{n} (resp. λ_{H}) the prime of H′_{n} (resp. of H) below λ_{n}. As g_{n}^{2} acts trivially on H′_{n}, the unique prime λ of K above ℓ splits completely in H′_{n}/K.

Corollary 2.7.13(2) for K′ = H says that

𝔭^{C_{5}+C_{6}} ∑_{w|ℓ} inv_{w} (s_{w} ∪ (res_{H/K_{χ}}(c(ℓ)))_{w}) = 0 ∈ O_{L}/𝔭^{m}O_{L},

where w runs through the set of primes of H above ℓ. All terms on the L.H.S. are equal to each other, since s (resp. res_{H/K_{χ}}(c(ℓ))) lies in the χ-eigenspace (resp. χ^{−1}-eigenspace) for the action of Δ = Gal(H/K) ↠ Gal(K_{χ}/K) and the action of Δ on the set {w | ℓ} is transitive. As a result, we obtain the following annihilation relation:

[H : K] 𝔭^{C_{5}+C_{6}} inv_{λ_{H}} (s_{λ_{H}} ∪ (res_{H/K_{χ}}(c(ℓ)))_{λ_{H}} ) = 0 ∈ O_{L}/𝔭^{m}O_{L}.  (⋆)

The localisation

s_{λ_{H}} ∈ H_{ur}^{1}(H_{λ_{H}} , (T/𝔭^{m}T) ⊗ ω^{−1}) = H_{ur}^{1}((H′_{n})_{λ′_{n}}, T/𝔭^{m}T) = (T/𝔭^{m}T)/(g_{n}^{2} − 1) =
= (O_{L}/𝔭^{m}O_{L})e_{1} ⊕ (O_{L}/𝔭^{C_{2}}O_{L})e_{2}

is represented by

˜(g_{n}^{2}) = s̃(h′) = s̃(h′)_{1} e_{1} ⊕ s̃(h′)_{2} e_{2},  s̃(h′)_{i} ∈ O_{L}/𝔭^{m}O_{L}.

Combining (⋆) with Corollary 2.8.4, we obtain

(N(v_{0}) + 1 − a_{v_{0}})[F_{ω} : F][H : K] 𝔭^{C_{1}+3C_{2}+12C_{4}+C_{5}+C_{6}} s(h′)_{1} = 0 ∈ O_{L}/𝔭^{m}O_{L},

as claimed.

2.9.5. Corollary. For each sufficiently large m >> 0 we have 𝔭^{C+C_{7}+C_{8}} H_{f}^{1}(H, (T/𝔭^{m}T) ⊗ ω^{−1})^{(χ)} = 0.

Proof. Proposition 2.9.4(2) tells us that, for each s ∈ H_{f}^{1}(H, (T/𝔭^{m}T) ⊗ ω^{−1})^{(χ)}, the U_{n}-stable subgroup

𝔭^{C} (res(s))(G_{H_{n}} ⊗ Z/p^{m}Z) ⊂ (T/𝔭^{m}T) ⊗ ω^{−1}

is contained in (O_{L}/𝔭^{m}O_{L})e_{2}. By definition of C_{8}, it is killed by 𝔭^{C_{8}}, hence 𝔭^{C+C_{8}} res(s) = 0. As 𝔭^{C_{7}} Ker(res) = 0, the statement follows. 2.9.6. Proof of Theorem A. Corollary 2.9.5 implies that

𝔭^{C+C_{7}+C_{8}} H_{f}^{1}(H, (V/T) ⊗ ω^{−1})^{(χ)} = 0.

The kernel of the restriction map

H_{f}^{1}(K, (V/T) ⊗^{c}χ) ⟶ (H_{f}^{1}(H, (V/T) ⊗ ω^{−1}) ⊗ χ^{−1})^{Gal(H/K)}

is killed by [H : K]; thus

[H : K] 𝔭^{C+C_{7}+C_{8}} H_{f}^{1}(K, (V/T) ⊗^{c}χ) = 0,

which proves that both groups H_{f}^{1}(K, (V/T) ⊗ χ) ⟶^{∼} H_{f}^{1}(K, (V/T) ⊗^{c}χ) (see (0.6.1)) are finite (and equal to zero if 𝔭 does not belong to a certain finite set of primes of F, since the constants C_{i} = C_{i}(𝔭) are equal to zero for all but finitely many 𝔭).

Finally, the conditions (A1)–(A3) do not depend on χ and L_{𝔭}. It is legitimate, therefore, to analyse them by considering the minimal coefficient field L = L_{f} (the field generated over Q by the Hecke eigenvalues λ_{f}(v) (v ∤ 𝔫) of f). The corresponding analysis is carried out in Appendix B.5-6 in a more general context. The Galois representation V = V_{𝔭}(f)(1) corresponds to V_{𝔭}(π) from B.2.1 for π = π(f) ⊗ | · |; thus ∀v ∈ S_{∞} k_{v} = m = 2 in the notation of B.2.1. In particular, B.5.5(6) (resp. B.6.5(4),(6),(7)) implies that an element g_{𝔭} ∈ G_{F} satisfying (A1)–(A3) exists for 𝔭 belonging to a set of density 1 (resp. for 𝔭_{+} = L^{+} ∩ 𝔭 belonging to a set of density 1/2, where L^{+} is the maximal totally real subfield of L) if f does not have (resp. has) complex multiplication. This completes the proof of Theorem A (and of Theorem A’, which is its immediate consequence).

2.10. Proof of Theorem B

In this section we assume that f ∈ S_{2}(𝔫, 1) (hence the corresponding automorphic representation π(f) of GL_{2}(A_{F}) as well) has trivial character. We take L = L_{f}. 2.10.1. For any totally imaginary quadratic extension K/F we have

L(f_{K}, 1_{K}, s) = L(f, s) L(f ⊗ η, s), L(π(f) × 1_{K}, s) = L(π(f), s) L(π(f) ⊗ η, s),

using the notation of 0.5 for the trivial character 1_{K} : A_{K}^{×}/K^{×}A_{F}^{×} ⟶ C^{×} (and writing η = η_{K/F}).

2.10.2. Proposition. The following conditions are equivalent.

(1) π(f) = JL(π′) is associated by the Jacquet-Langlands correspondence to an irreducible (cuspidal) automorphic representation π′ of B_{A}^{′×}/F_{A}^{×}, where B′ is a quaternion algebra over F such that B′ ⊗ R ⟶^{∼} M_{2}(R)×H^{[F:Q]−1} ( ⟺ V_{𝔭}(f) occurs in H_{et}^{1}(M_{H} ⊗_{F} \overline{F} , L_{𝔭}) for a suitable (compactified) Shimura curve M_{H} over F arising from B′ ⟺ there exists a simple quotient A_{f} of the Jacobian J(M_{H}) satisfying (0.10.1));

(2) 2 ∤ [F : Q] or there exists a finite prime v of F such that π(f)_{v} is not a principal series representation;

(3) there exists a totally imaginary quadratic extension K/F such that ε(π(f) × 1_{K}, \frac{1}{2}) = −1;

(4) there exists a finite set Σ of finite primes of F and for each v ∈ Σ a character μ_{v} : F_{v}^{×} ⟶ {±1} with the following property: for each totally imaginary quadratic extension K/F satisfying ∀v ∈ Σ (η_{K/F})_{v} = μ_{v} we have ε(π(f) × 1_{K}, \frac{1}{2}) = −1.

Proof. The equivalence (1) ⟺ (2) is standard. If π(f)_{v} is a principal series representation for each v ∉ S_{∞}, then ε(π(f)_{v} × 1_{K}, \frac{1}{2}) = η_{v}(−1) for such v (Proposition 2.1.1), hence ε(π(f) × 1_{K}, \frac{1}{2}) = ∏_{v|∞} η_{v}(−1) = (−1)^{[F:Q]}, which proves the implication (3) ⟹ (2). It remains to show that (2) ⟹ (4). If 2 ∤ [F : Q], then we can take Σ = {w | 𝔫} and μ_{w} = 1 for each w ∈ Σ. If 2 | [F : Q] and π(f)_{v} is a twist of the Steinberg representation by an unramified character ( ⟺ ord_{v}(𝔫) = 1), then we can take Σ = {w | 𝔫}, μ_{w} = 1 if w ∈ Σ − {v} and μ_{v} = the unramified quadratic character of F_{v}^{×}. More generally, if 2 | [F : Q] and π(f)_{v} is not a principal series representation, let D be the quaternion division algebra over F_{v} and let π′_{v} be the (finite-dimensional) smooth representation of D^{×}/F_{v}^{×} corresponding to π(f)_{v} by the Jacquet-Langlands correspondence. Take any non-zero vector x in the representation space of π′_{v}. The stabiliser of x is open in D^{×}; it contains an element d ∈ D^{×} which does not belong to F_{v}^{×}. The commutative F_{v}- subalgebra E := F_{v}[d] ⊂ D is a field of degree two over F_{v} and, by construction, Hom_{E^{×}}(1_{E}, π′_{v}) ≠ 0. As π′_{v} decomposes under the action of E^{×} into a direct sum of finitely many one-dimensional representations, we also have Hom_{E^{×}}(π′_{v}, 1_{E}) ≠ 0, hence ε(π(f)_{v} × 1_{E}, \frac{1}{2}) = −η_{E/F_{v}}(−1), by Proposition 2.1.1(1). If we take Σ = {w | 𝔫}, μ_{w} = 1 if w ∈ Σ − {v} and μ_{v} = η_{E/F_{v}}, then

ε(π(f) × 1_{K}, \frac{1}{2}) = ∏_{w∉Σ} 1 ( ∏_{w∈Σ−\{v\}} η_{w}(−1)) (−η_{v}(−1)) = − ∏_{w|∞} η_{w}(−1) = −(−1)^{[F:Q]} = −1,

for every totally imaginary quadratic extension K/F satisfying η_{w} = μ_{w} for all w ∈ Σ, where η = η_{K/F}.

2.10.3. Proposition. If K from 2.10.1 satisfies ord_{s=1} L(f_{K}, 1_{K}, s) = 1, then there exists a (compactified) Shimura curve M_{H_{F^{×}}} (for an open compact subgroup H ⊂ B̂^{′×}), a simple quotient A_{f} of the Jacobian J(M_{H_{F^{×}}}) satisfying (0.10.1), a finite subextension K_{0}/K of (K^{ab})^{rec_{K}(A_{F}^{×})}/K and a CM point x = [z′, 1]_{H_{F^{×}}} ∈ M_{H_{F^{×}}}(K_{0}) whose image y ∈ A_{f}(K_{0}) under α : M_{H_{F^{×}}} ⟶ J(M_{H_{F^{×}}}) ⟶ A_{f} (where the first map is given by a suitable integral multiple of the Hodge class [Z1, p. 30], [CV 1, 3.5], [N1, 1.19]) has the following properties:

(1) the point y_{K} := Tr_{K_{0}/K}(y) ∈ A_{f}(K) is not torsion;

(2) the point c(y_{K}) + ε(π(f), \frac{1}{2}) y_{K} ∈ A_{f}(K) is torsion, where c is the non-trivial element of Gal(K/F).

Proof. The proof of Proposition 2.1.4 implies that there exists π′ as in Proposition 2.10.2(1) for the quaternion algebra B′ satisfying ∀v ∤ ∞ ε(π(f)_{v} × 1_{K}, \frac{1}{2}) = inv_{v}(B′) η_{v}(−1).

The generalised Gross-Zagier formula [Z2, Thm. 4.2.1] proved in [YZZ] states that the Néron-Tate distribution from [Z2, 4.2]

NT_{1,π′} ∈ Hom_{K^{×}}(π^{′∞}, C) ⊗ Hom_{K^{×}}(π^{′∞}, C)

is a non-zero multiple of the distribution β_{π′,1} defined in [Z2, 4.2], which implies that, for suitable H and A_{f}, the point y_{K} has non-zero height, hence is non-torsion. The action of c on y_{K} is given by c(P) = Tr_{K_{0}/K}(α([z′, n]_{H_{F^{×}}})), where n ∈ N_{B^{×}}(K^{×}) is an element of the normaliser of K^{×} in B^{×} (n ∉ K^{×}). It follows from Proposition 2.1.2 that n acts on Hom_{K^{×}}(π^{′∞}, C) by multiplication by

∏_{v∤∞} inv_{v}(B′) ε(π(f)_{v}, \frac{1}{2}) = ε(π(f), \frac{1}{2}) ∏_{v|∞} inv_{v}(B′) ε(π(f)_{v}, \frac{1}{2}) = −ε(π(f), \frac{1}{2}),

since ε(π(f)_{v}, \frac{1}{2}) = −1 for each v | ∞. As a result, the point c(y_{K}) + ε(π(f), \frac{1}{2}) y_{K} has trivial height pairing with y_{K}, hence also with c(y_{K}) and itself; it must be torsion, as claimed.

2.10.4. Corollary. If L(f, 1) ≠ 0 and if K/F is a totally imaginary quadratic extension such that f does not have CM by K and ord_{s=1}L(f_{K}, 1_{K}, s) = 1, then the groups A_{f}(F) and � (A_{f}/F) are finite.

Proof. It follows from Proposition 2.10.3(1) and [N1, Thm. 3.2] that the groups A_{f}(K)/O_{L} · y_{K} and � (A_{f}/K) are finite. As c(y_{K}) + y_{K} is torsion by Proposition 2.10.3(2), the group A_{f}(F) = A_{f}(K)^{c=1} is finite, and so is � (A_{f}/F) (since 2 · Ker(� (A_{f}/F) ⟶ � (A_{f}/K)) = 0). 2.10.5. Proof of Theorem B. (a), (b) Combining Proposition 2.10.2(4) with [FH, Thm. B.2] we obtain an extension K/F to which Corollary 2.10.4 applies. The exact sequence

0 ⟶ A_{f}(F) ⊗ L_{𝔭}/O_{L,𝔭} ⟶ H_{f}^{1}(F, V/T) ⟶ � (A_{f}/F)[𝔭^{∞}] ⟶ 0

(where T = T_{𝔭}(A_{f}) ⊂ V = V_{𝔭}(A_{f}) = V_{𝔭}(f)(1)) then implies that H_{f}^{1}(F, V/T) is finite (and equal to zero for all but finitely many 𝔭).

(c) Thanks to (a) we can assume that 2 | [F : Q]. If K/F is a totally imaginary quadratic extension in which all primes dividing 𝔫 split, then 2 | ord_{s=1}L(f_{K}, 1_{K}, s). According to [FH, Thm. B.1], there exists such a K for which K/F is ramified at two primes 𝔮_{1}, 𝔮_{2} not dividing 𝔫 lying above two distinct rational primes and L(f_{K}, 1_{K}, 1) = L(f, 1) L(f ⊗ η_{K/F} , 1) ≠ 0. The ramification assumption at 𝔮_{i} implies that K ⊄ F_{Γ_{𝔭}}F (V_{𝔭}) in the notation of B.5.1 (resp. K ⊄ K(f)F (V_{𝔭})) if f does not have (resp. has) CM. It follows from B.5.5(2),(4) (resp. B.6.5(2),(4)) that there exists g_{𝔭} ∈ G_{F} satisfying (A1)–(A3). Applying Theorem A for f, K and χ = 1 we obtain that the group H_{f}^{1}(K, V/T) is finite (and equal to zero if 𝔭 does not belong to a certain finite set).

As 2 · Ker (H_{f}^{1}(F, V/T) ⟶ H_{f}^{1}(K, V/T)) = 0, the same finiteness result holds over F.

2.10.6. Proof of Theorem B’. The statement (a) is an immediate consequence of Theorem B(a). In the case (b1) (resp. (b2)) there exists a finite prime v of F at which A_{f} does not have potentially good reduction (resp. such that A_{f} ⊗_{F} F_{v} does not acquire semistable reduction over any cyclic extension of F_{v}, by [AT, ch. 10, Thm. 5]); thus π(f)_{v} is a twist of a Steinberg representation (resp. is a supercuspidal representation), so Theorem B(b) applies. Finally, in the case (c1) (resp. (c2)) Theorem B(c) applies, thanks to B.4.12 (resp. B.6.5(2)).

Appendix A: Finite flat group schemes and their cohomology

In this appendix we recall basic facts about flat cohomology of finite flat group schemes over the ring of integers in a finite extension of Q_{p}. The main references are [Ma] and [Mi 2, §III.1].

A.1 Finite flat group schemes

Let R be a complete DVR of mixed characteristic (0, p), let K (resp. k) be its fraction field (resp. its residue field). A.1.1. Denote by Gr_{R} the (exact) category of commutative finite flat group schemes over R. The generic fibre H_{K} of any H ∈ Gr_{R} is a finite étale group scheme over K, which is determined by the G_{K}-module H_{K}(\overline{K}). A.1.2. The functor

H ↦ H(\overline{K}) = H_{K}(\overline{K})  (A.1.2.1)

from Gr_{R} into the category of discrete G_{K}-modules is faithful. In other words, the map

α_{H,H′} : Hom_{Gr_{R}}(H, H′) ⟶ Hom_{G_{K}}(H(\overline{K}), H′(\overline{K}))

is injective for any H, H′ ∈ Gr_{R}. Moreover,

Coker(α_{H,H′}) ⊗ Z[1/p] = 0.

If the absolute ramification index e = v_{R}(p) of K satisfies e < p−1, then the functor (A.1.2.1) is fully faithful [Ra 2, Cor. 3.3.6] (i.e., α_{H,H′} is bijective).

A.1.3. Question. In general, is there a constsnt s depending only on e and p such that

∀H, H′ ∈ Gr_{R} p^{s} Coker(α_{H,H′}) = 0 ?

A.1.4. According to [Bo, Thm. A,B] (see also [Li1, Thm. 1.0.5], [Li2, Thm. 2.4.2] and [VZ, Thm. 1]), the answer is “yes”. However, a much weaker statement (Corollary A.1.8-9) is sufficient for our purposes. A.1.5. Recall [Ra 2, Prop. 2.2.2] that the generic fibre H_{K} of any H ∈ Gr_{R} admits a maximal (resp. a minimal) prolongation H_{max} ∈ Gr_{R} (resp. H_{min} ∈ Gr_{R}). By definition, there are canonical morphisms in

Gr_{R}
H_{max} ⟶ H ⟶ H_{min}

inducing the identity on the generic fibre. Moreover,

Coker(α_{H_{max},H′_{max}}) = Coker(α_{H_{min},H′_{min}}) = 0.

A.1.6. For any Barsotti-Tate group H = (H_{n})_{n≥1} over R denote by

i_{n} : H_{m} ⟶ H_{m+n}, j_{n} : H_{m+n} ⟶ H_{m}

the standard transition morphisms (the composite maps i_{n}j_{n} and j_{n}i_{n} are given by multiplication by p^{n}).

A.1.7. Proposition. For each Barsotti-Tate group H′ = (H′_{n}) over R there is an integer a = a(R, H′) ≥ 0 such that, for each n ≥ 0, the minimal prolongation (H′_{n+a})_{min} ⟶ (H′_{n})_{min} of (j_{a})_{K} : (H′_{n+a})_{K} ⟶ (H′_{n})_{K} factors through a morphism

(H′_{n+a})_{min} ⟶^{ϕ} H′_{n} ⟶ (H′_{n})_{min}.

Proof. We follow the proof of [Ra 2, Prop. 2.3.1] for G(n) = (H′_{n})_{min}. The construction in [loc. cit. (b) - (d)] gives an integer i_{0} ≥ 0 and prolongations G(n)_{i_{0}} ⊂ G(n) of (H′_{i_{0}})_{K} (∀n ≥ i_{0}) such that the quotients H(n) := G(n + i_{0})/G(n + i_{0})_{i_{0}} (n ≥ 0) have the following property: for fixed n, the inductive system of the scheme-theoretical closure of H(n)_{K} = G(n)_{K} in H(n + j) (indexed by j) becomes stationary; its stationary values H_{n} form a Barsotti-Tate group (H_{n}) over R extending ((H′_{n})_{K}). The functor (H_{n}) ↦ (H_{n})_{K} from Barsotti-Tate groups over R to Barsotti-Tate groups over K being fully faithful [Ta, p. 180], there exists a compatible system of isomorphisms in Gr_{R}

u_{n} : H_{n} ⟶^{∼} H′_{n}.

Set a := i_{0}. The composite morphism

ϕ : (H′_{n+a})_{min} = G(n + a) ⟶ H(n) ⟶ H_{n} ⟶^{u_{n}} H′_{n}

has the required property, as it is a prolongation of the morphism (j_{a})_{K} : (H′_{n+a})_{K} ⟶ (H′_{n})_{K}.

A.1.8. Corollary. For each H ∈ Gr_{R} and each f ∈ Hom_{G_{K}}(H(\overline{K}), H′_{n+a}(\overline{K})) (n ≥ 0) the composite morphism

H(\overline{K}) ⟶^{f} H′_{n+a}(\overline{K}) ⟶^{j_{a}} H′_{n}(\overline{K})

is of the form α_{H,H′_{n}}(f̃) for some f̃ ∈ Hom_{Gr_{R}}(H, H′_{n}).

A.1.9. Corollary. For each H ∈ Gr_{R} and each m ≥ 0 p^{a} Coker(α_{H,H′_{m}}) = 0.

A.2 Flat cohomology

A.2.1. For any H ∈ Gr_{R} we denote by H_{fl}^{i}(R, H) the cohomology of the sheaf on the flat site of Spec(R) represented by H [Mi 1, II.1.7]. We can consider either the small or the big site, equipped with any of the following topologies: fpqf, fppf, syntomic; the cohomology groups H_{fl}^{i}(R, H) remain the same [Mi 1, III.3.4]. A.2.2. The functor (A.1.2.1) induces maps into Galois cohomology

H_{fl}^{i}(R, H) ⟶ H^{i}(K, H_{K}) := H^{i}(G_{K}, H_{K}(\overline{K})).

A.2.3. Local flat duality [Ma], [Mi 2, III.1]. Let H ∈ Gr_{R}, let H^{D} ∈ Gr_{R} be its Cartier dual. Assume that K is a finite extension of Q_{p}.

(1) If H is étale over R (e.g., if the order of H is prime to p), then H_{K}(\overline{K}) is an unramified G_{K}-module and H_{fl}^{i}(R, H) = H_{et}^{i}(R, H) = H^{i}(G_{k}, H_{K}(\overline{K})) = H_{ur}^{1}(K, H_{K}(\overline{K})).

(2) The map H_{fl}^{0}(R, H) ⟶^{∼} H^{0}(K, H_{K}) is an isomorphism.

(3) ∀i > 1 H_{fl}^{i}(R, H) = 0.

(4) The map H_{fl}^{1}(R, H) ↪ H^{1}(K, H_{K}) is injective.

(5) Under the (non-degenerate) Tate pairing

H^{1}(K, H_{K}) × H^{1}(K, H_{K}^{D}) ⟶^{∪} H^{2}(K, G_{m,K}) ⟶^{inv_{K}} Q/Z,

the orthogonal complement of H_{fl}^{1}(R, H) is equal to H_{fl}^{1}(R, H^{D}).

(6) For each i ≥ 0 H_{fl}^{i}(R, H) is a finite abelian group killed by the order of H (= the cardinality of H(\overline{K})).

A.2.4. Restriction, corestriction, conjugation. Assume that K is a finite extension of Q_{p} and K′/K is a finite extension; let R′ be the ring of integers in K′. For each H ∈ Gr_{R} the standard functoriality of the flat site gives rise to a “restriction map” res_{fl} which sits, thanks to A.2.3(5), in a commutative diagram with exact rows

0 ⟶ H_{fl}^{1}(R′, H_{R′}) ⟶ H^{1}(K′, H_{K′}) ⟶ Hom(H_{fl}^{1}(R′, H_{R′}^{D}), Q/Z) ⟶ 0
↑^{res_{fl}}  ↑^{res}  ↑
0 ⟶ H_{fl}^{1}(R, H) ⟶ H^{1}(K, H_{K}) ⟶ Hom(H_{fl}^{1}(R, H^{D}), Q/Z) ⟶ 0.

(A.2.4.1)

Applying A.2.3(5) to this diagram, we obtain a “corestriction map” cor_{fl} sitting in a commutative diagram with exact rows

0 ⟶ H_{fl}^{1}(R′, H_{R′}^{D}) ⟶ H^{1}(K′, H_{K′}^{D}) ⟶ Hom(H_{fl}^{1}(R′, H_{R′}), Q/Z) ⟶ 0
↓^{cor_{fl}}  ↓^{cor}  ↓
0 ⟶ H_{fl}^{1}(R, H^{D}) ⟶ H^{1}(K, H_{K}^{D}) ⟶ Hom(H_{fl}^{1}(R, H), Q/Z) ⟶ 0.

(A.2.4.2)

Similarly, any field automorphism σ : K′ ⟶ K′ fixing K gives rise, by functoriality, to a commutative diagram

\begin{matrix}H_{fl}^{1}(R′, H_{R′}) & ↪ & H^{1}(K′, H_{K′}) \\ ↓^{σ_{fl}} & & ↓^{σ} \\ H_{fl}^{1}(R′, H_{R′}) & ↪ & H^{1}(K′, H_{K′}). \end{matrix}  (A.2.4.3)

Moreover, σ_{fl} ∘ res_{fl} = res_{fl}, since σ ∘ res = res.

A.2.5. Proposition. The maps from A.2.4 have the following properties.

(1) cor_{fl} ∘ res_{fl} = [K′ : K] · id.

(2) If [K′ : K] is prime to p, then res_{fl} is injective, cor_{fl} is surjective and the left square in (A.2.4.1) is cartesian.

(3) More generally, res^{−1}(H_{fl}^{1}(R′, H_{R′}))/H_{fl}^{1}(R, H) is killed by the greatest common divisor d of [K′ : K] and the order of H.

(4) If K′/K is a Galois extension with Galois group Δ, then res_{fl} ∘ cor_{fl} = ∑_{σ∈Δ} σ_{fl}. The kernel and cokernel of the map

res_{fl} : H_{fl}^{1}(R, H) ⟶ H_{fl}^{1}(R′, H_{R′})^{Δ}

is killed by the integer d from (3).

Proof. This follows from the corresponding statements for Galois cohomology and from A.2.3(4),(6).

A.2.6. Proposition. Assume that K is a finite extension of Q_{p}. Let H = (H_{n}) be a Barsotti-Tate group over R, let H^{t} = (H_{n}^{D}) be the dual Barsotti-Tate group. Denote by T_{p}(H) := \underleftarrow{lim}_{n} H_{n}(\overline{K}) the Tate module

of H and set V_{p}(H) = T_{p}(H) ⊗_{Z_{p}} Q_{p}.

(1) V_{p}(H) is a crystalline representation of G_{K}.

(2) For each m, n ≥ 1 the map j_{m} : H_{m+n} ⟶ H_{n} induces an isomorphism

H_{fl}^{1}(R, H_{m+n}) ⊗ Z/p^{n}Z ⟶^{∼} H_{fl}^{1}(R, H_{n}).

(3) The Z_{p}-module X(H) := \underleftarrow{lim}_{n} H_{fl}^{1}(R, H_{n}) is of finite type and, for each n ≥ 1, there is a canonical

isomorphism X(H)/p^{n}H(H) ⟶^{∼} H_{fl}^{1}(R, H_{n}).

(4) The orthogonal complement of X(H) under the (non-degenerate) pairing

H^{1}(K, T_{p}(H)) × H^{1}(K, T_{p}(H^{t})) ⟶^{∪} H^{2}(K, T_{p}(G_{m})) = H^{2}(K, Z_{p}(1)) ⟶^{∼} Z_{p}

is equal to X(H^{t}).

(5) The subgroup X(H) ↪ \underleftarrow{lim}_{n} H^{1}(K, (H_{n})_{K}) = H^{1}(K, T_{p}(H)) is equal to the Bloch-Kato subspace

H_{f}^{1}(K, T_{p}(H)) = Ker (H^{1}(K, T_{p}(H)) ⟶ H^{1}(K, V_{p}(H) ⊗_{Q_{p}} B_{cris})) .

Proof. (1) This is a theorem of Fontaine [Fo, Thm. 6.2]. The statements (2) and (3) follow from the exact cohomology sequences attached to

\begin{matrix}0 ⟶ H_{m} ⟶^{i_{n}} H_{m+n} ⟶^{j_{m}} H_{n} ⟶ 0 \\ 0 ⟶ H_{n} ⟶^{i_{m}} H_{m+n} ⟶^{j_{n}} H_{m} ⟶ 0 \end{matrix}

and the vanishing of H_{fl}^{2}(R, H_{j}), while (4) is a consequence of A.2.3(5).

(5) We first prove the inclusion X(H) ⊆ H_{f}^{1}(K, T_{p}(H)). Let x = (x_{n}) ∈ X(H) ⊂ H^{1}(K, T_{p}(H)). Each element x_{n} ∈ H_{fl}^{1}(R, H_{n}) is represented by H̃_{n} ∈ Gr_{R} sitting in an exact sequence (of flat sheaves)

0 ⟶ H_{n} ⟶ H̃_{n} ⟶ Z/p^{n}Z ⟶ 0.

The generic fibres (H̃_{n})_{K} form a Barsotti-Tate group over K; it extends, thanks to [Ra 2, Prop. 2.3.1], to a Barsotti-Tate group H′ = (H′_{n}) over R. By construction, x is the extension class of the exact sequence of G_{K}-modules

0 ⟶ T_{p}(H) ⟶ T_{p}(H′) ⟶ Z_{p} ⟶ 0.

As V_{p}(H′) is a crystalline representation of G_{K} (by (1)), x ∈ H_{f}^{1}(K, T_{p}(H)), proving X(H) ⊆ H_{f}^{1}(K, T_{p}(H)). Combining the inclusion X(H^{t}) ⊆ H_{f}^{1}(K, T_{p}(H^{t})) with (4) and the equality [BK, Prop. 3.8]

H_{f}^{1}(K, T_{p}(H)) = H_{f}^{1}(K, T_{p}(H^{t}))^{⊥}

we obtain the converse inclusion X(H) ⊇ H_{f}^{1}(K, T_{p}(H)).

A.2.7. Proposition. Let ℬ be a semi-abelian variety over R siting in an exact sequence

0 ⟶ 𝒯 ⟶ ℬ ⟶ 𝒜 ⟶ 0,

where 𝒯 (resp. 𝒜) is a torus (resp. an abelian scheme) over R.

(1) For each integer m ≥ 1 there is an exact sequence of flat sheaves on Spec(R)

0 ⟶ ℬ[m] ⟶ ℬ ⟶^{m} ℬ ⟶ 0.

(2) If K is a finite extension of Q_{p}, then H_{fl}^{1}(R, ℬ) = 0 and the coboundary map in the exact cohomology sequence of (1) induces an isomorphism

∂_{R} : ℬ(R) ⊗ Z/mZ ⟶^{∼} H_{fl}^{1}(R, ℬ[m]).

Proof. (1) [NM, Lemma 7.3.2(a)]. The vanishing statement in (2) is a consequence of the isomorphisms

H_{fl}^{1}(R, ℬ) ⟶^{∼} H_{et}^{1}(R, ℬ) ⟶^{∼} H^{1}(G_{k}, ℬ(\overline{k})) ⟶^{∼} 0

(the first one is a theorem of Grothendieck [G1, Thm. 11.7], the second one an elementary property of étale cohomology of Spec(R) [Mi 2, Prop. II.1.1(b)] and the third one a theorem of Lang [La, Prop. 3]). The exact cohomology sequence

0 ⟶ ℬ(R)[m] ⟶ ℬ(R) ⟶^{m} ℬ(R) ⟶ H_{fl}^{1}(R, ℬ[m]) ⟶ H_{fl}^{1}(R, ℬ) = 0

implies that ∂_{R} is an isomorphism.

A.2.8. Proposition. In the situation of A.1.7, for each H ∈ Gr_{R}, f ∈ Hom_{G_{K}}(H(\overline{K}), H′_{n+a}(\overline{K})) (n ≥ 0) and a finite extension K′/K with ring of integers R′,

Im (H_{fl}^{1}(R′, H_{R′}) ⟶ H^{1}(K′, H_{K′}) ⟶^{f_{∗}} H^{1}(K′, (H′_{n+a})_{K′}) ⟶^{j_{a∗}} H^{1}(K′, (H′_{n})_{K′})) ⊆ H_{fl}^{1}(R′, (H′_{n})_{R′}).

Proof. This follows from Corollary A.1.8 and the fact that the maps A.2.2 are functorial in H.

A.2.9. Proposition. There exists an integer a ≥ 0 depending on K with the following property. For each m ≥ 0 and for each H ∈ Gr_{R} such that there is an exact sequence of G_{K}-modules

E : 0 ⟶ μ_{p^{m}}(\overline{K}) ⟶ H(\overline{K}) ⟶ Z/p^{m}Z ⟶ 0,

the image of the extension class [E] ∈ H^{1}(K, μ_{p^{m}}) = K^{×} ⊗ Z/p^{m}Z in K^{×} ⊗ Z/p^{m−a}Z is contained in R^{×} ⊗ Z/p^{m−a}Z.

Proof. There is an exact sequence in Gr_{R}

0 ⟶ H_{1} ⟶ H ⟶ H_{2} ⟶ 0,

where H_{1} is the scheme-theoretical closure of (μ_{p^{m}})_{K} in H. Applying Proposition A.1.7 to the Barsotti-Tate group μ_{p^{∞}} over R we obtain an integer a ≥ 0 such that the surjection

H_{1}(\overline{K}) ⟶^{∼} μ_{p^{m}}(\overline{K}) ⟶^{j_{a}} μ_{p^{m−a}}(\overline{K})

extends to a morphism H_{1} ⟶ μ_{p^{m−a}} in Gr_{R}. Applying the same argument to H_{2}^{D} and dualising, we see that the inclusion

Z/p^{m−a}Z ⟶^{i_{a}} Z/p^{m}Z ⟶^{∼} H_{2}(\overline{K})

extends to a morphism Z/p^{m−a}Z ⟶ H_{2} in Gr_{R}. The fibre product H′ := H ×_{H_{2}} Z/p^{m−a}Z ∈ Gr_{R} sits in an exact sequence in Gr_{R}

0 ⟶ H_{1} ⟶ H′ ⟶ Z/p^{m−a}Z ⟶ 0.

Applying the same argument to H^{′D} and dualising we obtain an exact sequence in in Gr_{R}

0 ⟶ μ_{p^{m−a}} ⟶ H′ ⟶ Z/p^{m−a}Z ⟶ 0,

whose \overline{K}-valued points coincide with E ⊗ Z/p^{m−a}Z. It follows that the image of [E] in H^{1}(K, μ_{p^{m−a}}) = K^{×} ⊗Z/p^{m−a}Z is contained in the image of H_{fl}^{1}(R, μ_{p^{m−a}}) = R^{×} ⊗Z/p^{m−a}Z ⟶ H^{1}(K, μ_{p^{m−a}}), as claimed.

A.3 Raynaud extensions

A.3.1. Let J be an abelian variety over K with semi-abelian reduction, let 𝒥 be its Néron model over R. By definition, the connected component of the identity 𝒥_{s}^{∘} of the special fibre 𝒥_{s} of 𝒥 is a semi-abelian variety over k. A.3.2. The Raynaud extension [SGA 7/I, §7] attached to J is a smooth commutative group scheme 𝒢 over R with the following properties [FC, ch. II,III].

A.3.2.1. The connected component 𝒢^{∘} is a semi-abelian scheme over R: as in A.2.7 there is an exact sequence

0 ⟶ 𝒯 ⟶ 𝒢^{∘} ⟶ 𝒜 ⟶ 0.

A.3.2.2. The quotient 𝒢 /𝒢^{∘} is a finite étale group scheme over R whose special fibre is isomorphic to 𝒥/𝒥_{s}^{∘}. A.3.2.3. There exists a monomorphism Z ⟶ 𝒢_{K}^{∘} of étale group schemes over K, where Z_{K^{ur}} ⟶^{∼} Z^{rk(𝒯 )} and

𝒢^{∘}(O\overline{K}) ∩ Z(\overline{K}) = 0, and an isomorphism of G_{K}-modules

𝒢_{K}^{∘}(\overline{K})/Z(\overline{K}) ⟶^{∼} J(\overline{K}).

A.3.2.4. For any finite extension K′/K with ring of integers R′, the following properties are equivalent:

Z_{K′} is a constant group scheme ⟺ 𝒯_{R′} is a split torus ⟺ J_{K′} has split semiabelian reduction.

A.3.3. The exact Galois cohomology sequence of

0 ⟶ Z(\overline{K}) ⟶ 𝒢_{K}^{∘}(\overline{K}) ⟶ J(\overline{K}) ⟶ 0

over a finite extension K′ of K reads as

0 ⟶ Z(K′) ⟶ 𝒢_{K}^{∘}(K′) ⟶ J(K′) ⟶ H^{1}(G_{K′}, Z(\overline{K})).  (A.3.3.1)

If Z_{K′} is a constant group scheme, then G_{K′} acts trivially on Z(\overline{K}) and (A.3.3.1) reduces to

𝒢_{K}^{∘}(K′)/Z(\overline{K}) ⟶^{∼} J(K′).  (A.3.3.2)

A.3.4. For each integer m ≥ 1 there is an exact sequence of flat sheaves over Spec(R)

0 ⟶ 𝒯 [m] ⟶ 𝒢^{∘}[m] ⟶ 𝒜 [m] ⟶ 0

and an exact sequence of G_{K}-modules

0 ⟶ 𝒢^{∘}[m](\overline{K}) ⟶ J(\overline{K})[m] ⟶ Z(\overline{K}) ⊗ Z/mZ ⟶ 0.

For any finite extension K′ of K denote by

∂_{K′} : J(K′) ⊗ Z/mZ ↪ H^{1}(K′, J[m]) := H^{1}(G_{K′}, J(\overline{K})[m])

(as in 1.7.3) the injective map induced by the cohomology sequence of

0 ⟶ J[m] ⟶ J(\overline{K}) ⟶^{m} J(\overline{K}) ⟶ 0.

Note that there is a commutative diagram (where R′ is the ring of integers in K′)

𝒢^{∘}(R′) ⊗ Z/mZ ⟶ 𝒢_{K}^{∘}(K′) ⊗ Z/mZ ⟶ J(K′) ⊗ Z/mZ ⟵ 𝒯_{K}(K′) ⊗ Z/mZ
↓^{∂_{R′}}  ↓  ↓^{∂_{K′}}  ↓
H_{fl}^{1}(R′, 𝒢^{∘}[m]) ⟶ H^{1}(K′, 𝒢^{∘}[m]) ⟶ H^{1}(K′, J[m]) ⟵ H^{1}(K′, 𝒯 [m]).

(A.3.4.1)

If Z_{K′} is a constant group scheme, then the second map in the top row is surjective.

A.3.5. Proposition. Assume that K is a finite extension of Q_{p}. Let K′ be a finite extension of K, let R′ be its ring of integers.

(1) ∂_{K′} (Im (𝒢_{K}^{∘}(K′) ⊗ Z/mZ ⟶ J(K′) ⊗ Z/mZ)) ⊆ Im (H^{1}(K′, 𝒢^{∘}[m]) ⟶ H^{1}(K′, J[m])).

(2) ∂_{K′} (Im (𝒢^{∘}(R′) ⊗ Z/mZ ⟶ J(K′) ⊗ Z/mZ)) = Im (H_{fl}^{1}(R′, 𝒢^{∘}[m]) ⟶ H^{1}(K′, J[m])).

(3) The L.H.S. of (1) is contained in

∂_{K′} (Im (𝒢^{∘}(R′) ⊗ Z/mZ ⟶ J(K′) ⊗ Z/mZ)) + Im (H^{1}(K′, 𝒯 [m]) ⟶ H^{1}(K′, J[m])) .

Proof. The statements (1) and (2) follow from the commutative diagram (A.3.4.1) and A.2.7(2). The inclusion (3) is a consequence of the fact that 𝒢_{K}^{∘}(K′) = 𝒯_{K}(K′) + 𝒢^{∘}(R′).

A.3.6. Corollary. If J_{K′} has split semi-abelian reduction, then:

(1) Im(∂_{K′}) ⊆ Im (H^{1}(K′, 𝒢^{∘}[m]) ⟶ H^{1}(K′, J[m])).

(2) Im(∂_{K′}) ⊆ Im (H_{fl}^{1}(R′, 𝒢^{∘}[m]) ⟶ H^{1}(K′, J[m])) + Im (H^{1}(K′, 𝒯 [m]) ⟶ H^{1}(K′, J[m])).

Appendix B: Galois images for cohomological Hilbert modular forms

In this appendix we collect basic statements about the images of Galois representations attached to Hilbert modular forms of regular weight. For elliptic modular forms, these results were proved by Ribet [Ri 1-4, 6] and Momose [Mo]. The case of Hilbert modular forms seems to be well-known, but we have not been able to find a good reference.

B.1 Twisted endomorphisms and the Brauer group

B.1.1. Assume that we are given the following data: a field L, a group G, an L[G]-module V (of finite dimension over L) satisfying End_{L[G]}(V) = L, and a finite group Γ ⊂ Aut(L) of field automorphisms of L.

We assume, in addition, that the isomorphism class of V is Γ-invariant in the sense that, for each σ ∈ Γ, there is an isomorphism of L[G]-modules

α_{σ} : ^{σ}V ⟶^{∼} V.  (B.1.1.1)

B.1.2. In concrete terms, after choosing an L-basis of V the action of G on V will be given by a group homomorphism

ρ : G ⟶ Aut_{L}(V) ⟶^{∼} GL_{n}(L) (n = dim_{L}(V))

and the isomorphisms (B.1.1.1) by matrices α_{σ} ∈ GL_{n}(L) satisfying

∀g ∈ G  α_{σ} ·^{σ}ρ(g) = ρ(g) · α_{σ}  ( ⟺ Ad(α_{σ})(^{σ}ρ(g)) = ρ(g)).  (B.1.2.1)

For each couple σ, τ ∈ Γ the composite map

α_{σ} ∘^{σ}α_{τ} ∘ α_{στ}^{−1} ∈ Aut_{L[G]}(V)

is given by multiplication by a scalar β_{σ,τ} ∈ L^{×} and the function β = (β_{σ,τ}) ∈ Z^{2}(Γ, L^{×}) is a 2-cocycle with cohomology class

[β] = δ([P (α)]) ∈ H^{2}(Γ, L^{×}) = H^{2}(L/L^{Γ}, L^{×}) ⊂ Br(L^{Γ}),

where

δ : H^{1}(Γ, P GL_{n}(L)) ⟶ H^{2}(Γ, L^{×})

is the coboundary arising from the exact sequence

1 ⟶ L^{×} ⟶ GL_{n}(L) ⟶^{P} P GL_{n}(L) ⟶ 1

and P (α) = (P (α_{σ})) ∈ Z^{1}(Γ, P GL_{n}(L)). If we choose another basis of V, then P (α) will be replaced by a cohomologous cocycle. Moreover, n[β] = 0 ∈ H^{2}(Γ, L^{×}). B.1.3. Twisted action, twisted endomorphisms. The formula

tw(σ)f := Ad(α_{σ})(^{σ}f) = α_{σ} ∘^{σ}f ∘ α_{σ}^{−1} (f ∈ End_{L}(V) ⟶^{∼} M_{n}(L))

defines a twisted action of Γ on W := End_{L}(V) (by morphisms of L^{Γ}-algebras). Let us call the L^{Γ}-subalgebra of endomorphisms invariant by the twisted action of Γ

End_{L}(V)^{tw(Γ)} ⊂ End_{L}(V)

the algebra of “twisted endomorphisms”. According to (B.1.2.1), the morphism of L-algebras

L[G] ⟶ End_{L}(V)  (B.1.3.1)

given by the G-action is obtained from a morphism of L^{Γ}-algebras

L^{Γ}[G] ⟶ End_{L}(V)^{tw(Γ)}  (B.1.3.2)

by extension of scalars via the multiplication map

m : L ⊗_{L^{Γ}} End_{L}(V)^{tw(Γ)} ⟶ End_{L}(V),  m(c ⊗ f) = cf.  (B.1.3.3)

B.1.4. Proposition. The map (B.1.3.3) is an isomorphism of L-algebras. The ring End_{L}(V)^{tw(Γ)} is a central simple algebra over L^{Γ} whose class in the Brauer group of L^{Γ} is equal to

[End_{L}(V)^{tw(Γ)}] = [β] ∈ H^{2}(Γ, L^{×}) ⊂ Br(L^{Γ})

(in particular, this class is killed by n). If, in addition, V is a simple L[G]-module, then the maps (B.1.3.1) and (B.1.3.2) are surjective.

Proof. As H^{1}(Γ, Aut_{L}(W)) = 1, the 1-cocycle (Ad(α_{σ})) ∈ Z^{1}(Γ, Aut_{L}(W)) is a coboundary: there exists an L-linear automorphism φ of W = End_{L}(V), necessarily of the form φ(f) = a∘f ∘b^{−1} for some a, b ∈ Aut_{L}(V), such that

∀σ ∈ Γ ∀f ∈ W Ad(α_{σ})f = φ^{−1} ∘^{σ}φ(f).

As a result,

∀σ ∈ Γ ∀f ∈ W φ(^{tw(σ)}f) = ^{σ}(φ(f)),

hence

End_{L}(V)^{tw(Γ)} = φ^{−1}(M_{n}(L^{Γ})) = {a^{−1} ∘ f ∘ b | M_{n}(L^{Γ})}

(note that φ is a morphism of L^{Γ}-algebras iff a = b iff [P (α)] = 0 ∈ H^{1}(Γ, P GL_{n}(L)) iff [β] = 0 ∈ H^{2}(Γ, L^{×})).

The map (B.1.3.3) is a surjective morphism of L-algebras, since its image contains a^{−1} ∘L⊗_{L^{Γ}} M_{n}(L^{Γ})∘ b = M_{n}(L). A dimension count implies that m is bijective. In other words, End_{L}(V)^{tw(Γ)} is an L/L^{Γ}-form of the matrix algebra M_{n}(L^{Γ}), hence it is a central simple algebra over L^{Γ}. Its Brauer class in H^{2}(Γ, L^{×}) is represented by the 2-cocycle b = (b_{σ,τ}) defined as follows: for each σ ∈ Γ there exists a_{σ} ∈ Aut(V) ⟶^{∼} GL_{n}(L) such that m ∘ (σ ⊗ id) = Ad(a_{σ}) ∘ (σ ∘ m); then b_{σ,τ} = a_{σ}^{σ}a_{τ} a_{στ}^{−1} ∈ L^{×}. By definition of the twisted action, we can take a_{σ} = α_{σ}, hence b = β.

If V is a simple L[G]-module, then (B.1.3.1) is surjective by Burnside’s theorem [CR, Thm. 3.32]. As (B.1.3.1) is obtained from (B.1.3.2) by extension of scalars and m is an isomorphism, the map (B.1.3.2) must also be surjective.

B.1.5. Proposition. The centraliser A of the L^{Γ}-subalgebra End_{L}(V)^{tw(Γ)} of End_{L^{Γ}}(V) ⟶^{∼} M_{n|Γ|}(L^{Γ}) is a central simple algebra over L^{Γ} satisfying M_{n}(A)^{op} ⟶^{∼} M_{|Γ|}(End_{L}(V)^{tw(Γ)}). In particular, dim_{L^{Γ}}(A) = |Γ|^{2} and [A] = −[End_{L}(V)^{tw(Γ)}] ∈ Br(L^{Γ}).

Proof. According to [Sc, Thm. 4.5], A is a central simple algebra over L^{Γ}. The remaining statements follow from the isomorphism of L^{Γ}-algebras [Sc, Thm. 4.5]

A ⊗_{L^{Γ}} End_{L}(V)^{tw(Γ)} ⟶^{∼} End_{L^{Γ}}(V), a ⊗ b ↦ ab.

B.1.6. Proposition. The normaliser {g ∈ End_{L}(V)^{×} | g ∘ End_{L}(V)^{tw(Γ)} ∘ g^{−1} = End_{L}(V)^{tw(Γ)}} of the subring End_{L}(V)^{tw(Γ)} ⊂ End_{L}(V) is equal to (End_{L}(V)^{tw(Γ)})^{×}L^{×} = L^{×}(End_{L}(V)^{tw(Γ)})^{×}.

Proof. One inclusion is obvious. To prove the opposite one, let g be an element of the normaliser. By definition,

∀σ ∈ Γ ∀f ∈ End_{L}(V)^{tw(Γ)} g ∘ f ∘ g^{−1} = ^{tw(σ)}(g ∘ f ∘ g^{−1}) = ^{tw(σ)}g ∘ f ∘^{tw(σ)}g^{−1},

which implies that g^{−1} ∘ ^{tw(σ)}g centralises L ⊗_{L^{Γ}} End_{L}(V)^{tw(Γ)} = End_{L}(V), hence ^{tw(σ)}g = c_{σ}g for some c_{σ} ∈ L^{×}. As ^{tw(σ)}a = ^{σ}a for all a ∈ L, the function {c_{σ}} ∈ Z^{1}(Γ, L^{×}) is a 1-cocycle. It follows that c_{σ} = ^{σ}bb^{−1} for some b ∈ L^{×}; thus g = (gb^{−1})b with gb^{−1} ∈ (End_{L}(V)^{tw(Γ)})^{×}.

B.2 Automorphic representations and Galois representations

Let F be a totally real number field. As in the main body of this article, denote by S_{∞} (resp. by S_{p}) the set of all infinite primes (resp. all primes above a rational prime p) of F. B.2.1. Automorphic representations. Fix an irreducible cuspidal automorphic representation π of GL_{2}(A_{F}) whose infinity type is of the form π_{∞} = ⊗_{v|∞}π_{v}, where π_{v} is a discrete series representation of weight k_{v} and algebraic central character, with all k_{v} ≥ 2 of the same parity.

Denote by ω : A_{F}^{×}/F^{×} ⟶ C^{×} the central character of π. Unlike in [N2, ch. 12] or in 0.2 we do not insist on ω being unitary ( ⟺ of finite order). Our assumptions imply that ω = | · |^{m}φ, where m ∈ Z and φ : A_{F}^{×}/F^{×} ⟶ C^{×} is a character of finite order satisfying φ_{v}(−1) = (−1)^{k_{v}} = (−1)^{m} for all v ∈ S_{∞}. Denote by

S = {v | π_{v} ≠ unramified principal series} ⊃ S_{∞}

the (finite) ramification set of π. B.2.2. Fields of moduli. As in [W1], set

𝒮 (π_{v}) = {σ ∈ Aut(C) | ^{σ}π_{v} ⟶^{∼} π_{v}}  (v ∉ S_{∞})
𝒮 (π) = {σ ∈ Aut(C) | ^{σ}π ⟶^{∼} π}

and define the field of moduli of π_{v} and π, respectively, to be

Q(π_{v}) := C^{𝒮 (π_{v})} , Q(π) := C^{𝒮 (π)} .

Note that, for each σ ∈ Aut(C), the representation ^{σ}π is also of the form considered in B.2.1, with infinity type (^{σ}π)_{∞} = π_{∞} and central character ^{σ}ω = | · |^{m} ^{σ}φ.

Let c ∈ Aut(C) be the complex conjugation. The existence of a hermitian scalar product on the space of cusp forms implies that, for each σ ∈ Aut(C),

c(^{σ}π ⊗ | · |^{−m/2}) ⟶^{∼} { \atop (^{σ}π ⊗ | · |^{−m/2})} ⟶^{∼} ^{σ}π ⊗ | · |^{−m/2} ⊗ φ^{−1},

hence

cσπ ⟶^{∼} ^{σ}π ⊗^{σ}φ^{−1} ⟶^{∼} ^{σc}π.  (B.2.2.1)

B.2.3. Proposition [W1]. (1) Q(π) is a number field which is either totally real or a CM field. If φ = 1, then Q(π) is totally real.

(2) Q(π) ⊃ Q(ω) := Q(Im(ω)) = Q(Im(φ)).

(3) Q(π) ⊃ Q(π_{v}) for all v ∉ S_{∞}.

(4) Q(π) is the compositum of {Q(π_{v})}_{v∉Σ}, for any finite set of primes Σ ⊃ S of F.

(5) For each v ∉ S_{∞}, π_{v} has a model over Q(π_{v}).

(6) π has a model over Q(π).

(7) ∀v ∉ S Q(π_{v}) = Q(λ_{π}(v), ω(v)), where λ_{π}(v) is the eigenvalue of the Hecke operator T (v) on the spherical line in π_{v}. Explicitly, π_{v} = π(μ_{1}, μ_{2}) acts by right translations on the space

B(μ_{1}, μ_{2}) = \{f : GL_{2}(F_{v}) ⟶ C | f(( \begin{matrix}a & b \\ 0 & d \end{matrix} ) g) = μ_{1}(a)μ_{2}(d)|ad^{−1}|_{v}^{1/2}f(g)\},

where μ_{i} : F^{×} ⟶ C^{×} are unramified characters, and λ_{π}(v) = (μ_{1}(v) + μ_{2}(v))(Nv)^{1/2}.

B.2.4. In the situation of B.2.3(7), Q(π_{v}) is equal to the field of definition of the local Hecke polynomial

P^{H}(π_{v}, X) = 1 − λ_{π}(v)X + ω(v)(Nv)X^{2},

which is related to the local L-factor

L(π_{v}, s) = [(1 − μ_{1}(v)(Nv)^{−s})(1 − μ_{2}(v)(Nv)^{−s})]^{−1}

by the following relation:

L(π_{v}, s − \frac{1}{2}) = P^{H}(π_{v}, (Nv)^{−s})^{−1}.

B.2.5. Galois representations. From now on (until the end of Appendix B) we are going to denote the number field Q(π) by L. It is known, thanks to the work of many people culminating in [T1] (see also [C1]), that for each finite prime 𝔭 of L above a rational prime p there exists a representation V_{𝔭}(π) of G_{F} which is two-dimensional over L_{𝔭}, unramified outside S ∪ S_{p} and which satisfies

∀v ∉ S ∪ S_{p} det(1 − X Fr_{geom}(v) | V_{𝔭}(π)) = P^{H}(π_{v}, X).

In other words, {V_{𝔭}(π)}_{𝔭} is a strongly compatible system of L-rational representations of G_{F} whose Lfunction is equal to

L^{S_{∞}}(π, s − \frac{1}{2}) = ∏_{v∤∞} L(π_{v}, s − \frac{1}{2}).

It follows from B.2.4 that

∀σ ∈ Aut(C) V_{σ(𝔭)}(^{σ}π) ⟶^{∼} V_{𝔭}(π) ⊗_{L_{𝔭},σ} σ(L)_{σ(𝔭)}.

Let us identify characters of finite order of A_{F}^{×}/F^{×} with characters of G_{F} as in 0.4; then

Λ^{2}V_{𝔭}(π) ⟶^{∼} L_{𝔭}(−1) ⊗ L_{𝔭}(m) ⊗ φ = L_{𝔭}(m − 1) ⊗ φ.  (B.2.5.1)

If f ∈ S_{k}(𝔫, φ) is a Hilbert modular newform of parallel even weight k as in [N2, 12.3] and π = π(f) is normalised as in [loc. cit.] by ω = φ, then

V_{𝔭}(π) = V_{𝔭}(f)(k/2 − 1).

In particular, V_{𝔭}(π) = V_{𝔭}(f) for newforms f ∈ S_{2}(𝔫, φ) considered in the main body of this article.

We define the (semi-simplified) residual representation \overline{V_{𝔭}(π)}^{ss} of V_{𝔭}(π) to be the semi-simplification of T_{𝔭}(π)/𝔭T_{𝔭}(π), for any G_{F} -stable O_{L,𝔭}-lattice T_{𝔭}(π) ⊂ V_{𝔭}(π) (up to isomorphism, \overline{V_{𝔭}(π)}^{ss} does not depend on the choice of the lattice).

B.2.6. Proposition [T2]. (1) V_{𝔭}(π) is an absolutely irreducible L_{𝔭}[G_{F}]-module.

(2) For all but finitely many 𝔭 the residual representation \overline{V_{𝔭}(π)}^{ss} is an absolutely irreducible O_{L}/𝔭[G_{F}]- module.

B.2.7. Proposition. The following properties are equivalent:

(1) There exists a finite extension E/F such that V_{𝔭}(π)|_{G_{E}} is not an absolutely irreducible L_{𝔭}[G_{E}]-module.

(2) There exists a totally imaginary quadratic extension K(π)/F such that π ⊗η ⟶^{∼} π, where η = η_{K(π)/F} : Gal(K(π)/F) ⟶^{∼} {±1} is the quadratic character corresponding to K(π)/F.

(3) There exists K(π)/F as in (2) such that π = I_{K(π)/F} (ψ) is obtained by automorphic induction from a Hecke character ψ : A_{K(π)}^{×}/K(π)^{×} ⟶ C^{×} of infinity type ψ_{w}(z) = z^{a_{w}} \overline{z}^{b_{w}} (a_{w}, b_{w} ∈ Z, |a_{w} − b_{w}| = k_{v} − 1 and a_{w} + b_{w} = m − 1 for each w | v | ∞). [If they are satisfied, we say that π has complex multiplication by K(π).]

Proof. Ribet’s argument [Ri 3, Thm. 4.5] for F = Q combined with [He, Thm. 2] works in general (see [N1, Prop. 6.2.1] for a special case).

B.2.8. Proposition-Definition [Ri 2, p. 788]. Let E be a finite extension of F contained in \overline{F}. The Frobenius field M_{E} attached to π over E is the subfield of L generated over Q by the traces Tr(Fr_{geom}(w) | V_{𝔭}(π)) ∈ L (which do not depend on 𝔭), for w running through all primes of E not dividing S. We have M_{F} = L.

B.3 Inner twists

Let π, L = Q(π) and V_{𝔭}(π) be as in B.2.1 and B.2.5. Denote by L^{+} the maximal real subfield of L. B.3.1. In order to simplify the notation we are going to write V_{𝔭} := V_{𝔭}(π). The direct sum V_{p} := ⊕_{𝔭|p}V_{𝔭} is a free module of rank 2 over L ⊗ Q_{p} = ∏_{𝔭|p} L_{𝔭}. The action of G_{F} defines a morphism of (L ⊗ Q_{p})-algebras

(L ⊗ Q_{p})[G_{F}] ⟶ End_{L⊗Q_{p}}(V_{p}) = ∏_{𝔭|p} End_{L_{𝔭}}(V_{𝔭}) ⟶^{∼} M_{2}(L ⊗ Q_{p}) = ∏_{𝔭|p} M_{2}(L_{𝔭}).

B.3.2. Proposition-Definition. An inner twist of π is a pair (σ, χ), where σ : L ↪ C and χ : A_{F}^{×}/F^{×} ⟶ C^{×} is a character such that there exists an isomorphism ^{σ}π ⟶^{∼} π ⊗ χ (more precisely, such that there exists an isomorphism ^{σ′}π ⟶^{∼} π ⊗ χ for some – hence for each – field automorphism σ′ ∈ Aut(C) extending σ). Each such a pair has the following properties.

(1) The character χ is unramified outside S.

(2) ∀v ∉ S ^{σ}λ_{π}(v) = χ(v) λ_{π}(v).

(3) χ^{2} = ^{σ}ω/ω = ^{σ}φ/φ.

(4) χ = φ^{i}μ, where i ∈ Z and μ^{2} = 1 (⟹ Im(χ) ⊆ L).

(5) σ(L) ⊆ L, hence σ ∈ Aut(L/Q).

(6) For each rational prime p there is an isomorphism of (L ⊗ Q_{p})[G_{F}]-modules ^{σ}V_{p} ⟶^{∼} V_{p} ⊗ χ (recall from B.2.5 that we identify χ with a character χ : G_{F} ⟶ L^{×} ⊂ C^{×}).

Proof. The arguments from [Mo, §1] and [Ri 4, §3] apply.

B.3.3. Proposition. (1) The inner twists of π form a group Γ under (σ, χ) · (σ′, χ′) = (σ ∘ σ′, χ ·^{σ}χ′).

(2) The map “forget χ” is a group homomorphism Γ ⟶ Aut(L/Q), (σ, χ) ↦ σ, whose kernel is trivial (resp. is cyclic of order 2 generated by (id, η_{K(π)/F})) if π has no CM (resp. if π has CM by K(π) as in B.2.7). Consequently, Γ is a finite group.

(3) The image of Γ in Aut(L/Q) is an abelian group.

(4) If φ ≠ 1, then (c, φ^{−1}) ∈ Γ (if, in addition, π has no CM, then L is not totally real).

(5) For each 𝔭 | p in L, the subgroup Γ_{𝔭} := {(σ, χ) ∈ Γ | σ(𝔭) = 𝔭} ⊂ Γ is equal to

{(σ, χ) | σ ∈ Aut(L_{𝔭}/Q_{p}), χ : G_{F} ⟶ L_{𝔭}^{×}, ^{σ}V_{𝔭} ⟶^{∼} V_{𝔭} ⊗ χ}.

Proof. Again, everything works as in [Mo, §1] and [Ri 4, §3].

B.3.4. Proposition-Definition. From now on, until the end of B.5, we assume that π has no CM. Under this assumption we can identify Γ with a (commutative) subgroup of Aut(L/Q) and write its elements as (σ, χ_{σ}), since χ is determined by σ. For any subgroup Δ ⊂ Γ denote by F_{Δ} the fixed field of the open subgroup ⋂_{σ∈Δ} Ker(χ_{σ}) ⊆ G_{F}.

(1) F_{Δ} is a finite abelian extension of F unramified outside S.

(2) Δ ⊂ Δ′ ⟺ F_{Δ} ⊂ F_{Δ′}.

(3) F_{Δ}F_{Δ′} = F_{ΔΔ′}.

(4) F_{\{1\}} = F, F_{\{1,c\}} = F_{φ} := \overline{F}^{Ker(φ)}.

(5) F_{Γ} = F_{φ}(\sqrt{a_{1}}, …, \sqrt{a_{r}}) for some a_{j} ∈ F_{φ}^{×}.

(6) F_{φ} is totally real (resp. totally complex) if 2 | m (resp. if 2 ∤ m).

(7) If 2 | m, then an intermediate field F ⊂ F′ ⊂ F_{Γ} is totally real ⟺ it is not totally complex.

(8) If the prime 𝔭_{Γ} := 𝔭 ∩ L^{Γ} of L^{Γ} splits completely in L^{+}/L^{Γ}, then Γ_{𝔭} ⊂ {1, c} and F_{Γ_{𝔭}} ⊂ F_{φ}.

Proof. Easy exercise.

B.3.5. Definition. Set Γ_{triv} = {1} (resp. {1, c}) if φ = 1 (resp. if φ ≠ 1). We say that π has no non-trivial inner twist if Γ = Γ_{triv} ( ⟺ F_{Γ} = F_{φ}). Similarly, we say that V_{𝔭} has no non-trivial inner twist if Γ_{𝔭} ⊂ Γ_{triv} ( ⟺ F_{Γ_{𝔭}} ⊂ F_{φ}).

B.3.6. Proposition-Definition [Ri 2]. For any field embedding σ : L ↪ \overline{Q}_{p} let V_{σ} be the \overline{Q}_{p}[G_{F}]-module V_{p} ⊗_{L⊗Q_{p},σ⊗id} \overline{Q}_{p} = V_{𝔭} ⊗_{L_{𝔭},σ} \overline{Q}_{p}, where 𝔭 | p is the prime of L induced by σ. Let E be a finite extension of F contained in \overline{F}, let σ, τ : L ↪ \overline{Q}_{p} be field embeddings. The following conditions are equivalent.

(1) σ|_{M_{E}} = τ|_{M_{E}}.

(2) The \overline{Q}_{p}[G_{E}]-modules V_{σ}|_{G_{E}} and V_{τ} |_{G_{E}} are isomorphic.

(3) [In the case when E/F is a Galois extension.] There exists a character χ : Gal(E/F) ⟶ \overline{Q}_{p}^{×} and an isomorphism of \overline{Q}_{p}[G_{F}]-modules V_{σ} ⟶^{∼} V_{τ} ⊗ χ.

Proof. [Ri 2, Lemma 4.4.5], [Ri 4, Proof of Thm. 4.7], [Ch, Prop. 5.4].

B.3.7. Corollary. ⋂_{E} M_{E} = M_{F_{Γ}} = L^{Γ}.

B.4 Image of the Galois representation V_{p} (the non CM case)

The assumptions of B.3 are in force. In particular, π does not have CM. B.4.1. Fix a rational prime p and denote by

ρ_{p} : G_{F} ⟶ End_{L⊗Q_{p}}(V_{p}) ⟶^{∼} GL_{2}(L ⊗ Q_{p})

the morphism given by the action of G_{F} on V_{p}.

The restriction of ρ_{p} to G_{F_{Γ}} gives rise to a semi-local version of the situation considered in B.1 (for G = G_{F_{Γ}}): for each σ ∈ Γ there is an isomorphism of (L ⊗ Q_{p})[G_{F_{Γ}}]-modules α_{σ} : ^{σ}V ⟶^{∼} V. The corresponding subalgebra of twisted endomorphisms

D(p) := End_{L⊗Q_{p}}(V_{p})^{tw(Γ)} ⊂ End_{L⊗Q_{p}}(V_{p}) ⟶^{∼} ∏_{𝔭|p} M_{2}(L_{𝔭})

is an Azumaya algebra with centre (L ⊗ Q_{p})^{Γ} = L^{Γ} ⊗ Q_{p} ⟶^{∼} ∏_{𝔭_{Γ}|p}(L^{Γ})_{𝔭_{Γ}}. It satisfies

(L ⊗ Q_{p}) ⊗_{(L^{Γ}⊗Q_{p})} D(p) ⟶^{∼} End_{L⊗Q_{p}}(V_{p})

and its class in

H^{2}(Γ, (L ⊗ Q_{p})^{×}) = ⨁_{𝔭_{Γ}|p} H^{2}(Γ_{𝔭}, L_{𝔭}^{×}) ⊂ ⨁_{𝔭_{Γ}|p} Br((L^{Γ})_{𝔭_{Γ}})

is killed by 2 (above, 𝔭_{Γ} runs through all primes of L^{Γ} above p, 𝔭 | 𝔭_{Γ} is any prime of L above 𝔭_{Γ} and Γ_{𝔭} = Gal(L_{𝔭}/(L^{Γ})_{𝔭_{Γ}}) ⊂ Γ is – as in B.3.3(5) – the decomposition group of 𝔭 in L/L^{Γ}). In other words,

D(p) = ⨁_{𝔭_{Γ}|p} D(𝔭_{Γ}), D(𝔭_{Γ}) is a quaternion algebra over (L^{Γ})_{𝔭_{Γ}}.

B.4.2. Proposition. Let p be any rational prime.

(1) The restriction of ρ_{p} to G_{F_{Γ}} defines a surjective morphism of L ⊗ Q_{p}-algebras (L ⊗ Q_{p})[G_{F_{Γ}}] ⟶ End_{L⊗Q_{p}}(V_{p}), which is obtained by extension of scalars from a surjective morphism of L^{Γ} ⊗ Q_{p}-algebras (L^{Γ} ⊗ Q_{p})[G_{F_{Γ}}] ⟶ D(p).

(2) If 2 | m, then ρ_{p}(G_{F_{φ}}) ⊆ GL_{2}(L^{+} ⊗ Q_{p}), in a suitable basis of V_{p} (this is also true if π has complex multiplication).

Proof. (1) Combine B.1.4 with B.2.6(1).

(2) The argument of [Ri 3, Cor. 5.2] applies.

B.4.3. Corollary. For each rational prime p the G_{F_{Γ}}-action on V_{p} factors through

ρ_{p} : G_{F_{Γ}} ⟶ {x ∈ D(p)^{×} | Nrd(x) ∈ χ_{cycl,p}(G_{F_{Γ}})^{m−1}},

where m ∈ Z is as in B.2.1 and χ_{cycl,p} : G_{Q} ⟶ Z_{p}^{×} is the cyclotomic character.

Proof. As F_{Γ} ⊃ F_{φ}, the formula (B.2.5.1) implies that (Λ^{2}V_{p})|_{G_{F_{Γ}}} = χ_{cycl,p}^{m−1}|_{G_{F_{Γ}}} .

B.4.4. Corollary. For each rational prime p the Lie algebra (over Q_{p}) 𝔤_{p} := Lie(ρ_{p}(G_{F_{Γ}})) ⊂ D(p) is contained in 𝔥_{p} := {x ∈ D(p) | Trd(x) ∈ (m − 1)Q_{p}}.

B.4.5. Theorem. For each rational prime p we have 𝔤_{p} = 𝔥_{p}.

Proof. The arguments in [Mo, Thm. 4.1] and [Ri 4, Prop. 4.5] apply almost word by word. For the reader’s convenience we repeat the main points. There is an isomorphism of \overline{Q}_{p}-algebras

D(p) ⊗_{Q_{p}} \overline{Q}_{p} ⟶^{∼} ∏_{σ:L^{Γ}↪\overline{Q}_{p}} M_{2}(\overline{Q}_{p})

such that, for each field embedding σ : L^{Γ} ↪ \overline{Q}_{p}, the composite morphism

ρ_{σ} : G_{F_{Γ}} ⟶^{ρ_{p}} D(p)^{×} ↪ (D(p) ⊗_{Q_{p}} \overline{Q}_{p})^{×} ⟶^{∼} ∏_{σ:L^{Γ}↪\overline{Q}_{p}} GL_{2}(\overline{Q}_{p}) ⟶^{pr_{σ}} GL_{2}(\overline{Q}_{p})

corresponds to the action of G_{F_{Γ}} on V_{σ′}|_{G_{F_{Γ}}} , where σ′ : L ↪ \overline{Q}_{p} is any embedding extending σ (see B.3.6 above). As V_{σ′}|_{G_{E}} is an irreducible representation of G_{E}, for any finite extension E/F, the Lie algebra (over \overline{Q}_{p}) 𝔤_{σ} := Lie(ρ_{σ}(G_{F_{Γ}})) ⊂ 𝔤l_{2}(\overline{Q}_{p}) is a reductive Lie subalgebra acting irreducibly on \overline{Q}_{p}^{2}; thus 𝔤_{σ} contains 𝔰l_{2}(\overline{Q}_{p}).

The Lie subalgebra 𝔤_{p} ⊗_{Q_{p}} \overline{Q}_{p} ⊂ ∏_{σ:L^{Γ}↪\overline{Q}_{p}} 𝔤_{σ} has the following property: for two distinct embeddings

σ, τ : L^{Γ} ↪ \overline{Q}_{p} the \overline{Q}_{p}[G_{F_{Γ}}]-modules V_{σ′}|_{G_{F_{Γ}}} and V_{τ′}|_{G_{F_{Γ}}} are not isomorphic, by B.3.6. This implies that (pr_{σ} × pr_{τ})(𝔤_{p} ⊗_{Q_{p}} \overline{Q}_{p}) contains 𝔰l_{2}(\overline{Q}_{p}) × 𝔰l_{2}(\overline{Q}_{p}); this is enough to conclude that 𝔤_{p} ⊗_{Q_{p}} \overline{Q}_{p} contains ∏_{σ:L^{Γ}↪\overline{Q}_{p}} 𝔰l_{2}(\overline{Q}_{p}), by [Ri 4, Lemma 4.6]. Finally, the image of 𝔤_{p} ⊗_{Q_{p}} \overline{Q}_{p} in ∏_{σ:L^{Γ}↪\overline{Q}_{p}} 𝔤l_{2}(\overline{Q}_{p})/𝔰l_{2}(\overline{Q}_{p}) is

given by det(V_{p})|_{G_{F_{Γ}}} = χ_{cycl,p}^{m−1}|_{G_{F_{Γ}}} .

B.4.6. Corollary. (1) For each rational prime p the G_{F} -action on V_{p} factors through

ρ_{p} : G_{F} ⟶ {x ∈ D(p)^{×}(L ⊗ Q_{p})^{×} | Nrd(x) ∈ χ_{cycl,p}(G_{F})^{m−1}Im(φ)}.

(2) For each rational prime p and each finite extension E/F_{Γ} we have

Im (Q_{p}[G_{E}] ⟶ End_{Q_{p}}(V_{p})) = D(p).

Proof. (1) As G_{F_{Γ}} is a normal subgroup of G_{F}, the image of G_{F} under ρ_{p} normalises 𝔤_{p} +(L⊗Q_{p})id = D(p). The statement follows from a semi-local version of B.1.6.

(2) D(p) contains the L.H.S., which in turn contains the Q_{p}-algebra generated by the Lie algebra 𝔤_{p} = 𝔥_{p}, namely D(p).

B.4.7. Definition. let 𝔭 be a finite prime of L. We say that the representation V_{𝔭} is quaternionic if the quaternion algebra D(𝔭_{Γ}) (where 𝔭_{Γ} = 𝔭 ∩ L^{Γ}) is a division algebra (this depends only on the Γ-orbit of 𝔭).

B.4.8. Proposition. (1) If 𝔭 is unramified in L/L^{Γ} and V_{𝔭} is quaternionic, then the residual representation \overline{V_{𝔭}(π)}^{ss} is reducible.

(2) The set of 𝔭 for which V_{𝔭} is quaternionic is finite.

(3) There exists a quaternion algebra D over L^{Γ} such that D ⊗ Q_{p} ⟶^{∼} D(p) for each rational prime p (hence D(𝔭_{Γ}) ⟶^{∼} D ⊗_{L^{Γ}} (L^{Γ})_{𝔭_{Γ}} for each finite prime 𝔭_{Γ} of L^{Γ}).

Proof. Thanks to B.2.6(2) it is enough to prove (1). Denote by O the (unique) maximal order of the quaternion division algebra D(𝔭_{Γ}). As L_{𝔭}/(L^{Γ})_{𝔭_{Γ}} is unramified, the maximal (bilateral) ideal of O is of the form aO = Oa, where a^{2} is a uniformiser of (L^{Γ})_{𝔭_{Γ}}, hence of L_{𝔭}. According to B.4.6, the image of the compact group G_{F} in Aut_{L_{𝔭}}(V_{𝔭}) is contained in the maximal compact subgroup O^{×}O_{L,𝔭}^{×} of D(𝔭_{Γ})^{×}L_{𝔭}^{×}. The image of O under that map O ⊂ M_{2}(O_{L,𝔭}) ⟶ M_{2}(O_{L}/𝔭) is isomorphic to O/a^{2}O ⟶^{∼} k[ε]/(ε^{2}), where k = O/aO. This implies that, possibly after conjugation by an element of GL_{2}(O_{L}/𝔭), the image of O is contained in

R = \{( \begin{matrix}x & y \\ 0 & x \end{matrix} ) | x, y ∈ O_{L}/𝔭\} ⊂ M_{2}(O_{L}/𝔭).

Consequently, the image of O^{×}O_{L,𝔭}^{×}, hence of G_{F}, is contained in R^{×}, which proves (1). B.4.9. It is likely that there exists a (unique) quaternion algebra D as in B.4.8(3) which is totally indefinite (resp. totally definite) if 2 | m (resp. if 2 ∤ m). In the case F = Q this was proved in [Mo, Thm. 3.1.2].

B.4.10. Theorem. There exists a quaternion algebra D over L^{Γ} such that:

(1) For each rational prime p the representation ρ_{p} factors through

ρ_{p} : G_{F_{Γ}} ⟶ (D ⊗ Q_{p})^{×} ⊂ ((D ⊗_{L^{Γ}} L) ⊗ Q_{p})^{×} ⟶^{∼} GL_{2}(L ⊗ Q_{p}),
ρ_{p} : G_{F} ⟶ (D ⊗ Q_{p})^{×}(L ⊗ Q_{p})^{×} ⊂ GL_{2}(L ⊗ Q_{p}).

(2) For each rational prime p the image ρ_{p}(G_{F_{Γ}}) is an open subgroup of

{x ∈ (D ⊗ Q_{p})^{×} | Nrd(x) ∈ χ_{cycl,p}(G_{F_{Γ}})^{m−1}}.

(3) For all but finitely many rational primes p there is a basis of V_{p} over L ⊗ Q_{p} in which

ρ_{p}(G_{F_{Γ}}) = {x ∈ GL_{2}(O_{L^{Γ}} ⊗ Z_{p}) | det(x) ∈ (Z_{p}^{×})^{m−1}}.

Proof. (1) This is a combination of B.4.2(1), B.4.6 and B.4.8(3). The statement (2) is equivalent to B.4.5. Finally, (3) is proved exactly as in [Ri 6, Thm. 3.1]. B.4.11. Modular abelian varieties over F. Assume that m = 2 and k_{v} = 2 for each v ∈ S_{∞}. It is expected that there exists an abelian variety A over F with the following properties:

dim(A) = [L : Q],  End_{F} (A) = O_{L},  ∀p V_{p} = H_{et}^{1}(A ⊗_{F} \overline{F} , Q_{p})(1).

The existence of A is known if 2 ∤ [F : Q] or if there exists a finite prime v of F for which π_{v} is not a principal series representation (in this case A can be constructed as a quotient of the Jacobian of a suitable Shimura curve over F).

B.4.12. Proposition. (1) There exists a quaternion algebra D over L^{Γ} such that

End\overline{Q}(A) ⊗ Q = End_{F_{Γ}}(A) ⊗ Q ⟶^{∼} \{ \begin{matrix}M_{|Γ|}(L^{Γ}), & if D ⟶^{∼} M_{2}(L^{Γ}) \\ M_{|Γ|/2}(D), & if D is a division algebra \end{matrix}

and D ⊗ Q_{p} ⟶^{∼} D(p) for each rational prime p. [Recall that π does not have CM, by assumption; this means that A does not have CM, either.]

(2) V_{𝔭} = H_{et}^{1}(A ⊗_{F} \overline{F} , Q_{p})(1) ⊗_{L⊗Q_{p}} L_{𝔭} is quaternionic in the sense of B.4.7 iff the central simple algebra (End\overline{Q}(A) ⊗ Q) ⊗_{L^{Γ}} (L^{Γ})_{𝔭_{Γ}} (where 𝔭_{Γ} = 𝔭 ∩ L^{Γ}) has a non-zero class in Br((L^{Γ})_{𝔭_{Γ}}).

Proof. It is enough to prove (1). For any finite extension E/F_{Γ} the Faltings isogeny theorem tells us that End_{E}(A) ⊗ Q_{p} is equal to the centraliser of Im (Q_{p}[G_{E}] ⟶ End_{Q_{p}}(V_{p})) = D(p) in End_{Q_{p}}(V_{p}). As L^{Γ} ⊗ Q_{p} = Z(D(p)) ⊆ Z(End_{E}(A) ⊗ Q_{p}), we deduce that End_{E}(A) ⊗ Q_{p} coicides with the centraliser of D(p) in End_{L^{Γ}⊗Q_{p}}(V_{p}), hence is isomorphic to

End_{E}(A) ⊗ Q_{p} ⟶^{∼} \{ \begin{matrix}M_{|Γ|}(L^{Γ} ⊗ Q_{p}), & if D(p) ⟶^{∼} M_{2}(L^{Γ} ⊗ Q_{p}) \\ M_{|Γ|/2}(D(p)), & if not, \end{matrix}

thanks to a semi-local version of B.1.5. As L ⊂ End_{E}(A) ⊗ Q and Z(End_{E}(A) ⊗ Q_{p}) = L^{Γ} ⊗ Q_{p}, it follows that End_{E}(A) ⊗ Q is a central simple algebra over L^{Γ} which does not depend on E ⊃ F_{Γ}, whose class in Br(L^{Γ}) is killed by 2 and whose localisations at finite primes are given by the above formula. The statement (1) is implied by these properties.

B.5 Image of the Galois representation V_{𝔭} (the non CM case)

The assumptions of B.3 are in force. In particular, π does not have CM. Let K be a totally imaginary quadratic extension of F. B.5.1. For a prime 𝔭 of L above a rational prime p we denote by

ρ_{𝔭} : G_{F} ⟶ Aut_{L_{𝔭}}(V_{𝔭}) ⟶^{∼} GL_{2}(L_{𝔭})

the morphism defining the action of G_{F} on V_{𝔭} and by F (V_{𝔭}) := \overline{F}^{Ker(ρ_{𝔭})} the extension of F trivialising ρ_{𝔭}. As before, we denote by 𝔭_{Γ} = 𝔭 ∩ L^{Γ} the prime of L^{Γ} below 𝔭 and by Γ_{𝔭} ⊂ Γ the decomposition group of 𝔭 in the extension L/L^{Γ}.

B.5.2. Theorem. ρ_{𝔭}(G_{F_{φ}F_{Γ𝔭}}) is an open subgroup of {x ∈ D(𝔭_{Γ})^{×} | Nrd(x) ∈ (Z_{p}^{×})^{m−1}}. For all but finitely many 𝔭 there exists a basis of V_{𝔭} in which

ρ_{𝔭}(G_{F_{φ}F_{Γ𝔭}}) = {x ∈ GL_{2}(O_{L^{Γ},𝔭_{Γ}}) | det(x) ∈ (Z_{p}^{×})^{m−1}} (𝔭_{Γ} = 𝔭 ∩ L^{Γ}).

Proof. The proof of B.4.5 (resp. B.4.10(3)) applies with trivial modifications (taking into account B.3.3(5)).

B.5.3. Proposition. If the field F_{φ}F_{Γ_{𝔭}} is not totally complex ( ⟺ 2 | m and F_{Γ_{𝔭}} is totally real), then V_{𝔭} is not quaternionic.

Proof. Let c ∈ G_{F_{φ}F_{Γ𝔭}} be the complex conjugation with respect to some real prime of F_{φ}F_{Γ_{𝔭}}. The element ρ_{𝔭}(c) ∈ D(𝔭_{Γ})^{×} ⊂ GL_{2}(L_{𝔭}) has two distinct eigenvalues ±1 ∈ Q_{p} ⊂ Z(D(𝔭_{Γ})), which implies that D(𝔭_{Γ}) is not a division algebra. B.5.4. As in Theorem A in the Introduction to this article, consider the following conditions on g ∈ G_{F}:

(A1) g acts trivially on F_{φ} ( ⟺ φ(g) = 1); (A2) ρ_{𝔭}(g) ∈ GL_{2}(L_{𝔭}) has eigenvalues λ_{1}, λ_{2} ∈ L_{𝔭}^{×} satisfying λ_{1}^{2} = 1 ≠ λ_{2}^{2}; (A3) g does not act trivially on K.

B.5.5. Proposition. (1) Any g ∈ G_{F} satisfying (A1) and (A2) acts trivially on F_{φ}F_{Γ_{𝔭}}.

(2) There exists g ∈ G_{F} satisfying (A1) and (A2) ⟺ V_{𝔭} is not quaternionic.

(3) If there exists g ∈ G_{F} satisfying (A1)–(A3), then V_{𝔭} is not quaternionic and K ⊄ F_{φ}F_{Γ_{𝔭}}.

(4) If V_{𝔭} is not quaternionic and K ⊄ F_{φ}F_{Γ_{𝔭}}F (V_{𝔭}), then there exists g ∈ G_{F} satisfying (A1)–(A3).

(5) If 2 | m and F_{Γ_{𝔭}} is totally real, then there exists g ∈ G_{F} satisfying (A1)–(A3).

(6) If 2 | m and if 𝔭_{Γ} splits completely in L^{+}/L^{Γ}, then there exists g ∈ G_{F} satisfying (A1)–(A3).

(7) For all but finitely many 𝔭 satisfying K ⊄ F_{φ}F_{Γ_{𝔭}} there exists g ∈ G_{F} satisfying (A1)–(A3).

(8) If K ⊄ F_{Γ}, then for all but finitely many 𝔭 there exists g ∈ G_{F} satisfying (A1)–(A3).

(9) If K ⊂ F_{Γ}, then the set of primes 𝔭_{Γ} of L^{Γ} for which there exists (for each 𝔭 | 𝔭_{Γ} in L) an element g ∈ G_{F} satisfying (A1)–(A3) has density equal to at least 1 − [F_{φ} : F]/|Γ| ≥ 1/2.

(10) If 2 | m and if V_{𝔭} has no non-trivial inner twist in the sense of B.3.5, then there exists g ∈ G_{F} satisfying (A1)–(A3).

Proof. (1) As g satisfies (A1) and (A2), we have ±λ_{2} = det(ρ_{𝔭}(g)) = χ_{cycl,p}(g)^{m−1}φ(g) = χ_{cycl,p}(g)^{m−1} ∈ Q_{p}. If (σ, χ_{σ}) ∈ Γ_{𝔭}, then ^{σ}V_{𝔭} ⟶^{∼} V_{𝔭} ⊗ χ_{σ}, which implies that 0 ≠ λ_{1} + λ_{2} = ^{σ}(λ_{1} + λ_{2}) = χ_{σ}(g)(λ_{1} + λ_{2}), hence χ_{σ}(g) = 1, as claimed.

(2) If V_{𝔭} is not quaternionic, then D(𝔭_{Γ}) ⟶^{∼} M_{2}((L^{Γ})_{𝔭_{Γ}}) and B.5.2 implies that there exists g ∈ G_{F} which

acts trivially on F_{φ}F_{Γ_{𝔭}} and for which ρ_{𝔭}(g) = ( \begin{matrix}1 & 0 \\ 0 & u \end{matrix} ) with u ∈ Z_{p}^{×}, u^{2} ≠ 1. Conversely, if there exists

g ∈ G_{F} satisfying (A1) and (A2), then g acts trivially on F_{φ}F_{Γ_{𝔭}} (by (1)), hence ρ_{𝔭}(g) is an element of D(𝔭_{Γ})^{×} ⊂ GL_{2}(L_{𝔭}) with two distinct eigenvalues lying in Q_{p} ⊂ Z(D(𝔭_{Γ})); thus D(𝔭_{Γ}) cannot be a division algebra. (3), (4) Both statements follow immediately from (1) and (2).

(5) Firstly, the assumptions imply that V_{𝔭} is not quaternionic, thanks to B.3.4(6-7) and B.5.3. Secondly, we can choose an isomorphism D(𝔭_{Γ}) ⟶^{∼} M_{2}((L^{Γ})_{𝔭_{Γ}}) in such a way that ρ_{𝔭} : G_{F_{φ}F_{Γ𝔭}} ⟶ GL_{2}((L^{Γ})_{𝔭_{Γ}})

will map the complex conjugation c with respect to some real place of F_{φ}F_{Γ_{𝔭}} to the matrix ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ).

According to B.5.2 there exists h ∈ G_{F} which acts trivially on F_{φ}F_{Γ_{𝔭}}K and for which ρ_{𝔭}(h) = ( \begin{matrix}1 & 0 \\ 0 & a \end{matrix} ),

a ∈ Z_{p}^{×}. The element g = ch ∈ G_{F} then satisfies (A1)–(A3).

(6) Combine (5) with B.3.4(6) and B.3.4(8).

(7) Thanks to (4) it is enough to show that there are only finitely many 𝔭 ∤ 6 satisfying K ⊄ F_{φ}F_{Γ_{𝔭}} and K ⊂ F_{φ}F_{Γ_{𝔭}}F (V_{𝔭}). Fix a G_{F} -stable O_{L,𝔭}-lattice T ⊂ V_{𝔭}; then F (V_{𝔭}) = ⋃_{n≥1} F (T/𝔭^{n}T), where F (T/𝔭^{n}T) is the fixed field of Ker(G_{F} ⟶ Aut(T/𝔭^{n}T)). As F (V_{𝔭})/F (T/𝔭T) is a pro-p-extension and p ≠ 2, the field K satisfies F_{φ}F_{Γ_{𝔭}} ⊊ F_{φ}F_{Γ_{𝔭}}K ⊆ F_{φ}F_{Γ_{𝔭}}F (T/𝔭T). According to B.5.2 (see also [Di, Prop. 0.1(ii)]), the Galois group G := Gal(F_{φ}F_{Γ_{𝔭}}F (T/𝔭T)/F_{φ}F_{Γ_{𝔭}}) is equal to {x ∈ GL_{2}(k(𝔭_{Γ})) | det(x) ∈ (F_{p}^{×})^{m−1}}, for all but finitely many 𝔭. As p > 3, the commutator of G contains SL_{2}(k(𝔭_{Γ})) [Gr, Thm. 1.9], which implies that F_{φ}F_{Γ_{𝔭}} ⊊ F_{φ}F_{Γ_{𝔭}}K ⊆ F_{φ}F_{Γ_{𝔭}}(μ_{p}), which can happen only for finitely many p (since there are only finitely many possible values of F_{Γ_{𝔭}}).

(8) This is an immediate consequence of (7).

(9) If 𝔭 is unramified in L/L^{Γ}, then Γ_{𝔭} ⊂ Γ is cyclic, generated by the Frobenius element σ(𝔭_{Γ}) = Fr_{geom}(𝔭_{Γ}). If K ⊂ F_{φ}F_{Γ_{𝔭}}, then the restriction of χ_{σ(𝔭_{Γ})} to Gal(F_{Γ}/F_{φ}) ⟶^{∼} (Z/2Z)^{r} must be equal to the quadratic character η_{F_{φ}K/F_{φ}} : Gal(F_{Γ}/F_{φ}) ⟶ {±1} associated to the quadratic extension F_{φ}K/F_{φ}, which implies that χ_{σ(𝔭_{Γ})} = χ_{0}φ^{i} for a fixed character χ_{0} of Gal(F_{Γ}/F) extending η_{F_{φ}K/F_{φ}} and some integer i ∈ Z. It follows that χ_{σ(𝔭_{Γ})}, hence σ(𝔭_{Γ}) ∈ Γ, has only [F_{φ} : F] possible values for which K ⊂ F_{φ}F_{Γ_{𝔭}}. The remaining |Γ| − [F_{φ} : F] values correspond to a set of primes 𝔭_{Γ} of L^{Γ} of density 1 − [F_{φ} : F]/|Γ| ≥ 1/2 for which K ⊄ F_{φ}F_{Γ_{𝔭}}, for each 𝔭 | 𝔭_{Γ}; we conclude by (7).

(10) Combine (5) with B.3.4(6).

B.6 The case of complex multiplication

Assume that π = I_{K(π)/F} (ψ) is as in B.2.7 and that 2 | m. Let K be a totally imaginary quadratic extension of F.

B.6.1. The Hecke character ψ : A_{K(π)}^{×}/K(π)^{×} ⟶ C^{×} can be written as

ψ(x) = ψ_{alg}(x) r(x_{∞})^{−1},

where

ψ_{alg} : A_{K(π)}^{×} ⟶ L̃^{×}

is an algebraic Hecke character with values in a finite extension L̃ ⊂ C of L and

r : R_{K(π)/Q}G_{m} ⟶ R_{L′/Q}G_{m}

(for a suitable totally imaginary quadratic extension L′ ⊂ L̃ of L^{+}; L′ = L if φ ≠ 1) is a morphism of algebraic tori satisfying

∀x ∈ K(π)^{×} ψ_{alg}(x) = r(x).

The infinity type of ψ is related to r by the formula

∀x_{∞} ∈ (K(π) ⊗ R)^{×} ψ_{∞}(x_{∞}) = r(x_{∞})^{−1}.

B.6.2. For each rational prime p the action of G_{K(π)F_{φ}} ⊂ G_{F} on V_{p} is given (via the reciprocity map) by the character

ψ^{(p)} : A_{K(π)F_{φ}}^{×}/(K(π)F_{φ})^{×} ⟶ (L′ ⊗ Q_{p})^{×} = ∏_{𝔭′|p} L_{𝔭′}^{′×} ⊂ GL_{2}(L^{+} ⊗ Q_{p}) = ∏_{𝔭_{+}|p} GL_{2}(L_{𝔭_{+}}^{+}),

ψ^{(p)}(x) = ψ_{alg}(N(x)) r(N(x_{p}))^{−1}, N = N_{K(π)F_{φ}/K(π)}.

Consequently, the action of G_{K(π)F_{φ}} on V_{𝔭} is given by the projection ψ^{(𝔭)} of ψ^{(p)} to ∏_{𝔭′|𝔭_{+}}, where 𝔭_{+} = 𝔭∩L^{+}:

ψ^{(𝔭)} : A_{K(π)F_{φ}}^{×}/(K(π)F_{φ})^{×} ⟶ (L′ ⊗_{L^{+}} L_{𝔭_{+}}^{+})^{×} = ∏_{𝔭′|𝔭_{+}} L_{𝔭′}^{′×} ⊂ GL_{2}(L_{𝔭_{+}}^{+}).

In particular, if 𝔭_{+} splits in L′/L^{+}, then Im(ψ^{(𝔭)}) is contained (after conjugation) in the split Cartan subgroup (L_{𝔭_{+}}^{+})^{×} × (L_{𝔭_{+}}^{+})^{×} ⊂ GL_{2}(L_{𝔭_{+}}^{+}). If 𝔭_{+} does not split in L′/L^{+}, then Im(ψ^{(𝔭)}) is contained in a non-split Cartan subgroup L_{𝔭′}^{′×} ⊂ GL_{2}(L_{𝔭_{+}}^{+}), where 𝔭′ is the unique prime of L′ above 𝔭_{+}.

B.6.3. Proposition. (1) For each rational prime p the Galois image ρ_{p}(G_{K(π)F_{φ}}) = Im(ψ^{(p)}) ⊂ (L′ ⊗ Q_{p})^{×} ⊂ GL_{2}(L ⊗ Q_{p}) contains r (an open subgroup of (K(π) ⊗ Q_{p})^{×}).

(2) For all but finitely many rational primes p the image Im(ψ^{(p)}) contains r ((O_{K(π)} ⊗ Z_{p})^{×}).

Proof. (1) As Ker(ψ_{alg}) is open, there exists an open subgroup of (K(π) ⊗ Q_{p})^{×} on which ψ^{(p)} = r^{−1}.

(2) For all but finitely many p the norm map N : (O_{K(π)F_{φ}} ⊗ Z_{p})^{×} ⟶ (O_{K(π)} ⊗ Z_{p})^{×} is surjective and (O_{K(π)} ⊗ Z_{p})^{×} ⊂ Ker(ψ_{alg}). B.6.4. Consider the following conditions on g ∈ G_{F} corresponding to the CM case of Theorem A:

(A1) g acts trivially on F_{φ} ( ⟺ φ(g) = 1); (A2) ρ_{𝔭}(g) ∈ GL_{2}(L_{𝔭}) has eigenvalues λ_{1}, λ_{2} ∈ L_{𝔭}^{×} satisfying λ_{1}^{2} = 1 ≠ λ_{2}^{n} (∀n ≥ 1); (A3) g does not act trivially on K.

B.6.5. Proposition. (1) Any g ∈ G_{F} satisfying (A1) and (A2) acts trivially on F_{φ}K(π).

(2) There exists g ∈ G_{F} satisfying (A1) and (A2) ⟺ 𝔭_{+} splits in L′/L^{+} and Im(ψ^{(𝔭)}) ⊂ (L_{𝔭_{+}}^{+})^{×} ×(L_{𝔭_{+}}^{+})^{×} contains an open subgroup of Z_{p}^{×} × Z_{p}^{×} ⟺ 𝔭_{+} splits in L′/L^{+} and ψ^{(𝔭)} = ψ_{1} ⊕ ψ_{2}, where the characters ψ_{i} : A_{K(π)F_{φ}}^{×}/(K(π)F_{φ})^{×} ⟶ (L_{𝔭_{+}}^{+})^{×} are such that ψ_{2}(Ker(ψ_{1})) is infinite.

(3) If there exists g ∈ G_{F} satisfying (A1)–(A3), then K ⊄ F_{φ}K(π) ( ⟺ K(π) ⊄ F_{φ}K), 𝔭_{+} splits in L′/L^{+} and Im(ψ^{(𝔭)}) contains an open subgroup of Z_{p}^{×} × Z_{p}^{×}.

(4) If K ⊄ F_{φ}K(π)F (V_{𝔭}) and if there exists g′ ∈ G_{F} satisfying (A1) and (A2), then there exists g ∈ G_{F} satisfying (A1)–(A3).

(5) There exists a constant b(ψ) depending only on ψ such that for each 𝔭 for which there exists g ∈ G_{F} satisfying (A1) and (A2) the image Im(ψ^{(𝔭)}) contains a subgroup of Z_{p}^{×} × Z_{p}^{×} of index ≤ b(ψ).

(6) If 𝔭_{+} splits in L′/L^{+} and L_{𝔭_{+}}^{+} ⟶^{∼} Q_{p}, then there exists g ∈ G_{F} satisfying (A1)–(A2).

(7) If K(π) ⊄ F_{φ}K, then K ⊄ F_{φ}K(π)F (V_{𝔭}) holds for all but finitely many 𝔭.

(8) If K(π) ⊄ F_{φ}K, then for all but finitely many 𝔭 for which there is g′ ∈ G_{F} satisfying (A1) and (A2) there exists g ∈ G_{F} satisfying (A1)–(A3).

Proof. (1) If g ∈ G_{F} satisfying (A1) and (A2) acts non-trivially on F_{φ}K(π), then ρ_{𝔭}(g) lies in the normaliser of a Cartan subgroup C ⊂ GL_{2}(L_{𝔭_{+}}^{+}) but not in C itself. This implies that Tr(ρ_{𝔭}(g)) = 0, in contradiction with (A2).

(2) If g ∈ G_{F} satisfies (A1) and (A2), then both eigenvalues of ρ_{𝔭}(g) belong to L_{𝔭_{+}}^{+}, since ±λ_{2} = det(ρ_{𝔭}(g)) = χ_{cycl,p}(g)^{m−1} ∈ Z_{p}. As a result, any Cartan subgroup of GL_{2}(L_{𝔭_{+}}^{+}) containg ρ_{𝔭}(g) must be split; thus 𝔭_{+} splits in L′/L^{+}. Furthermore, Im(ψ^{(𝔭)}) contains ρ_{𝔭}(g^{2})^{Z_{p}} = {1} × (λ_{2}^{2})^{Z_{p}} and ρ_{𝔭}(cg^{2}c^{−1})^{Z_{p}} = (λ_{2}^{2})^{Z_{p}} × {1} (where c ∈ G_{F_{φ}}, c ∉ G_{K(π)F_{φ}}), hence it contains an open subgroup of Z_{p}^{×} × Z_{p}^{×}. The remaining implications are easy (using the fact that ψ_{1}ψ_{2} = χ_{cycl,p}^{m−1}|_{G_{K(π)Fφ}} and ψ_{2}(h) = ψ_{1}(chc^{−1}) for all h ∈ G_{K(π)F_{φ}}).

(3) and (4) are immediate consequences of (1) and (2).

(5) This follows from [Ri 5, Thm. 2.4] combined with B.6.3(2).

(6) If not, then ψ^{(𝔭)} = ψ_{1} ⊕ ψ_{2} with ψ_{i} : G_{K(π)F_{φ}} ⟶ Z_{p}^{×} such that ψ_{2}(Ker(ψ_{1})) is finite. As ψ_{1}ψ_{2} = χ_{cycl,p}^{m−1}|_{G_{K(π)Fφ}}, it follows that there exist integers a_{1}, a_{2}, b (b ≠ 0) such that ψ_{i}^{b} = χ_{cycl,p}^{a_{i}}|_{G_{K(π)Fφ}}, a_{1} + a_{2} = b(m−1). On the other hand, ψ_{2}(h) = ψ_{1}(chc^{−1}) as in (2) above, which implies that a_{1} = a_{2}, hence ψ_{i}^{2}/χ_{cycl,p}^{m−1} is a character of finite order of G_{K(π)F_{φ}}, which is impossible, since 2 ∤ (m−1) and ψ_{i} is a potentially crystalline representation at each prime of K(π)F_{φ} above p.

(7) The equivalences [K ⊂ F_{φ}K(π) ⟺ F_{φ}K = F_{φ}K(π) ⟺ F_{φ}K ⊃ K(π)] imply that K ⊄ F_{φ}K(π). If p ≠ 2 and K ⊂ F_{φ}K(π)F (V_{𝔭}), then F_{φ}K(π) ⊊ F_{φ}K(π)K ⊂ F_{φ}K(π)F (T/𝔭T), as in the proof of B.5.5(7). The Galois group Gal(F_{φ}K(π)F (T/𝔭T)/F_{φ}K(π)) injects into k(𝔭_{+})^{×} × k(𝔭_{+})^{×} (k(𝔭_{+}) = O_{L^{+}}/𝔭_{+}), with the non-trivial element of Gal(F_{φ}K(π)/F_{φ}) interchanging the two factors. It follows that F_{φ}K(π)F (T/𝔭T)/F_{φ}K(π) has at most one quadratic subextension which is a Galois extension of F_{φ}, namely the one contained in F_{φ}(μ_{p})K(π). Consequently, F_{φ}K(π)K ⊂ F_{φ}(μ_{p})K(π), which is possible for only finitely many p.

(8) Combine (4) and (7).

B.6.6. Question. Is there an explicit criterion for deciding whether Im(ψ^{(𝔭)}) contains an open subgroup of Z_{p}^{×} × Z_{p}^{×}?

References

[AT] E. Artin, J. Tate, Class field theory, Second ed., Addison-Wesley, Redwood City, 1990. [BD 1] M. Bertolini, H. Darmon, A rigid analytic Gross-Zagier formula and arithmetic applications, Ann. of

Math. 146 (1997), 111–147. [BD 2] M. Bertolini, H. Darmon, Iwasawa’s Main Conjecture for Elliptic Curves over Anticyclotomic Z_{p}-

extensions, Ann. of Math. 162 (2005), 1–64.

[Bl] D. Blasius, Hilbert modular forms and the Ramanujan conjecture, in: Noncommutative geometry and number theory, Aspects Math. E37, Vieweg, Wiesbaden, 2006, pp. 35–56.

[BK] S. Bloch, K. Kato, L-functions and Tamagawa numbers of motives, in: The Grothendieck Festschrift I, Progress in Mathematics 86, Birkhäuser, Boston, Basel, Berlin, 1990, pp. 333–400.

[Bo] M. V. Bondarko, The generic fiber of finite group schemes; a “finite wild” criterion for good reduction of abelian varieties (Russian), Izv. Ross. Akad. Nauk Ser. Mat. 70 (2006), no. 4, 21–52. English translation in: Izv. Math. 70 (2006), no. 4, 661–691.

[BLR] B. Boston, H.W. Lenstra, K. Ribet, Quotients of group rings arising from two-dimensional representations, C. R. Acad. Sci. Paris Sér. I 312 (1991), 323–328.

[BC] J.-F. Boutot, H. Carayol, Uniformisation p-adique des courbes de Shimura: les théorèmes de Čerednik et de Drinfeld, in: Astérisque 196-197, Soc. Math. de France, Paris, 1991, pp. 45–158.

[BZ] J.-F. Boutot, T. Zink, The p-adic uniformization of Shimura curves, preprint, 1995.

[Br] C. Breuil, Groupes p-divisibles, groupes finis et modules filtrés, Ann. of Math. 152 (2000), 489–549.

[C1] H. Carayol, Sur les représentations p-adiques associées aux forms modulaires de Hilbert, Ann. Sci. E.N.S. (4) 19 (1986), 409–468.

[C2] H. Carayol, Sur la mauvaise réduction des courbes de Shimura, Compositio Math. 59 (1986), 151–230.

[CR] C.W. Curtis, I. Reiner, Methods of Representation Theory, Vol. I, Wiley, New York, 1981. [Če] I.V. Čerednik, Uniformization of algebraic curves by discrete arithmetic subgroups of P GL_{2}(k_{w}) with

compact quotient spaces, Mat. Sbornik 29 (1976), 55–78.

[Ch] W. Chi, Twists of central simple algebras and endomorphism algebras of some abelian varieties, Math. Ann. 276 (1987), 615–632.

[Cd] M. Chida, Selmer groups and the central values of L-functions for modular forms, in preparation. [CV 1] C. Cornut, V. Vatsal, Nontriviality of Rankin-Selberg L-functions and CM points, in: L-functions and

Galois representations (Durham, July 2004), LMS Lecture Note Series 320, Cambridge Univ. Press, 2007, pp. 121–186.

[CV 2] C. Cornut, V. Vatsal, CM points and quaternion algebras, Documenta Math. 10 (2005), 263-309.

[Di] M. Dimitrov, Galois representations modulo p and cohomology of Hilbert modular varieties, Ann. Sci. E.N.S. 38 (2005), 505–551.

[Dr] V.G. Drinfeld, Coverings of p-adic symmetric domains, Funkc. Anal. i Priloz. 10 (1976), 29–40.

[Ed] B. Edixhoven, Appendix to [BD 1].

[FC] G. Faltings, C.L. Chai, Degeneration of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 22, Springer, Berlin, 1990.

[Fo] J.-M. Fontaine, Sur certains types de représentations p-adiques du groupe de Galois d’un corps local; construction d’un anneau de Barsotti-Tate, Ann. of Math. (2) 115 (1982), 529–577.

[FH] S. Friedberg, J. Hoffstein, Nonvanishing theorems for automorphic L-functions on GL(2), Ann. of Math. (2) 142 (1995), 385–423.

[G1] A. Grothendieck, Le groupe de Brauer III: exemples et compléments, in: Dix exposés sur la cohomologie des schémas (A. Grothendieck, N.H. Kupier, eds.), Advanced Studies in Pure Mathematics 3, North Holland, Amsterdam, 1968, 88–188.

[G2] A. Grothendieck, Modèles de Néron et monodromie, in: SGA 7 I, Exposé IX, Lect. Notes in Math. 288, Springer, Berlin, 1972, pp. 313–523.

[Gr] L.C. Grove, Classical groups and geometric algebra, Graduate Studies in Mathematics 39, Amer. Math. Soc., Providence, 2002.

[H] B. Howard, Bipartite Euler systems, J. reine angew. Math. 597 (2006), 1–25.

[He] G. Henniart, Représentations l-adiques abéliennes, in: Séminaire de Théorie des Nombres de Paris 1980/81, Progress in Math. 22, (M.-J. Bertin, ed.), Birkhäuser, Boston, 1982, 107–126.

[J] F. Jarvis, Mazur’s principle for totally real fields of odd degree, Compositio Math. 116 (1999), 39–79.

[JL] B. Jordan, R. Livné, Local diophantine properties of Shimura curves, Math. Ann. 270 (1985), 235–248.

[K] A. Kurihara, On some examples of equations defining Shimura curves and the Mumford uniformization, J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 25 (1979), 277–301.

[Ki] M. Kisin, Crystalline representations and F -crystals, in: Algebraic geometry and number theory,

Progress in Math. 253, Birkhäuser, Boston, 2006, 459–496.

[KoLo] V.A. Kolyvagin, D.Yu. Logachev, Finiteness of the Shafarevich-Tate group and the group of rational points for some modular abelian varieties (Russian), Algebra i Analiz 1 (1989), No. 5, 171–196. English translation: Leningrad Math. J. 1 (1990), No. 5, 1229–1253.

[La] S. Lang, Algebraic groups over finite fields, Amer. J. Math. 78 (1956), 555–563.

[Li1] T. Liu, Potentially good reduction of Barsotti-Tate groups, J. Number Th. 126 (2007), no. 2, 155–184.

[Li2] T. Liu, Torsion p-adic Galois representations and a conjecture of Fontaine, Ann. Sci. E.N.S. 40 (2007), no. 4, 633–674.

[Li3] T. Liu, Lattices in filtered (φ, N)-modules, J. Inst. Math. Jussieu, to appear.

[L1] M. Longo, Euler systems obtained from congruences between Hilbert modular forms, Rend. Sem. Math. Univ. Padova 118 (2007), 1–34.

[L2] M. Longo, On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields, Ann. Inst. Fourier 56 (2006), 689–733.

[L3] M. Longo, Anticyclotomic Iwasawa Main Conjecture for Hilbert modular forms, preprint, 2007.

[LRV] M. Longo, V. Rotger, S. Vigni, Special values of L-functions and the arithmetic of Darmon points,

arXiv:1004.3424

[LV1] M. Longo, S. Vigni, On the vanishing of Selmer groups for elliptic curves over ring class fields, J. Number Theory 130 (2010), 128–163.

[LV2] M. Longo, S. Vigni, Quaternion algebras, Heegner points and the arithmetic of Hida families, Manuscr. Math., to appear.

[Ma] B. Mazur, Local flat duality, Amer. J. Math. 92 (1970), 343–361. [Mi 1] J.S. Milne, Étale cohomology, Princeton Univ. Press, Princeton, 1980. [Mi 2] J.S. Milne, Arithmetic duality theorems, Persp. in Math. 1, Academic Press, Boston, 1986. [Mi 3] J.S. Milne, Canonical models of (mixed) Shimura varieties and automorphic vector bundles, in: Auto-

morphic forms, Shimura varieties, and L-functions, Vol. I (Ann Arbor, MI, 1988), Perspect. Math. 10, Academic Press, Boston, 1990, pp. 283–414.

[Mo] F. Momose, On the ℓ-adic representations attached to modular forms, J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 28 (1981), 89–109.

[Mu] D. Mumford, An analytic construction of degenerating curves over complete local rings, Compositio Math. 24 (1972), 129–174.

[N1] J. Nekovář, The Euler system method for CM points on Shimura curves, in: L-functions and Galois representations (Durham, July 2004), LMS Lecture Note Series 320, Cambridge Univ. Press, 2007, pp. 471–547.

[N2] J. Nekovář, Selmer complexes, Astérisque 310 (2006), Soc. Math. de France, Paris.

[N3] J. Nekovář, On the parity of ranks of Selmer groups III, Doc. Math. 12 (2007), 243–274. Erratum: Doc. Math. 14 (2009), 191–194.

[NM] S. Bosch, W. Lütkebohmert, M. Raynaud, Néron Models, Ergebnisse der Mathematik und Ihre Grenzgebiete (3) 21, Springer, 1990.

[PW] R. Pollack, T. Weston, On anticyclotomic μ-invariants of modular forms, Compositio Math., to appear.

[Pr] D. Prasad, Some applications of seesaw duality to branching laws, Math. Ann. 304 (1996), 1–20.

[R] A. Rajaei, On the levels of mod ℓ Hilbert modular forms, J. reine angew. Math. 537 (2001), 33–65. [Ra 1] M. Raynaud, Spécialisation du foncteur de Picard, Publ. Math. I.H.E.S. 38 (1970), 27–76. [Ra 2] M. Raynaud, Schémas en groupes de type (p, …, p), Bull. S.M.F. 102 (1974), 241–280. [Ri 1] K. Ribet, On ℓ-adic representations attached to modular forms, Invent. Math. 28 (1975), 245–275. [Ri 2] K. Ribet, Galois action on division points of Abelian varieties with real multiplications, Amer. J. Math.

98 (1976), 751–804. [Ri 3] K. Ribet, Galois representations attached to eigenforms with nebentypus, in: Modular forms in one

variable V (eds. J.-P. Serre, D.B. Zagier), Lect. Notes in Math. 601, Springer, Berlin, 1977, pp. 17–52. [Ri 4] K. Ribet, Twists of modular forms and endomorphisms of abelian varieties, Math. Ann. 253 (1980),

43–62. [Ri 5] K. Ribet, Division fields of Abelian varieties with complex multiplication, Mémoires S.M.F. (2) 2 (1980),

75–94. [Ri 6] K. Ribet, On ℓ-adic representations attached to modular forms II, Glasgow Math. J. 27 (1985), 185–194. [Ri 7] K. Ribet, On modular representations of Gal(\overline{Q}/Q) arising from modular forms, Invent. Math. 100

(1990), 431–476.

[Ru] K. Rubin, Euler systems, Annals of Mathematics Studies 147, Princeton University Press, Princeton, 2000.

[S] C.-H. Sah, Automorphisms of finite groups, J. Algebra 10 (1968), 47–68.

[Sa] H. Saito, On Tunnell’s formula for characters of GL(2), Comp. Math. 85 (1993), 99–108.

[Sc] W. Scharlau, Quadratic and Hermitian Forms, Grund. math. Wiss. 270, Springer, Berlin, 1985.

[Sk] C. Skinner, A note on p-adic Galois representations attached to Hilbert modular forms, Doc. Math. 14 (2009), 241–258.

[Ta] J. Tate, p-divisible groups, in: Proceedings of a conference on local fields (Driebergen, 1966), Springer, Berlin, 1967, pp. 158–183.

[T1] R. Taylor, On Galois representations associated to Hilbert modular forms, Inv. Math. 98 (1989), 265–280.

[T2] R. Taylor, On Galois representations associated to Hilbert modular forms II, in: Elliptic Curves, Modular Forms and Fermat’s Last Theorem (J. Coates, S.T. Yau, eds.), International Press, 1997, pp. 333–340.

[TZ] Y. Tian, S.-W. Zhang, in preparation.

[Tu] J. Tunnell, Local ε-factors and characters of GL_{2}, Amer. J. Math. 105 (1983), 1277–1308.

[V] Y. Varshavsky, P -adic uniformization of unitary Shimura varieties II, J. Diff. Geom. 49 (1998), 75–113.

[VZ] A. Vasiu, T. Zink, Boundedness results for finite flat group schemes over discrete valuation rings of mixed characteristics, arXiv:0911.2474

[Vi] M.-F. Vigneras, Arithmétique des algèbres de quaternions, Lect. Notes in Math. 800, Springer, Berlin, 1980.

[W1] J.-L. Waldspurger, Quelques propriétés arithmétiques de certaines formes automorphes sur GL(2), Compositio Math. 54 (1985), 121–171.

[W2] J.-L. Waldspurger, Sur les valeurs de certains fonctions L automorphes en leur centre de symétrie, Compositio Math. 54 (1985), 173–242.

[W3] J.-L. Waldspurger, Correspondances de Shimura et quaternions, Forum Math. 3 (1991), 219–307.

[YZZ] X. Yuan, S.-W. Zhang, W. Zhang, Heights of CM points I. Gross-Zagier formula, preprint, 2008. [Z1] S.-W. Zhang, Heights of Heegner points on Shimura curves, Ann. of Math. (2) 153 (2001), 27–147.

[Z2] S.-W. Zhang, Arithmetic of Shimura curves and the Birch and Swinnerton-Dyer conjecture, Lectures at CRM in Barcelona (October 2009), SCIENCE CHINA Mathematics 53 (2010), no. 3, 573–592.

Université Pierre et Marie Curie (Paris 6) Institut de Mathématiques de Jussieu Théorie des Nombres, Case 247 4, place Jussieu F-75252 Paris cedex 05 FRANCE