The Euler system method for CM points on Shimura curves

Jan Nekovář

Let E be an elliptic curve over Q of conductor N; let π : X_{0}(N) ⟶ E be a modular parametrization sending the cusp ∞ to the origin. If K = Q(\sqrt{D}) is an imaginary quadratic field of discriminant D < 0 satisfying the “Heegner condition” of Birch [Bi] (“each prime dividing N splits in K/Q”), a choice of an ideal N ⊂ O_{K} satisfying N \overline{N} = NO_{K} and (N , \overline{N}) = (1) determines a Heegner point

x_{1} = [C/O_{K} ⟶ C/N^{−1}] ∈ X_{0}(N)(K[1]),

defined over the Hilbert class field K[1] of K. Set y_{1} = π(x_{1}) ∈ E(K[1]) and y = Tr_{K[1]/K}(y) ∈ E(K). A fundamental result of Kolyvagin [Ko, Thm. A] (explained in a simplified setting in [Gr 1]) states that

(K)  if y ∉ E(K)_{tors}, then the abelian groups E(K)/Zy, � (E/K) are finite.

More precisely, Kolyvagin showed that |� (E/K)| divides a certain multiple of [E(K) : Zy]^{2}. Kolyvagin’s result (K) has been generalized in several directions: Kolyvagin and Logačev [Ko-Lo 1] considered higherdimensional simple quotients of J_{0}(N) instead of elliptic curves, as well as [Ko-Lo 2] quotients of Jacobians of some Shimura curves over totally real number fields. Bertolini and Darmon [Be-Da] proved, under some additional assumptions, a “mod p” version of (K) over ring class fields K[c] ((c, N) = 1). Tian and Zhang ([Ti], [Ti-Zh]) considered a common generalization of [Ko-Lo 2] and [Be-Da]. The more refined “quantitative” results of Kolyvagin were generalized by Bertolini [Be] and Howard [Ho 1,2].

The main result of the present article is the following generalization of (K), which is one of the ingredients in the proof of the parity conjecture for Selmer groups of Hilbert modular forms ([Ne, Ch. 12]).

Theorem. Let F be a totally real number field, X a Shimura curve over F, and A an F -simple quotient of the Jacobian of X, which corresponds to (the Galois conjugacy class of) a Hilbert modular form f with trivial character (A is an abelian variety of GL_{2}-type and End_{F} (A) is an order in a totally real number field of degree equal to dim(A)). Let x be a CM point on X by a totally imaginary quadratic extension K of F and α a character of the Galois group Gal(K(x)/K). Assume that A does not acquire CM over any totally imaginary quadratic extension K′ of F contained in K(x)^{Ker(α)} (i.e., f is not a θ-series associated to a Hecke character of K′). If the α-component of the image y of x in A is not torsion, then the α-components of A(K(x))/End_{F} (A) · y and � (A/K(x)) are finite.

See Theorem 3.2 for a precise formulation. The proof follows the main lines of the argument of [Be-Da]; the novelty lies in the fact that under our minimalist assumptions there is no “geometric” formula for the action of the complex conjugation on x. Unlike all previous works in this direction, our version of the Euler system argument (in particular, 7.6.4 below) does not require such a formula.

Theorem 3.2 is expected to be related to arithmetic properties of the Rankin-Selberg L-function L(f_{K} ⊗ α^{−1}, s) at the central point s = 1 ([Zh 1,2], [Co-Va 1]). The assumption that f has trivial character ensures that f_{K} ⊗α^{−1} is self-dual in the sense that there is a “symmetric” functional equation relating L(f_{K} ⊗α^{−1}, s) (which turns out to be equal to L(f_{K} ⊗ α, s) in our case) and L(f_{K} ⊗ α^{−1}, 2 − s). As explained in [Gr 2] (see also [Co-Va 1]), it is possible for L(f_{K} ⊗ α^{−1}, s) to be self-dual even if f has non-trivial character ω, provided that ω is equal to the restriction of α (considered as a character of A_{K}^{∗}/K^{∗}) to A_{F}^{∗} /F^{∗}. It is an interesting question whether Theorem 3.2 can be generalized to such a more general self-dual setting.

A substantial part of the present article was written during the author’s visit at the COE programme at Nagoya University in spring 2005. The author is grateful to K. Fujiwara for inviting him to Nagoya. He would also like to thank H. Carayol, C. Cornut and M. Dimitrov for useful discussions, and the referee for helpful comments.

1. Quaternionic Shimura curves and their Jacobians In this section we recall basic properties of Shimura curves associated to quaternion algebras over totally real fields. The standard reference is [Ca 1] (see also [Zh 1,2]), but we follow the notations and sign conventions of [Co-Va 1,2] (with the parameter ϵ = 1). In particular, the reciprocity map of class field theory is normalized by making uniformizers correspond to geometric Frobenius elements. (1.1) Let F be a totally real number field of degree d = [F : Q]. Fix an archimedean prime τ_{1} of F and a finite set S_{B} of non-archimedean primes of F satisfying

|S_{B}| ≡ [F : Q] − 1 (mod 2).

Let B be the (unique) quaternion algebra over F ramified at the set

Ram(B) = {v|∞, v ≠ τ_{1}} ∪ S_{B}.

For each real embedding τ_{j} : F ↪ R (j = 1, …, d) put B_{τ_{j}} = B ⊗_{F,τ_{j}} R. We fix an isomorphism of R-algebras

B_{τ_{1}} = B ⊗_{F,τ_{1}} R ⟶^{∼} M_{2}(R)  (1.1.1)

(for j > 1, B_{τ_{j}} is isomorphic to the algebra of Hamilton quaternions, but there is no need to fix a specific isomorphism). Fix an embedding \overline{F} ↪ C extending τ_{1} : F ↪ R. (1.2) Let G be the algebraic group over Q satisfying G(A) = (B ⊗_{Q} A)^{∗}, for every commutative Q-algebra A. Denote by Z the centre of G and by nr : G(A) ⟶ (F ⊗_{Q} A)^{∗} the reduced norm. For j = 1, …, d, denote by G_{j} the algebraic group over R given by G_{j} = G ⊗_{F,τ_{j}} R; we have G_{R} ⟶^{∼} G_{1} × ⋯ × G_{d}. We denote, for any abelian group A, Â = A ⊗ Ẑ. Quaternionic Shimura curves are associated to the Shimura datum (G, X), where G is as above and X is the G(R)-conjugacy class of the morphism

h_{0} : S = Res_{C/R}(G_{m,C}) ⟶ G(R) = G_{1}(R) × ⋯ × G_{d}(R)
x + iy ↦ (( \begin{matrix}x & y \\ −y & x \end{matrix} ) , 1, …, 1) .

In concrete terms, X has a natural complex structure ([Mi 2, p. 320]) and the map

Ad(g)h_{0} = gh_{0}g^{−1} ↦ g(i) (g ∈ G(R))

(where g acts on C − R as in 1.3 below) defines a holomorphic isomorphism X ⟶^{∼} C − R. (1.3) The Shimura curves in question form a projective system {M_{H}} indexed by open compact subgroups H ⊂ G(A_{f}) = B̂^{∗}. Each curve M_{H} is smooth over Spec(F) and each transition map pr : M_{H} ⟶ M_{H′} (for H ⊂ H′) is finite and flat. The Riemann surface associated to M_{H}

M_{H}^{an} = (M_{H} ⊗_{F,τ_{1}} C) (C)

is identified with B^{∗}\(X × B̂^{∗}/H), where B^{∗} ⊂ B_{τ_{1}}^{∗} acts on X ⟶^{∼} C − R via the isomorphism B_{τ_{1}}^{∗} ⟶^{∼}

GL_{2}(R) induced by (1.1.1), and the standard action ( \begin{matrix}a & b \\ c & d \end{matrix} ) · z = \frac{az+b}{cz+d} of GL_{2}(R) on C − R. We denote

by [z, b] = [z, b]_{H} the point of M_{H}^{an} represented by a pair (z, b) ∈ X × B̂^{∗}. For H ⊂ H′, the transition map pr : M_{H} ⟶ M_{H′} corresponds to the natural projection

pr : M_{H}^{an} ⟶ M_{H′}^{an}, [z, b]_{H} ↦ [z, b]_{H′}.

In the “classical case” F = Q, S_{B} = ∅ ( ⟺ B = M_{2}(Q)), the curves M_{H} are the usual modular curves; we denote by

M_{H}^{∗} = M_{H} ∪ {cusps}_{H}

their standard smooth compactifications; they form again a projective system with finite flat transition maps. If B ≠ M_{2}(Q), then the curves M_{H} are proper over Spec(F); we put M_{H}^{∗} = M_{H}.

(1.4) The group G(A_{f}) = B̂^{∗} acts on the right on the projective systems {M_{H}} and {M_{H}^{∗}}. More precisely, the right multiplication by g ∈ B̂^{∗} induces an F -isomorphism [·g] : M_{H} ⟶^{∼} M_{g^{−1}Hg}; the corresponding holomorphic isomorphism

[·g] : M_{H}^{an} = B^{∗}\(X × B̂^{∗}/H) ⟶^{∼} M_{g^{−1}Hg}^{an} = B^{∗}\(X × B̂^{∗}/g^{−1}Hg)

is given by the formula [x, b] ↦ [x, bg].

In particular, the centre F̂^{∗} = Z(A_{f}) ⊂ B̂^{∗} acts on M_{H} (and M_{H}^{∗}) via its finite quotient F̂^{∗}/(F̂^{∗} ∩ H) = F̂^{∗}/(Ô_{F}^{∗} ∩ H); we denote by N_{H} ⊂ N_{H}^{∗} the corresponding quotient curves, which are again smooth over Spec(F) [Ka-Ma, p. 508] (the curve M_{H} (resp., N_{H}) was denoted by Y (resp., X) in [Zh 1]). The Riemann surface associated to N_{H}

N_{H}^{an} = (N_{H} ⊗_{F,τ_{1}} C) (C)

is identified with B^{∗}\(X ×B̂^{∗}/HF̂^{∗}). In particular, N_{H} (and N_{H}^{∗}) depend only on the subgroup HF̂^{∗} ⊂ B̂^{∗}. In abstract terms, the projective system {N_{H}} is associated to the Shimura datum (G/Z, X).

Note that, for each open compact subgroup H ⊂ B̂^{∗}, there exists an O_{F} -order R ⊂ B such that R̂^{∗} ⊂ HÔ_{F}^{∗}. In particular, R̂^{∗}F̂^{∗} ⊂ HF̂^{∗}, hence the curves N_{R^{∗}} (where R runs through all O_{F} -orders of B) form a co-final subsystem of {N_{H}}. (1.5) Each curve M_{H}^{∗} (hence also N_{H}^{∗}) is irreducible ([Ca 1, 1.3], [Co-Va 1, 3.2]), but not necessarily geometrically irreducible. Denote by F (M_{H}) (resp., F (N_{H})) the algebraic closure of F in the function field of M_{H}^{∗} (resp., of N_{H}^{∗}). The reciprocity law [De, 3.9], [Mi 2, II.5.1] (with the sign corrected, [Mi 3, 1.10]) implies that the fields F (N_{H}) ⊆ F (M_{H}) are abelian over F and that the reciprocity map

rec_{F} : F̂^{∗} ⟶ Gal(F^{ab}/F)

induces isomorphisms (depending on a choice of embedding F (M_{H}) ↪ F^{ab})

F_{+}^{∗}\F̂^{∗}/nr(H) ⟶^{∼} Gal(F (M_{H})/F), F_{+}^{∗}\F̂^{∗}/F̂^{∗2}nr(H) ⟶^{∼} Gal(F (N_{H})/F),

where F_{+}^{∗} = Ker(F^{∗} ⟶ π_{0}((F ⊗_{Q} R)^{∗})) denotes the subgroup of totally positive elements of F^{∗}. (1.6) Differentials and automorphic forms of “weight 2” ([Co-Va 1, 3.6]) Denote by Ω^{an} the sheaf of holomorphic 1-forms on any given Riemann surface. As in the classical case B = M_{2}(Q), the space of global holomorphic 1-forms

\underrightarrow{lim}_{H} Γ((M_{H}^{∗})^{an}, Ω^{an}),

equipped with a canonical left action of B̂^{∗} given by

[·g]^{∗} : Γ((M_{g^{−1}Hg}^{∗})^{an}, Ω^{an}) ⟶ Γ((M_{H}^{∗})^{an}, Ω^{an}),  (1.6.1)

can be naturally identified with a certain space of automorphic forms on G(A) = G(R) × B̂^{∗} = B_{A}^{∗}.

More precisely, for each compact open subgroup H ⊂ B̂^{∗} we have^{(1)}

G(A) = ∐_{α∈C} G(Q) · (G(R)^{+} × αH),

(1) As the case B = M_{2}(Q) is well-known, we discuss only the case B ≠ M_{2}(Q), when M_{H}^{∗} = M_{H}.

where

G(R)^{+} = GL_{2}^{+}(R) × G_{2}(R) × ⋯ × G_{d}(R), G(Q)^{+} = G(Q) ∩ G(R)^{+} = Ker (B^{∗}⟶^{nr}F^{∗} ⟶ F^{∗}/F_{+}^{∗})

and C ⊂ B̂^{∗} is a fixed (finite) set of representatives of the double cosets G(Q)^{+}\G(A_{f})/H. Put, for each α ∈ C,

Γ_{α} = αHα^{−1} ∩ G(Q)^{+} ⊂ G(R)^{+}, \overline{Γ}_{α} = Im (Γ_{α} ⟶ P GL_{2}^{+}(R)) .

Writing H = {z ∈ C | Im(z) > 0} for the complex upper half plane, the map

\begin{matrix}∐_{α∈C} \overline{Γ}_{α}\H & ⟶^{∼} M_{H}^{an} = (M_{H}^{∗})^{an} \\ \overline{Γ}_{α} · g(i) & ↦ [gh_{0}g^{−1}, α]_{H} \end{matrix} (g ∈ GL_{2}^{+}(R))

is a holomorphic isomorphism. Using the standard notation

j(g, z) = det(g)^{−1/2}(cz + d),  g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}^{+}(R),  z ∈ H,

the formula

f_{ω}(γ(g^{+}, αh)) = j(g_{1}^{+})^{−2}f_{α}(g_{1}^{+}(i)), (γ ∈ G(Q), g^{+} = (g_{1}^{+}, g_{2}, …, g_{d}) ∈ G(R)^{+}, h ∈ H)

defines a bijection

Γ((M_{H}^{∗})^{an}, Ω^{an}) ⟶^{∼} ⨁_{α∈C} Γ(\overline{Γ}_{α}\H, Ω^{an}) ⟶^{∼}  S_{2}^{H}
ω  ↦  (f_{α}(z) dz)_{α∈C}  ↦ (f_{ω} : G(A) ⟶ C) ,

(1.6.2)

where S_{2}^{H} denotes the space of functions f : G(A) ⟶ C satisfying

f (γ g z_{∞} (( \begin{matrix}cos(θ) & − sin(θ) \\ sin(θ) & cos(θ) \end{matrix} ) , g_{2}, …, g_{d}) h) = e^{−2iθ}f(g) \begin{matrix}( & γ ∈ G(Q), g ∈ G(A), z_{∞} ∈ Z(R), \\ θ ∈ R, g_{j} ∈ G_{j}(R) (j > 1), h ∈ H) \end{matrix}

and such that, for each g ∈ G(A), the function

x + iy ↦ \frac{1}{y} f (g (( \begin{matrix}y & x \\ 0 & 1 \end{matrix} ) , 1, …, 1))

is holomorphic on H (in the case B = M_{2}(Q) one has to impose an additional cuspidality condition on the function f; this is automatic if B ≠ M_{2}(Q)).

Passing to the inductive limit with respect to all open compact subgroups of B̂^{∗}, we obtain a B̂^{∗}-equivariant bijection

\underrightarrow{lim}_{H} Γ((M_{H}^{∗})^{an}, Ω^{an}) ⟶^{∼} S_{2} := ⋃_{H} S_{2}^{H},  (1.6.3)

with B̂^{∗} acting (on the left) on S_{2} by right translations: (b · f)(g) = f(gb) (which means that S_{2}^{H} coincides, as the notation suggests, with the space of H-invariant elements of S_{2}).

Note that the centre Z(A_{f}) = F̂^{∗} acts on S_{2}^{H} via its finite quotient F̂^{∗}/(F̂^{∗} ∩ H) = F̂^{∗}/(Ô_{F}^{∗} ∩ H), hence (1.6.2) induces bijections

Γ((N_{H}^{∗})^{an}, Ω^{an}) ⟶^{∼} S_{2}^{H_{F^{∗}}} = {f ∈ S_{2}^{H} | (∀g ∈ G(A)) (∀z ∈ Z(A)) f(gz) = f(g)}.  (1.6.4)

(1.7) Automorphic representations. The multiplicity one theorem for automorphic forms on B_{A}^{∗} ([Ge], VI.2) implies that the space S_{2} (as an admissible smooth representation of B̂^{∗}) decomposes as a countable direct sum

S_{2} ⟶^{∼} ⨁_{π=σ_{2}⊗π_{f}} π_{f},  (1.7.1)

where π runs through all irreducible cuspidal (this is automatic if B ≠ M_{2}(Q)) unitary automorphic representations of B_{A}^{∗} whose archimedean component π_{∞} is isomorphic to

σ_{2} = ( \begin{matrix}the weight 2 holomorphic discrete \\ series representation of G_{1}(R) \end{matrix} ) ⊗ ( \begin{matrix}the trivial representation of \\ G_{2}(R) × ⋯ × G_{d}(R) \end{matrix} ) .

In particular, (1.6.4) induces an isomorphism of admissible smooth representations of B̂^{∗}/F̂^{∗}

\underrightarrow{lim}_{H} Γ((N_{H}^{∗})^{an}, Ω^{an}) ⟶^{∼} S_{F^{∗}} ; 2 ⟶^{∼} ⨁_{{π=σ_{2}⊗π_{f} \atop ωπ =1}} π_{f},  (1.7.2)

where ω_{π} : Z(A) ⟶ C^{∗} denotes the central character of π. (1.8) The Hecke algebra. For each non-archimedean prime v of F, let dg_{v} be the (bi-invariant) Haar measure on B_{v}^{∗} normalized so that ∫_{H_{v}} dg_{v} = 1 for one (hence for every) maximal compact subgroup H_{v} ⊂ B_{v}^{∗}. The product dg = ∏ dg_{v} then defines a bi-invariant Haar measure on B̂^{∗}. We denote by vol(X) = ∫_{X} dg the corresponding volume. The Hecke algebra

H(B̂^{∗}) = C_{c}^{∞}(B̂^{∗}, C) = {α : B̂^{∗} ⟶ C | α is locally constant, with compact support}

with the convolution product

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

is isomorphic to the direct limit (over all open compact subgroups H ⊂ B̂^{∗}) of the double coset algebras

\begin{matrix}\underrightarrow{lim}_{H} C[H\B̂^{∗}/H] & ⟶^{∼} H(B̂^{∗}) \\ HbH & ↦ vol(H)^{−1} 1_{HbH} \end{matrix}

The product structure on C[H\B̂^{∗}/H] is as in [Sh 2, 3.1], [Miy, (2.7.3)] (hence is the opposite to that in [Co-Va 1, 3.4, 3.11]).

Any smooth representation (π, V) of B̂^{∗} gives rise to a left action of H(B̂^{∗}) on V, given by the formula

α · v = ∫_{B^{∗}} α(h)π(h)v dh (α ∈ H(B̂^{∗}), v ∈ V).

In particular, α ∈ H(B̂^{∗}) acts on S_{2} by

(α · f)(g) = ∫_{B^{∗}} α(h)f(gh) dh.

Note that, for any open compact subgroups H, H′ ⊂ B̂^{∗} and b ∈ B̂^{∗}, the action of 1_{HbH′} maps S_{2}^{H′} to S_{2}^{H}.

(1.9) Hecke correspondences. Let H, H′ be open compact subgroups of B̂^{∗} and g ∈ B̂^{∗}. The diagram

\xymatrix{M_{H∩gH′g^{−1}}^{∗} \ar[r]^{[·g]} \ar[d]_{pr} & M_{g^{−1}Hg∩H′}^{∗} \ar[d]_{_{pr_{′}}} \\ M_{H}^{∗} \ar@{-->}[r]^{[HgH′]} & M_{H′}^{∗}}  (1.9.1)

defines a multivalued map (a “Hecke correspondence”)

\xymatrix{[HgH′] : M_{H}^{∗} \ar@{-->}[r] & M_{H′}^{∗}.}

On non-cuspidal complex points,

\xymatrix{[HgH′] : M_{H}^{an} = B^{∗}\(X × B̂^{∗}/H) \ar@{-->}[r] & M_{H′}^{an} = B^{∗}\(X × B̂^{∗}/H′)}

is given by the formula

[x, b]_{H} ↦ ∑_{i}[x, bg_{i}]_{H′},  HgH′F^{∗} = ∐_{i} g_{i}H′F^{∗}.  (1.9.2)

The degree of [HgH′] is the degree of pr, namely (HF^{∗} : (H∩gH′g^{−1})F^{∗}) (which is equal to (H : H∩gH′g^{−1}), if O_{F}^{∗} ⊂ H ∩ H′). If z ∈ Z(A_{f}) = F̂^{∗}, then we have

[HzH] = [·z] : M_{H}^{∗} ⟶ M_{H}^{∗},  [H′zH′] = [·z] : M_{H′}^{∗} ⟶ M_{H′}^{∗},  [·z] ∘ [HgH′] = [HgH′] ∘ [·z].

For any “reasonable” cohomology theory H_{?}^{∗}, the correspondences [HgH′] induce maps

[HgH′]^{∗} : H_{?}^{q}(M_{H′}^{∗}) ⟶ H_{?}^{q}(M_{H}^{∗}), [HgH′]_{∗} : H_{?}^{q}(M_{H}^{∗}) ⟶ H_{?}^{q}(M_{H′}^{∗}).

As the transpose of the diagram (1.9.1) is given by

\xymatrix{M_{g^{−1}Hg∩H′}^{∗} \ar[r]^{[·g^{−1}]} \ar[d]_{_{pr_{′}}} & M_{H∩gH′g^{−1}}^{∗} \ar[d]_{pr} \\ M_{H′}^{∗} \ar@{-->}[r]^{[H′g^{−1}H]} & M_{H}^{∗},}

we have

[HgH′]_{∗} = [H′g^{−1}H]^{∗},  [HgH′]^{∗} = [H′g^{−1}H]_{∗}  (1.9.3)

If we replace H, H′ by HF̂^{∗}, H′F̂^{∗}, then we obtain Hecke correspondences

\xymatrix{N_{H}^{∗} \ar@{-->}[r] & N_{H′}^{∗}}

with similar properties. (1.10) Action on holomorphic differentials. In the situation of (1.9.1), we deduce from (1.9.2) and the B̂^{∗}-equivariance of (1.6.3) that the map

[HgH′]^{∗} : Γ((M_{H′}^{∗})^{an}, Ω^{an}) ⟶ Γ((M_{H}^{∗})^{an}, Ω^{an})

corresponds to the action of 1_{HgH′}, suitably normalized:

\begin{matrix}S_{2}^{H′} & ⟶ S_{2}^{H} \\ f′ & ↦ vol(H′)^{−1} 1_{HgH′} · f′. \end{matrix}

If z ∈ Z(A_{f}) = F̂^{∗} and H = H′, then

[HzH′]^{∗} = [·z]^{∗} : Γ((M_{H}^{∗})^{an}, Ω^{an}) ⟶ Γ((M_{H}^{∗})^{an}, Ω^{an})

corresponds to the action of z on S_{2}^{H}, which we denote by [z]. (1.11) Adjoints. Up to a scalar multiple, the hermitian scalar product

⟨ω_{1}, ω_{2}⟩ = \frac{i}{2} ∫_{(M_{H}^{∗})^{an}} ω_{1} ∧ \overline{ω_{2}}  (ω_{1}, ω_{2} ∈ Γ((M_{H}^{∗})^{an}, Ω^{an}))  (1.11.1)

corresponds, via the isomorphism (1.6.2), to the hermitian scalar product

⟨f_{1}, f_{2}⟩ =  ∫_{G(Q)\G(A)/Z_{∞}K_{∞}^{+}} f_{1}(g)\overline{f_{2}(g)} dg  (f_{1}, f_{2} ∈ S_{2}^{H}),

where

Z_{∞}K_{∞}^{+} = \{(( \begin{matrix}z & 0 \\ 0 & z \end{matrix} ) ( \begin{matrix}cos(θ) & − sin(θ) \\ sin(θ) & cos(θ) \end{matrix} ) , g_{2}, …, g_{d}) | z ∈ R^{∗}, θ ∈ R, g_{j} ∈ G_{j}(R)\} ⊂ G(R)^{+}.

For α ∈ H(B̂^{∗}) and f_{1}, f_{2} ∈ S_{2}, we have

⟨α · f_{1}, f_{2}⟩ = ⟨f_{1},^{t}α · f_{2}⟩,  ^{t}α(g) = \overline{α(g^{−1})}  (1.11.2)

(i.e., ^{t}α is the adjoint of α). For example,

t1_{HgH′} = 1_{H′g^{−1}H}.  (1.11.3)

In particular, if H = H′ = H_{v}H^{v}, where v ∉ S_{B}, H_{v} ⊂ B_{v}^{∗} is a maximal compact subgroup and b_{v} ∈ B_{v}^{∗} ⊂ B̂^{∗}, then

H_{v}b_{v}^{−1}H_{v} = nr(b_{v})^{−1} H_{v}b_{v}H_{v},

as H_{v} is isomorphic to GL_{2}(O_{F,v}) and

GL_{2}(O_{F,v}) ( \begin{matrix}t_{1}^{−1} & 0 \\ 0 & t_{2}^{−1} \end{matrix} ) GL_{2}(O_{F,v}) = (t_{1}t_{2})^{−1}GL_{2}(O_{F,v}) ( \begin{matrix}t_{2} & 0 \\ 0 & t_{1} \end{matrix} ) GL_{2}(O_{F,v}) =

= (t_{1}t_{2})^{−1}GL_{2}(O_{F,v}) ( \begin{matrix}t_{1} & 0 \\ 0 & t_{2} \end{matrix} ) GL_{2}(O_{F,v}),

hence

t1_{Hb_{v}H} = [nr(b_{v})^{−1}] ∘ 1_{Hb_{v}H} = 1_{Hb_{v}H} ∘ [nr(b_{v})^{−1}]  (1.11.4)

in this case. (1.12) Hecke operators and cup products. Fix an open compact subgroup H ⊂ B̂^{∗}. There exists a finite set S ⊃ S_{B} of non-archimedean primes of F such that H = H_{S}H^{S}, where

H_{S} ⊂ ∏_{v∈S} B_{v}^{∗},  H^{S} = ∏_{v∉S} H_{v} = a maximal compact subgroup of G(A_{f})^{S} = ∏′_{v∉S} B_{v}^{∗}.  (1.12.1)

More precisely, for each non-archimedean prime v ∉ S there exists a (unique) maximal O_{F,v}-order R(v) ⊂ B_{v} such that H_{v} = R(v)^{∗} (in other words, there exists an isomorphism of groups B_{v}^{∗} ⟶^{∼} GL_{2}(F_{v}) inducing an isomorphism H_{v} ⟶^{∼} GL_{2}(O_{F,v})). For such v ∉ S, we define the Hecke correspondence T (v) as

T (v) = [Hb_{v}H] : \xymatrix{M_{H}^{∗} \ar@{-->}[r] & M_{H}^{∗}} ,  (1.12.2)

for any element b_{v} ∈ R(v)∩B_{v}^{∗} ⊂ B_{v}^{∗} ⊂ B̂^{∗} satisfying ord_{v}(nr(b_{v})) = 1. For example, we can fix a uniformizer ϖ_{v} of O_{F,v} and take the element b_{v} which corresponds, under some isomorphism H_{v} ⟶^{∼} GL_{2}(O_{F,v}), to the matrix

( \begin{matrix}ϖ_{v} & 0 \\ 0 & 1 \end{matrix} ) .

The Hecke operator T (v) is defined as the induced map

T (v) := [Hb_{v}H]^{∗} : Γ((M_{H}^{∗})^{an}, Ω^{an}) ⟶ Γ((M_{H}^{∗})^{an}, Ω^{an}).

According to 1.10, the action of T (v) on Γ((M_{H}^{∗})^{an}, Ω^{an}) ⟶^{∼} S_{2}^{H} is given by

T (v) = [Hb_{v}H]^{∗} corresponds to vol(H)^{−1} 1_{Hb_{v}H}.

Combining 1.10 with (1.11.4), we see that the adjoint ^{t}T (v) of T (v) with respect to the hermitian scalar product (1.11.1) corresponds to

vol(H)^{−1} (^{t}1_{Hb_{v}H}) = vol(H)^{−1} 1_{Hb_{v}^{−1}H} = vol(H)^{−1} [ϖ_{v}^{−1}] ∘ 1_{Hb_{v}H},

hence

tT (v) = [Hb_{v}^{−1}H]^{∗} = [ϖ_{v}^{−1}] ∘ T (v) = T (v) ∘ [ϖ_{v}^{−1}]  (1.12.3)

Fix a prime number ℓ and define

W := H^{1}((M_{H}^{∗})^{an}, Q), W_{C} := Γ((M_{H}^{∗})^{an}, Ω^{an}), W_{ℓ} := H_{et}^{1}(M_{H}^{∗} ⊗_{F} \overline{F} , Q_{ℓ}), W_{dR} := H_{dR}^{1}(M_{H}^{∗}/F).

The comparison isomorphisms

W ⊗_{Q} C ⟶^{∼} W_{C} ⊕ \overline{W_{C}},  W ⊗_{Q} Q_{ℓ} ⟶^{∼} W_{ℓ}  (1.12.4)

are B̂^{∗}-equivariant, hence Q[H\B̂^{∗}/H]-equivariant (the second isomorphism is defined by using the embedding \overline{F} ↪ C that was fixed in 1.1). Up to a scalar multiple, the hermitian pairing (1.11.1)

W_{C} ⊗_{C} \overline{W_{C}} ⟶ C

is equal to the C-linear extension of the cup product

( , ) : W ⊗_{Q} W ⟶^{∪}H^{2}((M_{H}^{∗})^{an}, Q)⟶^{T r}Q(−1)

(note that W_{C} and \overline{W_{C}} are isotropic subspaces of W ⊗_{Q} C with respect to ( , )⊗_{Q} C). Similarly, ( , )⊗_{Q} Q_{ℓ} is identified with the étale cup product

( , )_{ℓ} : W_{ℓ} ⊗_{Q_{ℓ}} W_{ℓ}⟶^{∪}H_{et}^{2}(M_{H}^{∗} ⊗_{F} \overline{F} , Q_{ℓ})⟶^{T r}Q_{ℓ}(−1).

Taking into account (1.12.3), this implies that the adjoint of T (v) := [Hb_{v}H]^{∗} acting on W (resp., on W_{ℓ}) with respect to the pairing ( , ) (resp., to ( , )_{ℓ}) is equal to [ϖ_{v}^{−1}] ∘ T (v) = T (v) ∘ [ϖ_{v}^{−1}].

(1.13) Integral models. Integral models of the curves M_{H} were studied, in an increasing level of generality, by Deligne-Rapoport [De-Ra], Katz-Mazur [Ka-Ma] and Carayol [Ca 1]. Fix a non-archimedean prime v ∉ S_{B} of F; denote by O(v) the localization of O_{F} at v. Assume that H ⊂ B̂^{∗} factorizes as H = H_{v}H^{v}, where H_{v} (resp., H^{v}) is an open compact subgroup of B_{v}^{∗} (resp., of G(A_{f})^{v} = {g ∈ B̂^{∗} | g_{v} = 1}). In the classical case B = M_{2}(Q), the curve M_{H} is a coarse moduli space of elliptic curves with an H-level structure, for schemes over Spec(Q). Katz-Mazur [Ka-Ma] extended this moduli problem to schemes over Spec(O(v)), using the theory of Drinfeld bases. They obtained a regular model M_{H} ⟶ Spec(O(v)) of M_{H}, which they compactified - somewhat artificially - to a regular model M_{H}^{∗} of M_{H}^{∗}, proper over Spec(O(v)). Similar techniques apply in the case F = Q, B ≠ M_{2}(Q), when M_{H} is a coarse moduli space of abelian surfaces with quaternionic multiplication with a suitable level structure [Bu]; one obtains proper regular models M_{H} = M_{H}^{∗} ⟶ Spec(O(v)) of M_{H} = M_{H}^{∗}. If F ≠ Q, then M_{H} has no longer a moduli interpretation, but it can be related to a unitary Shimura curve parametrizing a certain class of abelian varieties of dimension 4[F : Q]. Carayol [Ca 1] extended this moduli problem to schemes over Spec(O(v)) and described, in the case when H^{v} is sufficiently small (this condition depends on H_{v}), a regular model M_{H} = M_{H}^{∗} of M_{H} = M_{H}^{∗}, proper over Spec(O(v)). If H_{v} is a maximal open compact subgroup of B_{v}^{∗}, then M_{H}^{∗} is smooth over Spec(O(v)). As explained in [Co-Va 2, 3.1.3], Carayol’s results ^{(1)} can be used to construct regular models M_{H}^{∗} of M_{H}^{∗}, proper over Spec(O(v)), even if H^{v} is not sufficiently small: fix a sufficiently small open compact normal subgroup H^{′v} of H^{v} and put H′ = H_{v}H^{′v} ⊂ B̂^{∗}. The right action of the finite group H/H′ on M_{H′}^{∗} extends naturally to an O(v)-linear action on M_{H′}^{∗}; one defines M_{H}^{∗} as the quotient of M_{H′}^{∗} by this action of H/H′. The model M_{H}^{∗} is regular, proper over Spec(O(v)), and independent on the choice of H^{′v}. Taking an additional quotient of M_{H}^{∗} by the action of the finite abelian group F̂^{∗}/(F̂^{∗} ∩ H), we obtain a regular model N_{H}^{∗} of N_{H}^{∗}, proper over Spec(O(v)). If H_{v} is a maximal open compact subgroup of B_{v}^{∗}, then M_{H}^{∗} and N_{H}^{∗} are again smooth over Spec(O(v)) (by [Ka-Ma, p. 508]). (1.14) The Eichler-Shimura congruence relation [Sh 1, Thm. 2.2.3], [Oh, Thm. 3.4.3], [Ca 1, 10.3]. In the situation of (1.12.1), fix v ∉ S and a prime number ℓ different from the residue characteristic of v. As H_{v} is a maximal compact subgroup of B_{v}^{∗}, the model M_{H}^{∗} is proper and smooth over Spec(O(v)), hence the proper and smooth base change theorems yield a canonical isomorphism

W_{ℓ} ⟶^{∼} H_{et}^{1}(M_{H}^{∗∘} ⊗_{κ(v)} \overline{κ(v)}, Q_{ℓ}) =: W_{ℓ}^{∘},

where κ(v) is the residue field of v and M_{H}^{∗∘} = M_{H}^{∗} ⊗_{O(v)} κ(v) the special fibre of M_{H}^{∗}. In particular, the natural action of G_{F} = Gal(\overline{F} /F) on W_{ℓ} is unramified at v. As the diagram (1.9.1) for g = b_{v} from (1.12.2) naturally extends to the integral models, the graph of the Hecke correspondence T (v) (which is a divisor on M_{H}^{∗} ×_{F} M_{H}^{∗}) naturally extends to a divisor on M_{H}^{∗} ×_{O(v)} M_{H}^{∗}; denote by T (v)^{∘} ⊂ M_{H}^{∗∘} ×_{κ(v)} M_{H}^{∗∘} its special fibre. The generalized Eichler-Shimura congruence relation states that

T (v)^{∘} = Γ_{φ} + Γ_{[ϖ_{v}·]} ∘^{t}Γ_{φ},  (1.14.1)

where Γ_{φ} denotes the graph of the Frobenius morphism φ = φ_{v} : M_{H}^{∗∘} ⟶ M_{H}^{∗∘} and ^{t}Γ_{φ} its transpose. As the contravariant action of Γ_{φ} (resp., of ^{t}Γ_{φ}) on W_{ℓ}^{∘} coincides with the Galois action of F = Fr(v)_{geom} (resp., with the action of ^{t}F = (Nv)F^{−1}), it follows that

(1 − XF)(1 − X[ϖ_{v}](Nv)F^{−1}) = 1 − XT (v) + X^{2}(Nv)[ϖ_{v}] ∈ End_{Q_{ℓ}}(W_{ℓ}^{∘})[X].

As [ϖ_{v}](Nv)F^{−1} is the adjoint of F with respect to the pairing ( , )_{ℓ}, it follows from (1.12.4) and the compatibility of the various pairings discussed in 1.12 that

det(1 − XF | W_{ℓ}^{∘}) = det(1 − XT (v) + X^{2}(Nv)[ϖ_{v}] | W_{C}).  (1.14.2)

(1) Note that Carayol [Ca 1,2] uses different sign conventions. This is discussed in detail in [Co-Va 2, 3.3.1].

For the curve N_{H}^{∗} and its model N_{H}^{∗}, the congruence relation (1.14.1) simplifies ([Zh 1, Prop. 1.4.10]) to

T (v)^{∘} = Γ_{φ} +^{t}Γ_{φ}.  (1.14.3)

(1.15) The L-function of M_{H}^{∗}. Combining the discussion from 1.14 (the notations and assumptions of which are still in force) with the decomposition (1.7.1), we obtain

W_{C} ⟶^{∼} S_{2}^{H} ⟶^{∼} ⨁_{π=σ_{2}⊗π_{f}} π_{f}^{H}.

If π = σ_{2} ⊗ π_{f} = ⊗′_{v}π_{v} is an irreducible unitary cuspidal automorphic representation of B_{A}^{∗} with π_{f}^{H} ≠ 0, then T (v) acts on the one-dimensional space π_{v}^{H_{v}} of spherical vectors (recall that v ∉ S, by assumption) by a scalar λ_{π}(v) ∈ C. The relation (1.14.2) then gives a formula for the local Euler factor

L_{v}(h^{1}(M_{H}^{∗}), s) := det(1 − (Nv)^{−s} Fr(v)_{geom} | W_{ℓ}^{I_{v}})^{−1},

namely

L_{v}(h^{1}(M_{H}^{∗}), s) = ∏_{π=σ_{2}⊗π_{f}}(1 − λ_{π}(v)(Nv)^{−s} + ω_{π}(v)(Nv)^{1−2s})^{− dim(π_{f}^{H})} = ∏_{π=σ_{2}⊗π_{f}} L_{v}(π, s − 1/2)^{dim(π_{f}^{H})},

(1.15.1)

where ω_{π} : A_{F}^{∗} /F^{∗} ⟶ C^{∗} denotes the central character (of finite order) of π. Taking a product of (1.15.1) over all v ∉ S, we obtain an equality of partial L-functions

L^{S}(h^{1}(M_{H}^{∗}), s) = ∏_{π=σ_{2}⊗π_{f}} L^{S}(π, s − 1/2)^{dim(π_{f}^{H})},  (1.15.2)

where

L^{S}(h^{1}(M_{H}^{∗}), s) = ∏_{v∉S} L_{v}(h^{1}(M_{H}^{∗}), s), L^{S}(π, s) = ∏_{v∉S} L_{v}(π, s).

Above, the automorphic L-functions are normalized as in [Ja-La] (the centre of symmetry of the functional equation lies at s = 1/2). It is a much deeper fact that the extreme left and right terms in (1.15.1) are equal even for v ∈ S. For v ∈ S − S_{B} this is proved in [Ca 2]; for v ∈ S_{B} one can reduce to the previous case by switching to a quaternion algebra unramified at v (possibly after a cyclic base change to a suitable totally real number field in which v splits completely). The Jacquet-Langlands correspondence [Ja-La, §14] associates to each (irreducible, cuspidal) π = σ_{2} ⊗ π_{f} with π_{f}^{H} ≠ 0 an irreducible cuspidal representation JL(π) of GL_{2}(A_{F}), which corresponds to a Hilbert modular newform of weight (2, …, 2) and central character ω_{π}. This representation is characterized by the property

(∀w ∉ Ram(B)) JL(π)_{w} ⟶^{∼} π_{w} as representations of GL_{2}(F_{w}) ⟶^{∼} B_{w}^{∗} (⟹ L_{w}(JL(π), s) = L_{w}(π, s))

(where w is a prime of F). In particular,

L^{S}(JL(π), s) = L^{S}(π, s).

In fact, the equality L_{w}(JL(π), s) = L_{w}(π, s) holds also for w ∈ Ram(B) ([Ja-La], Rmk. before Thm. 14.4). (1.16) The Jacobian of M_{H}^{∗} and N_{H}^{∗}. As in [Co-Va 1, 3.3], we define the Jacobian of M_{H}^{∗} (resp., N_{H}^{∗}) to be the abelian variety over F

J(M_{H}^{∗}) := Pic^{∘}(M_{H}^{∗}/F) = Res_{F (M_{H})/F} (Pic^{∘}(M_{H}^{∗}/F (M_{H})))
J(N_{H}^{∗}) := Pic^{∘}(N_{H}^{∗}/F) = Res_{F (N_{H})/F} (Pic^{∘}(N_{H}^{∗}/F (N_{H})))

(recall that F (M_{H}) (resp., F (N_{H})) denotes the field of constants of M_{H}^{∗} (resp., of N_{H}^{∗})). As Pic^{∘}(−/F) is both covariant and contravariant for finite flat morphisms of proper smooth curves over Spec(F), each diagram (1.9.1) induces morphisms of abelian varieties

[HgH′]_{∗} = [H′g^{−1}H]^{∗} : J(M_{H}^{∗}) ⟶ J(M_{H′}^{∗}),  [HgH′]^{∗} = [H′g^{−1}H]_{∗} : J(M_{H′}^{∗}) ⟶ J(M_{H}^{∗}) (1.16.1)

(and similarly for N_{H}^{∗}). Each Jacobian J(M_{H}^{∗}) has a canonical principal polarization. It follows from (1.16.1) that the corresponding Rosati involution ι acts by

ι ([HgH]_{∗}) = [HgH]^{∗} = [Hg^{−1}H]_{∗},  ι ([HgH]^{∗}) = [HgH]_{∗} = [Hg^{−1}H]^{∗}.  (1.16.2)

One can consider (1.16.2) as a “motivic” version of (1.11.3). We let the Hecke correspondences act on the Jacobians covariantly. As [·g] ∘ [·g′] = [·g′g], the map

\begin{matrix}Z[H\B̂^{∗}/H]^{op} & ⟶ End(J(M_{H}^{∗})) \\ [HgH] & ↦ [HgH]_{∗} \end{matrix}  (1.16.3)

is a ring homomorphism (recall that the multiplication on Z[H\B̂^{∗}/H] used here is the opposite of the one considered in [Co-Va 1, 3.4, 3.11]). The space Γ((M_{H}^{∗})^{an}, Ω^{an}) is canonically isomorphic to the cotangent space (T_{0}J(M_{H}^{∗})^{an})^{∨} of the complex torus J(M_{H}^{∗})^{an} at the origin. For each Hecke correspondence \xymatrix{[HgH′] : M_{H}^{∗} \ar@{-->}[r] & M_{H′}^{∗}} , the induced map

[HgH′]^{∗} : Γ((M_{H′}^{∗})^{an}, Ω^{an}) ⟶ Γ((M_{H}^{∗})^{an}, Ω^{an})

corresponds to the dual of the differential of the morphism

[HgH′]_{∗} : J(M_{H}^{∗}) ⟶ J(M_{H′}^{∗})

at the origin. (1.17) The “physical” Hecke algebra. In the situation of (1.12.1), the spherical Hecke algebra

T_{H}^{S} := Z[H^{S}\G(A_{f})^{S}/H^{S}] ⊂ Z[H\B̂^{∗}/H]

is commutative; we let it act on the Jacobian J(M_{H}^{∗}) by the rule (1.16.3), which defines a ring homomorphism

θ : T_{H}^{S} = (T_{H}^{S})^{op} ⊂ Z[H\B̂^{∗}/H]^{op} ⟶ End(J(M_{H}^{∗})).  (1.17.1)

As a ring, T_{H}^{S} is generated by the double cosets [Hb_{v}H] and [Hϖ_{v}^{±1}H] (v ∉ S), where b_{v} is as in 1.12. For each v ∉ S, the endomorphisms

T (v) := θ([Hb_{v}H]), [ϖ_{v}^{±1}] := θ([Hϖ_{v}^{±1}H])

of J(M_{H}^{∗}) induce the eponymous endomorphisms of Γ((M_{H}^{∗})^{an}, Ω^{an}) (which were defined in 1.12)) via the contravariant action on the cotangent space at the origin. In other words, the action of T_{H}^{S} on S_{2}^{H} ⟶^{∼} Γ((M_{H}^{∗})^{an}, Ω^{an}) factors through the morphism θ. Similarly, we let T (v) and [ϖ_{v}^{±1}] act on the spaces W, W_{dR} and W_{ℓ} defined in 1.12 by contravariant functoriality, i.e. by [Hb_{v}H]^{∗} and [Hϖ_{v}^{±1}H]^{∗}. The commutative ring

T := θ (T_{H}^{S}) ⊂ End(J(M_{H}^{∗}))

(the “physical” Hecke algebra) is a free Z-module of finite rank, generated (as a ring) by the endomorphisms T (v) and [ϖ_{v}^{±1}] (for v ∉ S). It follows from (1.16.2) that

(∀v ∉ S)  ι (T (v)) = [ϖ_{v}^{−1}] T (v),  ι ([ϖ_{v}^{±1}]) = [ϖ_{v}^{∓1}],  (1.17.2)

hence T is a commutative subring of End(J(M_{H}^{∗})) stable by the Rosati involution ι. The positivity property of this involution implies ([Mu, §21]) that the Q-algebra T ⊗ Q is isomorphic to a finite product of number fields

T ⊗ Q ⟶^{∼} ∏_{j∈J} L_{j}, J = J_{1} ∪ J_{2},

where each factor L_{j} is ι-stable and

(∀j ∈ J_{1}) L_{j} is totally real and ι acts trivially on L_{j}
(∀j ∈ J_{2}) L_{j} is a CM field and ι acts on L_{j} by complex conjugation.

(1.17.3)

For each j ∈ J, we denote by θ_{j} the ring homomorphism

θ_{j} : T_{H}^{S} ⟶^{θ}T ⊂ T ⊗ Q ↠ L_{j}.  (1.17.4)

The image of θ_{j} is an order in L_{j} and T has finite index in ∏_{j} Im(θ_{j}).

If π = σ_{2} ⊗ π_{f} is an irreducible cuspidal unitary representation of B_{A}^{∗} satisfying π_{f}^{H} ≠ 0, then T_{H}^{S} acts on π_{f}^{H} through a ring homomorphism λ_{π} : T_{H}^{S} ⟶ C, which factors as

λ_{π} : T_{H}^{S} ⟶^{θ_{j}}L_{j} ↪^{σ} C,

for a unique pair (j, σ) ∈ J × Hom(L_{j}, C). The strong multiplicity one theorem for automorphic representations of B_{A}^{∗} ([Ge], VI.2) implies that

[π = σ_{2} ⊗ π_{f} ≠ π′ = σ_{2} ⊗ π′_{f}, π_{f}^{H} ≠ 0 ≠ π_{f}^{′H} ⟹ λ_{π} ≠ λ_{π′}.]  (1.17.5)

The canonical map J(M_{H}^{∗}) ⟶ J(N_{H}^{∗}) (induced by the projection M_{H}^{∗} ⟶ N_{H}^{∗}) is T_{H}^{S}-equivariant and surjective; it follows that T_{H}^{S} ⊗ Q acts on J(N_{H}^{∗}) ⊗ Q (in the category of abelian varieties up to isogeny) by a certain quotient of T ⊗ Q. Let ℓ be a prime number. For each prime ℒ | ℓ of L_{j}, denote by V_{ℒ} (θ_{j}) the ℒ -adic Galois representation of G_{F,S} associated to JL(π) (which is a Hilbert modular form of parallel weight (2, …, 2) over F). This is a two-dimensional, absolutely irreducible ([Tay, Prop. 3.1]) representation of G_{F,S} over (L_{j})_{ℒ} , characterized by the property

(∀v ∉ S) det(1 − X Fr(v)_{geom} | V_{ℒ} (θ_{j})) = 1 − θ_{j}(T (v))X + ω_{π}(v)(Nv)X^{2}.

(1.18) Proposition (Decomposition of J(M_{H}^{∗}) and J(N_{H}^{∗}) up to isogeny). (i) For each pair (j, σ) ∈ J ×Hom(L_{j}, C) there exists a unique irreducible cuspidal unitary representation π = σ_{2}⊗π_{f} of B_{A}^{∗} satisfying π_{f}^{H} ≠ 0, λ_{π} = σ ∘ θ_{j}. We denote π = π(σ ∘ θ_{j}). (ii) For each prime number ℓ, the T ⊗ Q_{ℓ}[G_{F}]-module W_{ℓ} = H_{et}^{1}(M_{H}^{∗} ⊗_{F} \overline{F} , Q_{ℓ}) is isomorphic to

W_{ℓ} ⟶^{∼} ⨁_{j∈J} ( ⨁_{ℒ |ℓ}V_{ℒ} (θ_{j}))^{a_{j}},

where a_{j} = dim_{C}(π(σ ∘ θ_{j})_{f}^{H}) ≥ 1, ℒ runs through all primes of L_{j} above ℓ, and T acts on V_{ℒ} (θ_{j}) via θ_{j} and the embedding L_{j} ↪ (L_{j})_{ℒ} . (iii) There exists a T-linear isogeny u : J(M_{H}^{∗}) ⟶ ∏_{j∈J} A_{j}^{a_{j}} defined over F, where each a_{j} ≥ 1 is as in (ii), A_{j} is an abelian variety over F of dimension dim(A_{j}) = [L_{j} : Q] satisfying O_{L_{j}} = End_{F} (A_{j}) (hence A_{j} is F -simple) on which T acts via θ_{j}, H_{et}^{1}(A_{j} ⊗_{F} \overline{F} , Q_{ℓ}) is isomorphic to ⨁_{ℒ |ℓ} V_{ℒ} (θ_{j}) as an T⊗Q_{ℓ}[G_{F}]-module and there is an equality of L-series (Euler factor by Euler factor)

L(A_{j}/F, s) = ∏_{σ:L_{j}↪C} L(π(σ ∘ θ_{j}), s − 1/2).

The abelian variety A_{j} is unique up to an O_{L_{j}}-linear isogeny. (iv) If j, j′ ∈ J, j ≠ j′, then A_{j} is not isogeneous (over F) to A_{j′}.

(v) For each j ∈ J_{1} (resp., j ∈ J_{2}) and each polarization of A_{j} defined over F, the corresponding Rosati involution acts on L_{j} = End_{F} (A_{j}) ⊗ Q trivially (resp., by the complex conjugation). (vi) There exists a T-linear isogeny u_{1} : J(N_{H}^{∗}) ⟶ ∏_{j∈J_{1}} A_{j}^{a_{j}} (defined over F) fitting into a commutative diagram (in which the right vertical arrow is the canonical projection)

J(M_{H}^{∗}) ⟶^{u} ∏_{j∈J} A_{j}^{a_{j}}
↓  ↓ ↓
J(N_{H}^{∗}) ⟶^{u_{1}} ∏_{j∈J_{1}} A_{j}^{a_{j}}.

Proof. All statements are well-known, but as we are not aware of a good reference, we sketch the argument. In (i), the uniqueness is a consequence of (1.17.5). In order to prove the existence of π, note that W = H^{1}(M_{H}^{∗}, Q) is a faithful T ⊗ Q-module, hence it is isomorphic to

W ⟶^{∼} ⨁_{j∈J} L_{j}^{⊕b_{j}}  (b_{j} > 0).  (1.18.1)

Both comparison isomorphisms

W ⊗_{Q} C ⟶^{∼} W_{C} ⊕ \overline{W_{C}}, W_{dR} ⊗_{F,τ_{1}} C ⟶^{∼} W_{C}

(where W_{C} = Γ((M_{H}^{∗})^{an}, Ω^{an}) and W_{dR} = H_{dR}^{1}(M_{H}^{∗}/F)) are T ⊗ C-linear. As τ_{1}(F) ⊂ R, the second one implies that the T⊗C-modules W_{C} and \overline{W_{C}} are isomorphic, which yields an isomorphism of T⊗C-modules

W_{C} ⟶^{∼} ⨁_{j∈J} (L_{j} ⊗_{Q} C)^{⊕a_{j}} , a_{j} = b_{j}/2 > 0.

The isomorphism

L_{j} ⊗_{Q} C ⟶^{∼} ∏_{σ:L_{j}↪C} C, a ⊗ z ↦ (σ(a)z)_{σ}

implies that each morphism σ ∘ θ_{j} occurs as λ_{π} in W_{C}, with multiplicity a_{j} > 0. In other words, π(σ ∘ θ_{j}) exists and

dim_{C} π(σ ∘ θ_{j})_{f}^{H} = a_{j}.

(ii) As W_{ℓ} is a semi-simple Q_{ℓ}[G_{F}]-module ([Fa, Satz 3]), it is sufficient to show that the traces of Fr(v)_{geom} (v ∉ S) acting on both sides coincide, which follows from the T ⊗ Q_{ℓ}-linear comparison isomorphism W ⊗_{Q} Q_{ℓ} ⟶^{∼} W_{ℓ} and the formulas (1.14.2) and (1.18.1). (iii) The existence of a T-linear isogeny J(M_{H}^{∗}) ⟶ ∏_{j∈J} A_{j}^{a_{j}} (defined over F) such that L_{j} ↪ End_{F} (A_{j})⊗ Q, T acts on A_{j} via θ_{j} and H_{et}^{1}(A_{j} ⊗_{F} \overline{F} , Q_{ℓ}) is isomorphic to ⨁_{ℒ |ℓ} V_{ℒ} (θ_{j}) follows from (ii) and Faltings’ isogeny theorem [Fa, Satz 4] (formerly known as Tate’s conjecture). The endomorphism ring End_{F} (A_{j}) contains Im(θ_{j}), which is an order in L_{j}. Replacing A_{j} by the F -isogeneous abelian variety Hom_{Im(θ_{j})}(O_{L_{j}}, A_{j}) (see [Co, Ch. 10] for a discussion of the formalism of O-transforms), we may assume that End_{F} (A_{j}) contains O_{L_{j}}.

Fix a prime number ℓ which splits in L_{j}/Q. If ℒ and ℒ′ are distinct primes above ℓ in L_{j}, then the Q_{ℓ}[G_{F}]- modules V_{ℒ} (θ_{j}) and V_{ℒ′} (θ_{j}) are not isomorphic, as L_{j} is generated (as a field extension of Q) by the traces θ_{j}(T (v)) and determinants ω_{π}(v) of the Frobenius elements Fr(v)_{geom} (v ∉ S) acting on V_{ℒ} (θ_{j}) (as well as on V_{ℒ′} (θ_{j})), and the two embeddings L_{j} ↪ (L_{j})_{ℒ} , L_{j} ↪ (L_{j})_{ℒ′} are distinct. Absolute irreducibility of V_{ℒ} (θ_{j}) yields an inclusion

L_{j} ⊗_{Q} Q_{ℓ} ↪ End_{F} (A_{j}) ⊗_{Q} Q_{ℓ} = End_{Q_{ℓ}[G_{F}]}( ⨁_{ℒ |ℓ}V_{ℒ} (θ_{j})) = ⨁_{ℒ |ℓ}Q_{ℓ},  (1.18.2)

which must be an equality, by comparing the dimensions of the two flank terms; this proves that L_{j} = End_{F} (A_{j}) ⊗ Q. As O_{L_{j}} ⊂ End_{F} (A_{j}), we must have End_{F} (A_{j}) = O_{L_{j}}; as its endomorphism ring is an integral domain, the abelian variety A_{j} is F -simple. The statement about the local L-factor L_{v}(A_{j}/F, s) follows from (1.15.2) (resp., the discussion following (1.15.2)) if v ∈ S (resp., if v ∉ S), combined with (i) and (ii), since L_{v}(h^{1}(M_{H}^{∗}), s) = L_{v}(J(M_{H}^{∗}), s). The uniqueness of A_{j} (up to an O_{L_{j}}-linear isogeny) follows from Faltings’ isogeny theorem and the equality in (1.18.2). (iv) The same argument as in the proof of (iii) shows that, if ℓ is a prime number split in both L_{j} and L_{j′} and ℒ , ℒ′ are primes above ℓ in L_{j}, L_{j′}, respectively, (not necessarily distinct), then the Q_{ℓ}[G_{F}]-modules V_{ℒ} (θ_{j}) and V_{ℒ′} (θ_{j′}) are not isomorphic, hence A_{j} is not isogeneous to A_{j′} (over F).

(v) As End_{F} (A_{j}) is commutative, the Rosati involution does not depend on the polarization. For the canonical polarization, the statement follows from (1.17.3). (vi) T_{H}^{S} ⊗Q acts on J(N_{H}^{∗})⊗Q through a certain quotient T′ ⊗Q of T⊗Q, which is necessarily isomorphic to ∏_{j∈J′}, for some subset J′ ⊂ J. As Γ((N_{H}^{∗})^{an}, Ω^{an}) ⟶^{∼} S_{2}^{H_{F^{∗}}} consists precisely of those automorphic forms from S_{2}^{H} which have trivial central character, the formula (1.17.2) implies that the Rosati involution acts trivially on T′ ⊗ Q, hence J′ ⊂ J_{1}. The arguments used in the proof of (i), (ii) show that J(N_{H}^{∗}) is

T-linearly isogeneous to ∏_{j∈J′} A_{j}^{a′_{j}}, where

(∀j ∈ J′) (∀σ : L_{j} ↪ C) a′_{j} = dim_{C} π(σ ∘ θ_{j})_{f}^{H_{F^{∗}}} > 0.

In particular, for each pair (j, σ) ∈ J′ × Hom(L_{j}, C), the representation π(σ ∘ θ_{j}) satisfies π(σ ∘ θ_{j})_{f}^{H_{F^{∗}}} ≠ 0, hence it has trivial central character, which in turn implies that

a′_{j} = dim_{C} π(σ ∘ θ_{j})_{f}^{H_{F^{∗}}} = dim_{C} π(σ ∘ θ_{j})_{f}^{H} = a_{j}.

Conversely, if j ∈ J_{1}, then the existence of the Weil pairing implies that G_{F,S} acts on Λ^{2}V_{ℒ} (θ_{j}) by the cyclotomic character ([Cas, Cor. 5.5]; [Ri, Lemma 4.5.1]), hence ω_{π} = 1. It follows that J′ = J_{1}. (1.19) The maps M_{H}^{∗} ⟶ J(M_{H}^{∗}) and N_{H}^{∗} ⟶ J(N_{H}^{∗}). In the classical case B = M_{2}(Q), the class of the divisor (∞) in Pic(M_{H}^{∗}) has the following properties: its pull-back to each irreducible component of M_{H}^{∗} ⊗_{Q} \overline{Q} has degree one and there exists an integer m ≥ 1 such that each Hecke operator [HbH]_{∗} acts on the class of m(∞) by multiplication by its degree. Indeed, we can take any m that annihilates the cuspidal group C(M_{H}^{∗}) ⊂ Pic(M_{H}^{∗}), which is defined as the subgroup generated by the divisors whose pull-backs to each irreducible component of M_{H}^{∗} ⊗_{Q} \overline{Q} have degree zero and are supported at the cusps. The group C(M_{H}^{∗}) is finite by the theorem of Manin-Drinfeld [Dr]. For B ≠ M_{2}(Q), Zhang [Zh 1, Introduction] (see also [Co-Va 1, 3.5]) constructed a certain class ξ(M_{H}^{∗}) ∈ Pic(M_{H}^{∗}/F) ⊗ Q (“the Hodge class”) such that the degree of the pull-back of ξ(M_{H}^{∗}) to each irreducible component of M_{H}^{∗} ⊗_{F} \overline{F} is equal to one. More precisely, ξ(M_{H}^{∗}) is constructed, for a suitable integer m ≥ 1,

as ξ(M_{H}^{∗}) = { \atop mξ(M_{H}^{∗})} ⊗ 1/m, where { \atop mξ(M_{H}^{∗})} ∈ Pic(M_{H}^{∗}/F) has degree m on each irreducible component of M_{H}^{∗} ⊗_{F} \overline{F} and each Hecke operator [HbH]_{∗} acts on it by multiplication by its degree ([Co-Va 1], Remark 3.7). In general, if

M_{H}^{∗} ⊗_{F} \overline{F} = ∐_{α} C_{α}

is the decomposition into irreducible components, let mδ_{α} ∈ Pic(C_{α}) be the pull-back to Pic(C_{α}) of the class of m(∞) (resp., of { \atop mξ(M_{H}^{∗})}) if B = M_{2}(Q) (resp., if B ≠ M_{2}(Q)). We denote by

ι : M_{H}^{∗} ⟶ J(M_{H}^{∗})

the morphism (defined over F), which is characterized by the formula

(∀α) (∀P ∈ C_{α}(\overline{F})) ι(P) = m(P) − mδ_{α}.

The same discussion applies to the curves N_{H}^{∗}; we obtain a morphism

ι : N_{H}^{∗} ⟶ J(N_{H}^{∗}).

2. CM points In this section we recall basic properties of CM points on the curves M_{H} and N_{H}. We follow the notation and conventions of Sect. 1. (2.1) Let K be a totally imaginary quadratic extension of F such that each prime v ∈ S_{B} either ramifies or is inert in K/F. This assumption implies that there exists an F -embedding (= an injective homomorphism of F -algebras) t : K ↪ B; we fix such an embedding and denote by t_{v} : K ⊗_{F} F_{v} ↪ B_{v} (resp., by t̂: K̂ ↪ B̂) the induced embedding of the completions (resp., of the finite adèles). As in 1.1, we fix an embedding \overline{K} ↪ C extending τ_{1} : F ⟶ R.

(2.2) Lemma. (i) There exists a unique point z ∈ C with Im(z) > 0, which is fixed by the action of

t(K^{∗}) ⊂ B^{∗} ⊂ B_{τ_{1}}^{∗} ⟶^{∼} GL_{2}(R).

(ii) We have {λ ∈ B^{∗} | λ(z) = z} = t(K^{∗}).

Proof. Easy exercise.

(2.3) Definition. The set of CM-points by K on the curve M_{H} (resp., on N_{H}) is the set

CM(M_{H}, K) = {x = [z, b] ∈ M_{H}(C) | b ∈ B̂^{∗}}
CM(N_{H}, K) = {x = [z, b] ∈ N_{H}(C) | b ∈ B̂^{∗}} = the image of CM(M_{H}, K) in N_{H}(C)

(these sets do not depend on the choice of t, as two different F -embeddings of K into B are conjugate by an element of B^{∗}, by the Skolem-Noether theorem).

(2.4) The Galois action on CM-points. The reciprocity law [De, 3.9], [Mi 2, II.5.1] (with the sign corrected, [Mi 3, 1.10]) states that CM(M_{H}, K) ⊂ M_{H}(K^{ab}) and that the Galois action of Gal(K^{ab}/K) on CM(M_{H}, K) is described, via the reciprocity map rec_{K} : K̂^{∗} ⟶ Gal(K^{ab}/K), by the following formula:

(∀a ∈ K̂^{∗}) rec_{K}(a) [z, b] = [z, t̂(a)b].

(2.5) Proposition. Denote by K(x) ⊂ K^{ab} the field of definition over K of a CM point (by K) x = [z, b] on the curve M_{H} (resp., on N_{H}). Then the reciprocity map rec_{K} induces an isomorphism rec_{K} :

K^{∗}\K̂^{∗}/t̂^{−1}(bHb^{−1}) ⟶^{∼} Gal(K(x)/K) (resp., rec_{K} : K^{∗}\K̂^{∗}/t̂^{−1}(bHF̂^{∗}b^{−1}) ⟶^{∼} Gal(K(x)/K)).

Proof. If x = [z, b] ∈ CM(M_{H}, K) and a ∈ K̂^{∗}, then we deduce from 2.4 and Lemma 2.2(ii) that

rec_{K}(a) [z, b] = [z, b] ⟺ (∃λ ∈ B^{∗}) (∃h ∈ H)  (z, t̂(a)b) = (λ(z), λbh) ∈ (C − R) × B̂^{∗}
⟺ (∃μ ∈ K^{∗}) (∃h ∈ H) t̂(a)b = t(μ)bh
⟺ a ∈ K^{∗} · t̂^{−1}(bHb^{−1}).

The proof for x ∈ CM(N_{H}, K) is similar.

(2.6) Ring class fields of K (2.6.1) From now on, we shall consider only CM points on the curves N_{H}. For such a point x = [z, b], the formula from Proposition 2.5 can be restated as an isomorphism

rec_{K} : K̂^{∗}/K^{∗}F̂^{∗}Z ⟶^{∼} Gal(K(x)/K),

where Z = t̂^{−1}(bHÔ_{F}^{∗} b^{−1}) ⊂ Ô_{K}^{∗} is an open (compact) subgroup of Ô_{K}^{∗} containing Ô_{F}^{∗}.

(2.6.2) In the special case when H = R̂^{∗}, for an O_{F} -order R ⊂ B, then Z = Ô^{∗} for some O_{F} -order O ⊂ K. Such an order is necessarily of the form O_{c} = O_{F} + cO_{K}, where c ⊂ O_{F} is a non-zero ideal of O_{F}. The corresponding abelian extension K[c]/K satisfying

rec_{K} : K̂^{∗}/K^{∗}F̂^{∗}Ô_{c}^{∗} ⟶^{∼} Gal(K[c]/K)

is called the ring class field of K of conductor c (note that Zhang [Zh 2] uses the same terminology, but Cornut and Vatsal [Co-Va 1,2] consider a slightly more general class of extensions). If c, c′ ⊂ O_{F} are non-zero ideals, then we have

Ô_{c}^{∗} · Ô_{c′}^{∗} = Ô_{gcd(c,c′)}^{∗},  K^{∗}F̂^{∗}Ô_{c}^{∗} ∩ K^{∗}F̂^{∗}Ô_{c′}^{∗} ⊇ K^{∗}F̂^{∗}Ô_{lcm(c,c′)}^{∗},  (2.6.2.1)

hence

K[c] ∩ K[c′] = K[gcd(c, c′)], K[c] K[c′] ⊆ K[lcm(c, c′)].

(2.6.3) Each prime of K not dividing cO_{K} is unramified in K[c]/K. If ℓ is a prime of F which does not divide c and which is inert in K/F, then ℓO_{K} splits completely in K[c]/K and each prime above ℓ is totally tamely ramified in K[cℓ]/K[c].

(2.6.4) In general, if Z ⊂ Ô_{K}^{∗} is an open subgroup containing Ô_{F}^{∗}, we denote by K_{Z} ⊂ K^{ab} the finite abelian extension of K satisfying

rec_{K} : K̂^{∗}/K^{∗}F̂^{∗}Z ⟶^{∼} Gal(K_{Z}/K).

It follows from the last paragraph in 1.4 and from Proposition 2.5 that we have

⋃_{Z} K_{Z} = ⋃_{c} K[c] =: K[∞], CM(N_{H}, K) ⊂ N_{H}(K[∞]).

(2.6.5) The reciprocity map rec_{K} is compatible with the action of Gal(K/F) = {1, ρ} (where ρ denotes the complex conjugation on \overline{K} ⊂ C) on both sides. As

(∀a ∈ K̂^{∗}) a · ρ(a) ∈ F̂^{∗},

it follows that K_{Z}/F is a Galois extension and the conjugation action of ρ on Gal(K_{Z}/K) is given by ρgρ^{−1} = g^{−1}. In other words,

Gal(K_{Z}/F) ⟶^{∼} Gal(K_{Z}/K) ⋊ {1, ρ}

is a generalized dihedral group. (2.7) We shall now describe the Galois group

Gal(K_{Z′}/K_{Z}) ⟶^{∼} K^{∗}F̂^{∗}Z/K^{∗}F̂^{∗}Z′,  (2.7.1)

where Z′ ⊂ Z ⊂ Ô_{K}^{∗} are two open subgroups containing Ô_{F}^{∗}. We use the elementary fact that, whenever A and B ⊃ C are subgroups of some abelian group, then the obvious inclusion maps give rise to an exact sequence

0 ⟶ \frac{A ∩ B}{A ∩ C} ⟶ \frac{B}{C} ⟶ \frac{AB}{AC} ⟶ 0,

from which we deduce

\begin{matrix}F̂^{∗} ∩ Ô_{K}^{∗} = Ô_{F}^{∗} ⟹ F̂^{∗} ∩ Z = F̂^{∗} ∩ Z′ = Ô_{F}^{∗} ⟹ Z/Z′ ⟶^{∼} F̂^{∗}Z/F̂^{∗}Z′ \\ F^{∗} ∩ (O_{K}^{∗} ∩ Z) = F^{∗} ∩ (O_{K}^{∗} ∩ Z′) = O_{F}^{∗} ⟹ (O_{K}^{∗} ∩ Z)/(O_{K}^{∗} ∩ Z′) ⟶^{∼} F^{∗}(O_{K}^{∗} ∩ Z)/F^{∗}(O_{K}^{∗} ∩ Z′) \end{matrix}

(2.7.2)

and, by taking A = K^{∗}, B = F̂^{∗}Z, C = F̂^{∗}Z′, an exact sequence

0 ⟶ \frac{K^{∗} ∩ F̂^{∗}Z}{K^{∗} ∩ F̂^{∗}Z′} ⟶ \frac{Z}{Z′} ⟶ Gal(K_{Z′}/K_{Z}) ⟶ 0.  (2.7.3)

The next step is to determine the abelian group K^{∗} ∩ F̂^{∗}Z (and its counterpart for Z′ instead of Z).

(2.8) Proposition. Put G = Gal(K/F) = {1, ρ}.

(i) Ker (N_{K/F} : O_{K}^{∗} ⟶ O_{F}^{∗} ) = (O_{K}^{∗})_{tors}. (ii) There is an isomorphism (K^{∗} ∩ F̂^{∗}Ô_{K}^{∗})/F^{∗}O_{K}^{∗} ⟶^{∼} Ker(Pic(O_{F}) ⟶ Pic(O_{K})). (iii) ([Wa, Thm. 10.3]) The abelian group Ker(Pic(O_{F}) ⟶ Pic(O_{K})) naturally injects into H^{1}(G, O_{K}^{∗}). The order of the abelian group H^{1}(G, O_{K}^{∗}) is equal to 1 or 2.

Proof. (i) As K is a CM field and F its maximal real subfield, the group Ker(N_{K/F} : O_{K}^{∗} ⟶ O_{F}^{∗}) is finite, hence contained in (O_{K}^{∗})_{tors}. On the other hand, each root of unity ζ ∈ (O_{K}^{∗})_{tors} satisfies ζ · ρ(ζ) = 1. (ii) If β ∈ K^{∗} ∩ F̂^{∗}Ô_{K}^{∗}, then, for each non-archimedean prime v of F, there exists α_{v} ∈ F_{v}^{∗} such that ord_{w}(β) = ord_{w}(α_{v}), for all primes w above v in K. Let I be the ideal of O_{F} with divisor equal to ∑_{v} ord_{v}(α_{v}) [v]; then IO_{K} = (β). It follows that the map

β ↦ the class of IO_{K}

defines a homomorphism

(K^{∗} ∩ F̂^{∗}Ô_{K}^{∗})/F^{∗}O_{K}^{∗} ⟶ Ker(Pic(O_{F}) ⟶ Pic(O_{K})),

the inverse of which is given by I ↦ β, if IO_{K} = (β). (iii) The exact sequence

0 ⟶ P_{K} ⟶ I_{K} ⟶ Pic(O_{K}) ⟶ 0

(where P_{K} (resp., I_{K}) denotes the group of principal (resp., of all) fractional ideals of K) and its counterpart over F give rise to a commutative diagram with exact rows

0 ⟶ P_{F} ⟶ I_{F} ⟶ Pic(O_{F}) ⟶  0
↓  ∩ ↓  ↓  ↓
0 ⟶ P_{K}^{G} ⟶ I_{K}^{G} ⟶ Pic(O_{K})^{G} ⟶ H^{1}(G, P_{K}).

The Snake Lemma then implies that the group Ker(Pic(O_{F}) ⟶ Pic(O_{K})) injects into Coker(P_{F} ⟶ P_{K}^{G}). On the other hand, the cohomology sequence of

0 ⟶ O_{K}^{∗} ⟶ K^{∗} ⟶ P_{K} ⟶ 0

yields an exact sequence

0 ⟶ O_{F}^{∗} ⟶ F^{∗} ⟶ P_{K}^{G} ⟶ H^{1}(G, O_{K}^{∗}) ⟶ 0,

hence Coker(P_{F} ⟶ P_{K}^{G}) ⟶^{∼} H^{1}(G, O_{K}^{∗}). The abelian group H^{1}(G, O_{K}^{∗}) is a quotient of the finite cyclic group Ker(N_{K/F} : O_{K}^{∗} ⟶ O_{F}^{∗}) = (O_{K}^{∗})_{tors}; on the other hand, it is annihilated by |G| = 2, which implies that |H^{1}(G, O_{K}^{∗})| ≤ 2.

(2.9) Proposition. If Z′ ⊂ Z are open subgroups of Ô_{K}^{∗} containing Ô_{F}^{∗}, then the natural inclusions give rise to an exact sequence

0 ⟶ \frac{O_{K}^{∗} ∩ Z}{O_{K}^{∗} ∩ Z′} ⟶ \frac{K^{∗} ∩ F̂^{∗}Z}{K^{∗} ∩ F̂^{∗}Z′} ⟶ U_{Z,Z′} ⟶ 0,

in which |U_{Z,Z′}| ≤ |Ker(Pic(O_{F}) ⟶ Pic(O_{K}))| ≤ 2.

Proof. Injectivity of the first arrow follows from the second isomorphism in (2.7.2) and the equality

F^{∗}(O_{K}^{∗} ∩ Z) ∩ F̂^{∗}Z′ = F^{∗}(O_{K}^{∗} ∩ Z ∩ F̂^{∗}Z′) = F^{∗}(O_{K}^{∗} ∩ Z ∩ Ô_{F}^{∗} Z′) = F^{∗}(O_{K}^{∗} ∩ Z′).

Define U_{Z,Z′} to be the cokernel of this injective map. Assume first that Ker(Pic(O_{F}) ⟶ Pic(O_{K})) = 0; then K^{∗} ∩ F̂^{∗}Ô_{K}^{∗} = F^{∗}O_{K}^{∗}, by Proposition 2.8(ii), which implies that

K^{∗} ∩ F̂^{∗}Z = F^{∗}O_{K}^{∗} ∩ F̂^{∗}Z = F^{∗}(O_{K}^{∗} ∩ F̂^{∗}Z) = F^{∗}(O_{K}^{∗} ∩ Ô_{F}^{∗} Z) = F^{∗}(O_{K}^{∗} ∩ Z)

(and similarly for Z′), hence U_{Z,Z′} = 0. If Ker(Pic(O_{F}) ⟶ Pic(O_{K})) ≠ 0, then there exists β ∈ K^{∗} such that K^{∗} ∩ F̂^{∗}Ô_{K}^{∗} = F^{∗}O_{K}^{∗} ∪ βF^{∗}O_{K}^{∗}, by Proposition 2.8(ii)-(iii); thus

K^{∗} ∩ F̂^{∗}Z = F^{∗}(O_{K}^{∗} ∩ Z) ∪ F^{∗}(βO_{K}^{∗} ∩ F̂^{∗}Z).

If u, u′ ∈ O_{K}^{∗} satisfy βu, βu′ ∈ F̂^{∗}Z, then u′/u ∈ O_{K}^{∗} ∩ F̂^{∗}Z = O_{K}^{∗} ∩ Z, which shows that |U_{Z,Z′}| ≤ 2.

(2.10) Proposition. Let I_{0} ⊂ O_{K} be the non-zero ideal I_{0} := lcm {(u − 1) | u ∈ (O_{K}^{∗})_{tors}, u ≠ 1}. If c ⊂ O_{F} is a non-zero ideal such that cO_{K} ∤ I_{0} and Z ⊂ Ô_{c}^{∗} is a subgroup, then K^{∗} ∩ F̂^{∗}Z = F^{∗} and O_{K}^{∗} ∩ Z = O_{F}^{∗}.

Proof. It is enough to consider the case Z = Ô_{c}^{∗}. Assume that β ∈ K^{∗} ∩ F̂^{∗}Ô_{c}^{∗}, β = ab, a ∈ F̂^{∗}, b ∈ Ô_{c}^{∗}. Then we have

u := β/ρ(β) = b/ρ(b) ∈ K^{∗} ∩ Ô_{c}^{∗} ⊂ O_{K}^{∗}.

More precisely,

u ∈ Ker (N_{K/F} : O_{K}^{∗} ⟶ O_{F}^{∗} ) ∩ \binom{}{O_{c}^{∗}}^{1−ρ} ⊆ (O_{K}^{∗})_{tors} ∩ Ker (Ô_{K}^{∗} ⟶ (O_{K}/cO_{K})^{∗});

the condition cO_{K} ∤ I_{0} then implies that u = 1, hence β ∈ F^{∗}. It follows that K^{∗} ∩ F̂^{∗}Ô_{c}^{∗} = F^{∗} and O_{F}^{∗} ⊂ O_{c}^{∗} ⊂ O_{K}^{∗} ∩ F̂^{∗}Ô_{c}^{∗} ⊂ O_{K}^{∗} ∩ F^{∗} = O_{F}^{∗}.

(2.11) Corollary. If Z′ ⊂ Z ⊂ Ô_{K}^{∗} are open subgroups containing Ô_{F}^{∗} such that Z ⊂ Ô_{c}^{∗}, for some non-zero ideal c ⊂ O_{F} satisfying cO_{K} ∤ I_{0}, then the Galois group Gal(K_{Z′}/K_{Z}) is isomorphic to Z/Z′.

Proof. Combine (2.7.3) with Propositions 2.9-10.

3. The main result We follow the notations defined in Sect. 1-2. (3.1) Fix the following data:

(3.1.1) An open compact subgroup H ⊂ B̂^{∗}. (3.1.2) An F -simple quotient A_{j} (j ∈ J_{1}) of J(N_{H}^{∗}) satisfying End_{F} (A_{j}) = O_{L_{j}} and a non-trivial T-linear

morphism J(N_{H}^{∗}) ⟶ A_{j} (defined over F). Denote by ι_{j} the composite morphism

ι_{j} : N_{H}^{∗} ⟶^{ι}J(N_{H}^{∗}) ⟶ A_{j}.

(3.1.3) A totally imaginary quadratic extension K of F such that each prime v ∈ S_{B} either ramifies or is inert

in K/F. (3.1.4) An F -embedding t : K ↪ B. (3.1.5) A CM point x = [z, b] ∈ CM(N_{H}, K); denote by K(x) its field of definition over K. (3.1.6) A character α : Gal(K(x)/K) ⟶ O_{L}^{∗}, where L is a number field containing the totally real field

L_{j} = End_{F} (A_{j}) ⊗ Q.

We denote

e_{α} = ∑_{σ∈Gal(K(x)/K)} α^{−1}(σ) σ ∈ O_{L}[Gal(K(x)/K)]

and, for any O_{L}[Gal(K(x)/K)]-module M,

M^{(α)} = {m ∈ M | ∀σ ∈ Gal(K(x)/K) σ(m) = α(σ)m}.

The character α factors through an injective character

β : Gal(K(α)/K) ↪ O_{L}^{∗},

where K(α) = K(x)^{Ker(α)} ⊂ K(x).

(3.2) Theorem. Given the data (3.1.1-6), put y_{j} = ι_{j}(x) ⊗ 1 ∈ A_{j}(K(x)) ⊗_{O_{L_{j}}} O_{L}. Assume that the following condition is satisfied: (⋆) The abelian variety A_{j} does not acquire complex multiplication over any totally imaginary quadratic extension K′ of F contained in K(α). If e_{α}(y_{j}) ∉ (A_{j}(K(x)) ⊗_{O_{L_{j}}} O_{L})_{tors}, then the following abelian groups (more precisely, O_{L}-modules) are finite:

(A_{j}(K(x)) ⊗_{O_{L_{j}}} O_{L})^{(α)} /O_{L} · e_{α}(y_{j}), (� (A_{j}/K(x)) ⊗_{O_{L_{j}}} O_{L})^{(α)}
(A_{j}(K(x)) ⊗_{O_{L_{j}}} O_{L})^{(α^{−1})} /O_{L} · ρ(e_{α}(y_{j})), (� (A_{j}/K(x)) ⊗_{O_{L_{j}}} O_{L})^{(α^{−1})}

(where the complex conjugation ρ acts on A_{j}(K(x)) ⊗_{O_{L_{j}}} O_{L} by ρ ⊗ id).

(3.3) Previously known results in this direction ([Ko], [Ko-Lo 1,2], [Be-Da], [Ho 1,2], [Be], [Ti], [Zh 3]) use several additional hypotheses which ensure, among other things, that there is a “geometric” formula relating ρ(y_{j}) to y_{j}. Our main observation is that the Euler system argument does not require such a geometric relation. (3.4) As explained in 6.2.1 below, the condition (⋆) can be reformulated as follows: the Hilbert modular forms that occur in the factorization of L(A_{j}/F, s) do not have CM by K′. (3.5) It will be convenient to rephrase Theorem 3.2 and its proof in terms of the abelian variety

A = A_{j} ⊗_{O_{L_{j}}} O_{L}, O_{L} ↪ End_{F} (A)

(see [Con, §7] for the general formalism of such tensor products). We have

y_{j} ∈ A_{j}(K(x)) ⊗_{O_{L_{j}}} O_{L} = A(K(x)), e_{α}(y_{j}) = e_{β}(y) ∈ A(K(α)),

where

e_{β} = ∑_{σ∈Gal(K(α)/K)} β^{−1}(σ) σ ∈ O_{L}[Gal(K(α)/K)],  y = Tr_{K(x)/K(α)}(y_{j}) ∈ A(K(α)).

Fix a maximal ideal 𝔭 ⊂ O_{L} and a sufficiently large integer M >> 0 such that the ideal 𝔭^{M} is principal. Fixing its generator (which will be, by abuse of language, also denoted by 𝔭^{M}), the Galois cohomology of

0 ⟶ A[𝔭^{M}] ⟶ A(\overline{F})⟶^{𝔭^{M}}A(\overline{F}) ⟶ 0

gives rise to the usual descent sequence

0 ⟶ A(K(α))/𝔭^{M} ⟶^{δ}S ⟶ � (A/K(α))[𝔭^{M}] ⟶ 0,

where we have denoted by S = Sel(A/K(α), 𝔭^{M}) the classical Selmer group for the 𝔭^{M}-descent on A. In order to prove Theorem 3.2, it will be enough to show, assuming that e_{β}(y) ∉ A(K(α))_{tors}, that

(∃C(𝔭) ≥ 0) (∀M >> 0)  𝔭^{C(𝔭)} · (S^{(β)}/O_{L} · δ(e_{β}(y))) = 0.  (3.5.1)
For all but finitely many 𝔭,  C(𝔭) = 0.  (3.5.2)

(3.6) Let us briefly outline the proof of (3.5.1), the main steps of which follow [Be-Da]. We denote O_{𝔭} = O_{L,𝔭} and use temporarily the notation “X ≐ Y ” (resp., “x .∈ X”) as a shorthand for 𝔭^{C}X = 𝔭^{C}Y (resp., for 𝔭^{C}x ∈ X), where C is a suitable constant, independent of M. (3.6.1) The Euler system. The first step is to construct an Euler system, by letting various Hecke operators act on the CM point x. Applying Kolyvagin’s derivative and e_{β}, one obtains the derived cohomology classes

κ_{1} = a suitable multiple of δ(e_{β}(y)) ∈ S^{(β)}, κ_{ℓ_{1}···ℓ_{r}} ∈ S_{{ℓ_{1},...,ℓ_{r}}}^{(β)} ⊂ H^{1}(K(α), A[𝔭^{M}])(β),

where ℓ_{1}, …, ℓ_{r} are distinct primes of F, inert in K/F, which satisfy “Kolyvagin’s condition” modulo a sufficiently high power of 𝔭. The subscript ℓ_{1}, …, ℓ_{r} indicates that the class κ_{ℓ_{1}···ℓ_{r}} satisfies the “Selmer” local conditions only at primes distinct from ℓ_{1}, …, ℓ_{r}. (3.6.2) The Weil pairing. Fix a polarization φ : A_{j} ⟶ Â_{j} defined over F; it gives rise to the Weil pairing

T_{p}(A_{j}) × T_{p}(A_{j}) ⟶ Z_{p}(1)

(where p is the residual characteristic of 𝔭). This extends naturally (see 5.19 below) to an O_{𝔭}-bilinear pairing

T_{𝔭}(A) × T_{𝔭}(A) ⟶ O_{𝔭}(1),

whose reduction modulo 𝔭^{M}

( , )_{M} : A[𝔭^{M}] × A[𝔭^{M}] ⟶ O_{𝔭}/𝔭^{M}(1)

is skew-symmetric, and its kernel is killed by deg(φ). (3.6.3) The first annihilation relation. For simplicity, assume that β ≠ \overline{β} := β^{−1}. Let s ∈ S^{(β)}. For each Kolyvagin prime ℓ, we have a class κ_{ℓ} ∈ S_{\{ℓ\}}^{(β)} and its complex conjugate ^{ρ}κ_{ℓ} ∈ S_{\{ℓ\}}^{(\overline{β})}. Applying the reciprocity law of global class field theory

(∀a ∈ H^{2}(K(α), O_{𝔭}/𝔭^{M}(1)))  ∑_{v} inv_{v}(a_{v}) = 0 ∈ O_{𝔭}/𝔭^{M}

to the cup product a = s ∪^{ρ}κ_{ℓ}, we obtain the relation

∑_{v} inv_{v}(s_{v} ∪ (^{ρ}κ_{ℓ})_{v}) = 0 ∈ O_{𝔭}/𝔭^{M},  (3.6.3.1)

to which only the primes of K(α) above ℓ contribute. Fixing one such a prime, say, λ, and using the fundamental relation 5.18 between the localizations (κ_{1})_{λ}, (κ_{ℓ})_{λ}, one can rewrite (3.6.3.1) as

[K(α) : K] (s_{λ}, (κ_{1})_{λ})_{M} = 0 ∈ O_{𝔭}/𝔭^{M}(1),  (3.6.3.2)

where one considers the localizations s_{λ}, (κ_{1})_{λ} as elements of

H_{ur}^{1}(K(α)_{λ}, A[𝔭^{M}]) ⟶^{∼} A[𝔭^{M}] (⟶^{∼} (O_{𝔭}/𝔭^{M})^{2}),

by using evaluation at the Frobenius element Fr(λ). The assumption e_{β}(y) ∉ A_{tors} implies that

⟨κ_{1}⟩ (= O_{L} · κ_{1}) ≐ O_{𝔭}/𝔭^{M}.

An application of the Čebotarev density theorem then shows that one can choose a Kolyvagin prime ℓ in such a way that

⟨(κ_{1})_{λ}⟩ ≐ O_{𝔭}/𝔭^{M}

(inside H_{ur}^{1}(K(α)_{λ}, A[𝔭^{M}]) ⟶^{∼} A[𝔭^{M}]). As the pairing ( , )_{M} on A[𝔭^{M}] is almost symplectic, the annihilation relation (3.6.3.2) yields

s_{λ} .∈ ⟨(κ_{1})_{λ}⟩^{⊥} ≐ ⟨(κ_{1})_{λ}⟩.  (3.6.3.3)

If we put

S̃^{(β)} = {s ∈ S^{(β)} | s_{λ} = 0},

then we deduce from (3.6.3.3) that

S^{(β)} ≐ ⟨κ_{1}⟩ ⊕ S̃^{(β)}.

(3.6.4) The second annihilation relation. Let s ∈ S̃^{(β)}. In order to complete the proof of (3.5.1), we must show that s ≐ 0. If ℓ′ ≠ ℓ is another Kolyvagin prime and λ′ a prime above ℓ′ in K(α), then the reciprocity law applied to the cup product s ∪^{ρ}κ_{ℓℓ′}

∑_{v} inv_{v}(s_{v} ∪ (^{ρ}κ_{ℓℓ′})_{v}) = 0 ∈ O_{𝔭}/𝔭^{M}

simplifies to

[K(α) : K] (s_{λ′}, (κ_{ℓ})_{λ′})_{M} = 0 ∈ O_{𝔭}/𝔭^{M}(1).  (3.6.4.1)

As (κ_{1})_{λ} (resp., (κ_{ℓ})_{λ}) is unramified (resp., totally ramified) at λ, we have

⟨(κ_{1})_{λ}, (κ_{ℓ})_{λ}⟩ ≐ (O_{𝔭}/𝔭^{M})^{2} ⊂ H^{1}(K(α)_{λ}, A[𝔭^{M}]),

hence

⟨κ_{1}, κ_{ℓ}⟩ ≐ (O_{𝔭}/𝔭^{M})^{2} ⊂ S_{{ℓ}}^{(β)}.

Another application of the Čebotarev density theorem shows that one can choose ℓ′ in such a way that

⟨(κ_{1})_{λ′}, (κ_{ℓ})_{λ′}⟩ ≐ (O_{𝔭}/𝔭^{M})^{2} = H_{ur}^{1}(K(α)_{λ′}, A[𝔭^{M}]) ⟶^{∼} A[𝔭^{M}]  (3.6.4.2)

and

⟨s_{λ′}⟩ ≐ ⟨s⟩.  (3.6.4.3)

The first annihilation relation (3.6.3.2) applies with ℓ′ instead of ℓ, hence

s_{λ′} .∈ ⟨(κ_{1})_{λ′}⟩^{⊥}.

On the other hand, the second annihilation relation (3.6.4.1) yields

s_{λ′} .∈ ⟨(κ_{ℓ})_{λ′}⟩^{⊥},

thus

s_{λ′} .∈ ⟨(κ_{1})_{λ′}, (κ_{ℓ})_{λ′}⟩^{⊥} ≐ A[𝔭^{M}]^{⊥} ≐ 0,

by (3.6.4.2), which in turns implies that s ≐ 0, by (3.6.4.3). This finishes the proof that S̃^{(β)} ≐ 0, hence S^{(β)} ≐ ⟨κ_{1}⟩ = ⟨δ(e_{β}(y))⟩.

4. The Euler system Assume that we are in the situation of 3.1. (4.1) According to Proposition 2.5, the field of definition K(x) of x over K is characterized by the isomorphism

rec_{K} : K̂^{∗}/K^{∗}F̂^{∗}Z ⟶^{∼} Gal(K(x)/K), Z = t̂^{−1}(bHÔ_{F}^{∗} b^{−1}).

It follows from the first formula in (2.6.2.1) that there exists a smallest non-zero ideal c(x) ⊂ O_{F} (with respect to divisibility) such that Z ⊇ Ô_{c(x)}^{∗}. Equivalently, K[c(x)] is the smallest ring class field of K (with respect to inclusion) containing K(x). (4.2) Fix S as in 1.12. The decomposition (1.12.1) implies that we have

Z = Z_{S}Z^{S},  Z_{S} ⊂ ∏_{v∈S} O_{K,v}^{∗},  Z^{S} = ∏_{v∉S} Z_{v},  (∀v ∉ S) Z_{v} = t_{v}^{−1}(b_{v}H_{v}O_{F,v}^{∗}b_{v}^{−1}) ⊂ O_{K,v}^{∗},

where we have denoted, slightly abusively,

O_{K,v} = O_{K} ⊗_{O_{F}} O_{F,v} = ⨁_{w|v} O_{K,w}.

For each v ∉ S, we have

ord_{v}(c(x)) = 0 ⟺ Z_{v} = O_{K,v}^{∗}.

(4.3) Definition. Let 𝒮 = ⋃_{r≥0} 𝒮_{r} be the following set of square-free ideals of O_{F}: 𝒮_{0} = {(1)},

𝒮_{1} = {ℓ ⊂ O_{F} | ℓ is a maximal ideal inert in K/F, ℓ ∉ S, ℓ ∤ (p)c(x), ℓO_{K} ∤ I_{0}},
(∀r > 1) 𝒮_{r} = {ℓ_{1} ⋯ ℓ_{r} | ℓ_{j} ∈ 𝒮_{1} distinct},

where I_{0} ⊂ O_{K} is the ideal defined in Proposition 2.10.

(4.4) Definition. For each 𝔫 ∈ 𝒮, let h(𝔫) ∈ B̂^{∗} be the following element: h((1)) = 1. For each ℓ ∈ 𝒮_{1}, we have H_{ℓ} = R(ℓ)^{∗}, for a maximal O_{F,ℓ}-order R(ℓ) ⊂ B_{ℓ}; let h(ℓ) ∈ R(ℓ) ∩ B_{ℓ}^{∗} ⊂ B_{ℓ}^{∗} ⊂ B̂^{∗} be any element satisfying ord_{ℓ}(nr(h(ℓ))) = 1. Finally, for 𝔫 = ℓ_{1} ⋯ ℓ_{r} (r > 1, ℓ_{j} ∈ 𝒮_{1}), we put h(𝔫) = h(ℓ_{1}) ⋯ h(ℓ_{r}) ∈ ∏_{j=1}^{r} B_{ℓ_{j}}^{∗} ⊂ B̂^{∗}. Having defined h(𝔫), we define CM points

(∀𝔫 ∈ 𝒮) x(𝔫) := [z, bh(𝔫)] ∈ CM(N_{H}, K).

(4.5) Proposition. For each 𝔫 ∈ 𝒮, the field of definition K(x(𝔫)) is contained in K[c(x)𝔫] and the reciprocity map induces an isomorphism

rec_{K} : K̂^{∗}/K^{∗}F̂^{∗}Z(𝔫) ⟶^{∼} Gal(K(x(𝔫))/K),

Z(𝔫) = t̂^{−1}(bh(𝔫)HÔ_{F}^{∗} h(𝔫)^{−1}b^{−1}) = Z ∩ Ô_{𝔫}^{∗} = Z_{S} ∏_{ℓ|𝔫} Z(𝔫)_{ℓ} ∏_{ℓ∉S(𝔫)} Z_{ℓ}, S(𝔫) = S ∪ {ℓ ∈ 𝒮_{1}, ℓ | 𝔫}.

The quotient Z/Z(𝔫) is naturally isomorphic to

Z/Z(𝔫) ⟶^{∼} ∏_{ℓ|𝔫} Z_{ℓ}/Z(𝔫)_{ℓ}, Z_{ℓ}/Z(𝔫)_{ℓ} ⟶^{∼} \frac{(O_{K}/ℓO_{K})^{∗}}{(O_{F} /ℓO_{F})^{∗}};

in particular, each group Z_{ℓ}/Z(𝔫)_{ℓ} (for ℓ ∈ 𝒮_{1}, ℓ | 𝔫) is cyclic of order Nℓ + 1.

Proof. The isomorphism comes from Proposition 2.5. As h(𝔫) = 1 for each v ∤ 𝔫, the abelian group Z(𝔫) decomposes as required, with Z(𝔫)_{v} = Z_{v} for each v ∤ S(𝔫) and

(∀ℓ | 𝔫, ℓ ∈ 𝒮_{1}) Z(𝔫)_{ℓ} = t_{ℓ}^{−1}(b_{ℓ}h(ℓ)R(ℓ)^{∗}h(ℓ)^{−1}b_{ℓ}^{−1}).

Applying Proposition 4.7(iv) below with E = F_{ℓ}, R = R(ℓ) ⊂ B_{ℓ} ⟶^{∼} M_{2}(F_{ℓ}), i = Ad(b_{ℓ})^{−1} ∘t_{ℓ} : K ⊗_{F} F_{ℓ} ↪ B_{ℓ} ⟶^{∼} M_{2}(F_{ℓ}) and r = 1 implies that Z(𝔫) = Z ∩ Ô_{𝔫}^{∗} and

Z_{ℓ}/Z(𝔫)_{ℓ} ⟶^{∼} \frac{(O_{K}/ℓO_{K})^{∗}}{(O_{F} /ℓO_{F})^{∗}} .

Finally, it follows from Z ⊇ Ô_{c(x)}^{∗} that Z(𝔫) ⊇ Ô_{c(x)}^{∗} ∩ Ô_{𝔫}^{∗} = Ô_{c(x)𝔫}^{∗}, hence K(x(𝔫)) ⊆ K[c(x)𝔫].

(4.6) Proposition. Let v be a non-archimedean prime of F which does not divide c(x).

(i) If 𝔫 ∈ 𝒮 and v ∤ 𝔫, then each prime of K above v is unramified in K(x(𝔫))/K. (ii) If 𝔫 ∈ 𝒮, v ∤ 𝔫 and v is inert in K/F, then vO_{K} splits completely in K(x(𝔫))/K. (iii) If 𝔫ℓ ∈ 𝒮_{r+1} (ℓ ∈ 𝒮_{1}, 𝔫 ∈ 𝒮_{r}, r ≥ 0), then each prime of K(x(𝔫)) above ℓ is totally tamely ramified in K(x(𝔫ℓ))/K(x(𝔫)).

Proof. (i) As K(x(𝔫)) ⊆ K[c(x)𝔫] and v ∤ c(x)𝔫, the prime v is unramified in K(x(𝔫))/K. (ii) The decomposition group of vO_{K} in the extension K(x(𝔫))/K is equal to the image of the composite map

K_{v}^{∗} := (K ⊗_{F} F_{v})^{∗} ⟶ K̂^{∗} ⟶ K̂^{∗}/K^{∗}F̂^{∗}Z(𝔫),

which factors through K_{v}^{∗}/O_{K,v}^{∗}F_{v}^{∗} = 0, as vO_{K} is unramified in this extension. (iii) The inertia group of this extension at any prime λ of K(x(𝔫)) above ℓ is equal to the image of the composite map

f : O_{K(x(𝔫)),λ}^{∗}⟶^{f_{1}}O_{K,ℓ}^{∗}⟶^{f_{2}} \frac{(O_{K}/ℓO_{K})^{∗}}{(O_{F} /ℓO_{F})^{∗}} = Z(𝔫ℓ)/Z(𝔫)⟶^{f_{3}}Gal(K(x(𝔫ℓ))/K(x(𝔫))).

The arrow f_{1} (given by the norm map) is surjective, since λ is unramified in K(x(𝔫))/K, and the arrows f_{2} and f_{3} are given by the canonical projections (cf. (2.7.3)). It follows that f is surjective, hence λ is totally ramified; the ramification is tame, as [K(x(𝔫ℓ)) : K(x(𝔫))] divides Nℓ + 1, which is prime to Nλ.

(4.7) Proposition. Let E be a finite extension of Q_{p}, E′ the unique unramified quadratic extension of E and π a common uniformizer of E and E′. Let R ⊂ M_{2}(E) be a maximal O_{E}-order and i : E′ ↪ M_{2}(E) an E-embedding. For r ≥ 1, put O_{E′,r}^{∗} := Ker (O_{E′}^{∗} ⟶ (O_{E′}/π^{r})^{∗}).

(i) R^{∗} is a maximal compact subgroup of M_{2}(E)^{∗} = GL_{2}(E). (ii) (∀γ ∈ GL_{2}(E)) i^{−1}(γR^{∗}γ^{−1}) ⊂ O_{E′}^{∗}. (iii) If i^{−1}(R^{∗}) = O_{E′}^{∗} and if g ∈ R satisfies ord_{π}(det(g)) = 1, then, for each integer r ≥ 1, the map of sets

i_{r} : O_{E′}^{∗}/O_{E}^{∗}O_{E′,r}^{∗} ⟶ R^{∗}/(R^{∗} ∩ g^{r}R^{∗}g^{−r})

induced by i is bijective. (iv) Under the assumptions of (iii), we have, for each r ≥ 1,

i^{−1}(g^{r}R^{∗}g^{−r}) = i^{−1}(R^{∗} ∩ g^{r}R^{∗}g^{−r}) = O_{E}^{∗}O_{E′,r}^{∗},

hence the reduction modulo π^{r} induces an isomorphism of abelian groups

\frac{i^{−1}(R^{∗})}{i^{−1}(g^{r}R^{∗}g^{−r})} ⟶^{∼} \frac{O_{E′}^{∗}}{O_{E}^{∗}O_{E′,r}^{∗}} ⟶^{∼} \frac{(O_{E′}/π^{r}O_{E′})^{∗}}{(O_{E}/π^{r}O_{E})^{∗}} .

In particular, reduction modulo π induces an isomorphism

\frac{i^{−1}(R^{∗})}{i^{−1}(gR^{∗}g^{−1})} ⟶^{∼} k_{E′}^{∗}/k_{E}^{∗},

where k_{E} and k_{E′} denote the residue fields of E and E′, respectively.

Proof. Without loss of generality, we can assume that R = M_{2}(O_{E}) and g = ( \begin{matrix}1 & 0 \\ 0 & π \end{matrix} ). In this case (i) is

well-known and (ii) follows from the properness of the map i (which implies that i^{−1}(γR^{∗}γ^{−1}) is a compact subgroup of E^{′∗}, hence is contained in O_{E′}^{∗}). In order to prove (iii) and (iv), note first that we have, for each r ≥ 0,

i^{−1}(g^{r}R^{∗}g^{−r}) =^{(ii)} O_{E′}^{∗} ∩ i^{−1}(g^{r}R^{∗}g^{−r}) = i^{−1}(R^{∗}) ∩ i^{−1}(g^{r}R^{∗}g^{−r}) = i^{−1}(R^{∗} ∩ g^{r}R^{∗}g^{−r}).

On the other hand, the intersection R_{r} := R ∩ g^{r}Rg^{−r} is equal to the Eichler order (of level π^{r})

R_{r} = \{( \begin{matrix}a & b \\ π^{r}c & d \end{matrix} ) | a, b, c, d ∈ O_{E}\} ⊂ M_{2}(O_{E}) = R,

which implies that i^{−1}(R_{r}) is an O_{E}-order in O_{E′}, hence i^{−1}(R_{r}) = O_{E} + π^{c(r)}O_{E′}, for some c(r) ≥ 0. As

i^{−1}(R^{∗}) = O_{E′}^{∗}, we must have i^{−1}(R) = O_{E′}. For r ≥ 1, the O_{E}-module i^{−1}(R)/i^{−1}(R_{r}) ⟶^{∼} O_{E}/π^{c(r)}

injects into R/R_{r} ⟶^{∼} O_{E}/π^{r}, hence c(r) ≤ r. If c(r) < r, then i(π^{r−1}O_{E′}) ⊆ R_{r}, which implies that

i(O_{E′}) ⊆ R ∩ π^{1−r}R_{r} = R_{1}. In other words, i induces a k_{E}-embedding of k_{E′} into the ring \{( \begin{matrix}a & b \\ 0 & d \end{matrix} ) |

a, b, d ∈ k_{E}}, which is impossible. This contradiction proves that c(r) = r, hence

i^{−1}(R^{∗} ∩ g^{r}R^{∗}g^{−r}) = i^{−1}(R_{r}^{∗}) = i^{−1}(R_{r} ∩ R^{∗}) = (O_{E} + π^{r}O_{E′}) ∩ O_{E′}^{∗} = (O_{E} + π^{r}O_{E′})^{∗} = O_{E}^{∗}O_{E′,r}^{∗}.

The remaining statements in (iii) and (iv) follow automatically.

(4.8) Proposition (The norm relation). Let 𝔫ℓ ∈ 𝒮_{r+1} (ℓ ∈ 𝒮_{1}, 𝔫 ∈ 𝒮_{r}, r ≥ 0).

(i) K(x(𝔫ℓ))/K(x(𝔫)) is a cyclic extension of degree (Nℓ + 1)/u(r), where u(r) = 1 for r > 0 and u(0) = (K^{∗} ∩ F̂^{∗}Z : F^{∗}) (= (O_{K}^{∗} ∩ Z : O_{F}^{∗}) or 2 (O_{K}^{∗} ∩ Z : O_{F}^{∗})). (ii) We have an equality of divisors on N_{H}^{∗} ⊗_{F} \overline{F}

T (ℓ) x(𝔫) = u(r) ∑_{σ∈Gal(K(x(𝔫ℓ)/K(x(𝔫)))} σ(x(𝔫ℓ)),

where T (ℓ) x(𝔫) = [Hb_{ℓ}H]_{∗}(x(𝔫)) (which is also equal to [Hb_{ℓ}H]^{∗}(x(𝔫)), as we work with the curve N_{H}^{∗}). (iii) The following equality holds in J(N_{H}^{∗})(K(x(𝔫))):

T (ℓ) (ι(x(𝔫))) = u(r) Tr_{K(x(𝔫ℓ))/K(x(𝔫))}(ι(x(𝔫ℓ))).

Proof. (i) We apply (2.7.3) to the extension K(x(𝔫ℓ))/K(x(𝔫)) = K_{Z(𝔫ℓ)}/K_{Z(𝔫)}. As 𝔫ℓO_{K} ∤ I_{0}, Proposition 2.10 implies that K^{∗} ∩ F̂^{∗}Z(𝔫ℓ) = F^{∗}. If r > 0, then K^{∗} ∩ F̂^{∗}Z(𝔫) = F^{∗} as well, hence a combination of (2.7.3) with Proposition 4.5 yields an isomorphism

\frac{(O_{K}/ℓO_{K})^{∗}}{(O_{F} /ℓO_{F})^{∗}} ⟶^{∼} Gal(K(x(𝔫ℓ))/K(x(𝔫))).

If r = 0 ( ⟺ 𝔫 = (1)), then the same argument shows that the kernel of the natural surjection

\frac{(O_{K}/ℓO_{K})^{∗}}{(O_{F} /ℓO_{F})^{∗}} ⟶ Gal(K(x(𝔫ℓ))/K(x(𝔫)))

has order equal to u(0) = (K^{∗} ∩ F̂^{∗}Z : F^{∗}), which is in turn equal to (O_{K}^{∗} ∩ Z : O_{F}^{∗}) or 2(O_{K}^{∗} ∩ Z : O_{F}^{∗}), thanks to Proposition 2.9. (ii) By definition, the divisor T (ℓ) x(𝔫) contains x(𝔫ℓ). Proposition 4.7(iii) for r = 1 together with the proof of (i) imply that T (ℓ) x(𝔫) coincides with the orbit of x(𝔫ℓ) under the action of Gal(K(x(𝔫ℓ))/K(x(𝔫))), with each point counted with multiplicity u(r). (iii) This is a consequence of (ii) and the fact that T (ℓ) acts on the class of m(∞) (resp., on { \atop mξ(N_{H}^{∗})}) by multiplication by deg T (ℓ) = Nℓ + 1.

(4.9) Proposition (The congruence relation). Let 𝔫ℓ ∈ 𝒮_{r+1} (ℓ ∈ 𝒮_{1}, 𝔫 ∈ 𝒮_{r}, r ≥ 0). For each prime λ | ℓ above ℓ in K(𝔫ℓ) the following congruence holds:

x(𝔫ℓ) ≡ Fr(ℓ)_{arith} x(𝔫) ≡ Fr(ℓ)_{geom} x(𝔫) (mod λ)

(considered as an equality in N_{H}^{∗}(κ(λ))).

Proof. Firstly, κ(λ)/κ(ℓ) is a quadratic extension, by Proposition 4.6(ii)-(iii), hence the actions of Fr(ℓ)_{arith} and Fr(ℓ)_{geom} on κ(λ) coincide. Combining this remark with the congruence relation (1.14.3), we deduce that the reduction modulo λ of each point in the support of T (ℓ) x(𝔫) (in particular, of x(𝔫ℓ)) is equal to the reduction of Fr(ℓ)_{arith} x(𝔫).

(4.10) Proposition-Definition. For each 𝔫 ∈ 𝒮, define the subfield K(x(𝔫))′ ⊆ K(x(𝔫)) by

K(x(𝔫))′ = \{ \begin{matrix}K(x), & 𝔫 = (1) \\ K(x(ℓ_{1})) ⋯ K(x(ℓ_{r})), & 𝔫 = ℓ_{1} ⋯ ℓ_{r} (r ≥ 1, ℓ_{i} ∈ 𝒮_{1}) \end{matrix}

and put G(𝔫) = Gal(K(x(𝔫))′/K(x)).

(i) For each ℓ ∈ 𝒮_{1}, the group G(ℓ) is cyclic, of order (Nℓ + 1)/u(0). (ii) For each 𝔫 = ℓ_{1} ⋯ ℓ_{r} ∈ 𝒮_{r}, the canonical map G(𝔫) ⟶ G(ℓ_{1}) × ⋯ × G(ℓ_{r}) is an isomorphism. (iii) For each 𝔫 ∈ 𝒮_{r} (r ≥ 0), we have [K(x(𝔫)) : K(x(𝔫))′] = u(r)u(0)^{r−1}.

Proof. (i) This is a special case of Proposition 4.8(i). (ii) The case r = 0 is trivial. For r ≥ 1 the statement follows from Lemma 4.11 below, applied to G = Z/Z(𝔫) = Z/Z(ℓ_{1} ⋯ ℓ_{r}), G_{j} = Z(𝔫/ℓ_{j})/Z(𝔫), G̃_{j} = Z(ℓ_{j})/Z(𝔫), A = (K^{∗} ∩ F̂^{∗}Z)/F^{∗} (we use (2.7.3) and Proposition 2.10 to view A ⊂ G, and then use Proposition 2.10 again for G̃_{i} ∩ A = 0). (iii) The case r = 0 is again trivial. For r ≥ 1 we have

[K(x(𝔫)) : K(x(𝔫))′] = |A|^{r−1} = u(0)^{r−1},

by the proof of Lemma 4.11.

(4.11) Lemma. Let G = G_{1} ⊕ ⋯ ⊕ G_{r} (r ≥ 1) be a finite abelian group, A ⊂ G a subgroup and π : G ⟶ G/A the canonical projection. Assume that, for each i = 1, …, r, we have G̃_{i} ∩ A = 0, where G̃_{i} = ⨁_{j≠i} G_{i} ⊂ G. Then the canonical homomorphism

π(G)/ ⋂_{i=1}^{r} π(G̃_{i}) ⟶ ⨁_{i=1}^{r} π(G)/π(G̃_{i})

is an isomorphism.

Proof. There is nothing to prove if r = 1, so we can assume that r > 1. As the homomorphism in question is injective and |π(G̃_{i})| = |G̃_{i}| = |G|/|G_{i}| (since G̃_{i} ∩ Ker(π) = 0 by assumption), it is enough to show that

|⋂_{i=1}^{r} π(G̃_{i})| =^{?} |π(G)| ∏_{i=1}^{r} \frac{|π(G̃_{i})|}{|π(G)|} = \frac{|G|}{|A|} ∏_{i=1}^{r} \frac{|A|}{|G_{i}|} = |A|^{r−1}.

Denote, for any subset I ⊂ {1, …, r},

G_{I} = ⨁_{i∈I} G_{i} ⊂ G,  I^{0} = {1, …, r} − I,  G̃_{I} = G_{I^{0}}.

Applying the Snake Lemma to the commutative diagrams with exact rows

0 ⟶ G̃_{i} ⟶ G ⟶^{π_{i}}  G_{i}  ⟶ 0
↓^{≀}  ↓^{π}  ↓
0 ⟶ π(G̃_{i}) ⟶ π(G) ⟶ π(G)/π(G̃_{i}) ⟶ 0

(where π_{i} : G ⟶ G_{i}, i = 1, …, r, denotes the natural projection with kernel G̃_{i}) and

0 ⟶ G̃_{{1,...,k}} ⟶ G̃_{{1,...,k−1}} ⟶^{π_{k}}  G_{k}  ⟶ 0
↓^{α_{k}}  ↓^{α_{k−1}}  ↓^{π}
0 ⟶ ⋂_{i=1}^{k} π(G̃_{i}) ⟶ ⋂_{i=1}^{k−1} π(G̃_{i}) ⟶ π(G)/π(G̃_{k})

(k = 2, …, r), we obtain isomorphisms A ⟶^{∼} π_{i}(A) (i = 1, …, r), inclusions

Ker(α_{r}) ⊆ Ker(α_{r−1}) ⊆ ⋯ ⊆ Ker(α_{1}) = 0

and exact sequences

0 ⟶ π_{k}(A) ⟶ Coker(α_{k}) ⟶ Coker(α_{k−1}) ⟶ 0 (k = 2, …, r),

which imply, by induction, the desired equality

|⋂_{i=1}^{r} π(G̃_{i})| = |Coker(α_{r})| = ∏_{i=2}^{r} |π_{i}(A)| = |A|^{r−1}.

(4.12) Definition of the Euler system. For each 𝔫 ∈ 𝒮_{r} (r ≥ 0), define the point y(𝔫) ∈ A(K(x(𝔫))′) = A_{j}(K(x(𝔫))′) ⊗_{O_{L_{j}}} O_{L} by

y(𝔫) = \frac{u(0)}{u(r)} Tr_{K(x(𝔫))/K(x(𝔫))′} ι(x(𝔫)) ⊗ 1

(in particular, y((1)) = ι(x) ⊗ 1 ∈ A(K(x))).

(4.13) Proposition (The Euler system relations). Let 𝔫ℓ ∈ 𝒮_{r+1} (ℓ ∈ 𝒮_{1}, 𝔫 ∈ 𝒮_{r}, r ≥ 0). Then:

(i) Tr_{K(x(𝔫ℓ))′/K(x(𝔫))′} y(𝔫ℓ) = a_{ℓ} y(𝔫), where a_{ℓ} = θ(T (ℓ)) ∈ O_{L_{j}} is the eigenvalue of the Hecke operator T (ℓ) acting on A_{j}. (ii) For each prime λ′ | ℓ above ℓ in K(x(𝔫ℓ))′, we have

y(𝔫ℓ) ≡ u(0) Fr(ℓ)_{arith} y(𝔫) ≡ u(0) Fr(ℓ)_{geom} y(𝔫) (mod λ′).

Proof. (i) This follows from Proposition 4.8(iii). (ii) As m(∞) (resp., { \atop mξ(N_{H}^{∗})}) is defined over F, it follows from (the proof of) Proposition 4.9 that we have, for each prime λ | λ′ above λ′ in K(x(𝔫ℓ)),

ι(x(𝔫ℓ)) ≡ Fr(ℓ)_{arith} ι(x(𝔫)) ≡ Fr(ℓ)_{geom} ι(x(𝔫)) (mod λ),

hence (using Proposition 4.10(iii))

y(𝔫ℓ) ≡ \frac{u(0)}{u(r + 1)} [K(x(𝔫ℓ)) : K(x(𝔫ℓ))′] (ι(x(𝔫ℓ)) ⊗ 1) ≡ u(0)^{r+1} ι(x(𝔫ℓ)) ⊗ 1 (mod λ)

and, similarly,

y(𝔫) ≡ u(0)^{r} ι(x(𝔫)) ⊗ 1 (mod λ),

which proves the desired congruence.

5. The derivative classes We continue to use the notation of Sect. 1-4. (5.1) Fix a prime number p, a prime ideal 𝔭 ⊂ O_{L} above p and a prime element ϖ ∈ O_{𝔭} = O_{L,𝔭}. Set M_{0} = ord_{𝔭}(u(0)). (5.2) The set 𝒮_{1}(M) of “Kolyvagin primes”

(5.2.1) Definition. For each integer M ≥ 1, define 𝒮_{1}(M) to be the set of maximal ideals ℓ ⊂ O_{F} satisfying the following conditions:

ℓ ∤ (p)c(x)d_{K/F} ,  ℓO_{K} ∤ I_{0},  ℓ ∉ S,

and the conjugacy class of the arithmetic Frobenius Fr(ℓ)_{arith} in the Galois group Gal(K(x)(A[𝔭^{M+M_{0}}])/F) coincides with the conjugacy class of the complex conjugation ρ. We also define

𝒮_{0}(M) = {(1)},  𝒮_{r}(M) = {ℓ_{1} ⋯ ℓ_{r} | ℓ_{i} ∈ 𝒮_{1}(M) distinct}  (r > 1).

(5.2.2) If ℓ ∤ (p)c(x)d_{K/F} and ℓ ∉ S, then the extension K(x)(A[𝔭^{M+M_{0}}])/F is unramified at ℓ, so it makes sense to consider the conjugacy class of Fr(ℓ)_{arith}. (5.2.3) By the Čebotarev density theorem, the set 𝒮_{1}(M) has positive density. (5.2.4) If ℓ ∈ 𝒮_{1}(M), then ℓ is inert in K/F. In particular, 𝒮_{1}(M) ⊆ 𝒮_{1} (hence 𝒮_{r}(M) ⊆ 𝒮_{r} for each r ≥ 0). (5.2.5) If ℓ ∈ 𝒮_{1}(M), then u(0) | (Nℓ + 1) (by Proposition 4.8(i)) and

1 − a_{ℓ}X + (Nℓ)X^{2} = det(1 − Fr(ℓ)_{arith}X | T_{𝔭}A) ≡ det(1 − ρX | T_{𝔭}A) = 1 − X^{2} (mod 𝔭^{M+M_{0}}),

which implies that the following congruences hold in O_{L}:

a_{ℓ} ≡ 0 (mod 𝔭^{M+M_{0}}), Nℓ + 1 ≡ 0 (mod u(0)𝔭^{M}).

(5.3) From now on, we fix a sufficiently large integer M >> 0 divisible by the order of 𝔭 in the ideal class group of O_{L}, and a generator of the principal ideal 𝔭^{M} (also to be denoted 𝔭^{M}, by abuse of notation).

(5.4) Definition. For each ℓ ∈ 𝒮_{1}(M), fix a generator σ_{ℓ} of the cyclic group G(ℓ) = Gal(K(x(ℓ))/K(x)) and define

Tr_{ℓ} = ∑_{i=0}^{|G(ℓ)|−1} σ_{ℓ}^{i}, D_{ℓ} = ∑_{i=0}^{|G(ℓ)|−1} iσ_{ℓ}^{i} ∈ Z[G(ℓ)];

these elements satisfy

(σ_{ℓ} − 1)D_{ℓ} = |G(ℓ)| − Tr_{ℓ} = (Nℓ + 1)/u(0) − Tr_{ℓ}.

For each 𝔫 = ℓ_{1} ⋯ ℓ_{r} ∈ 𝒮_{r}(M) (ℓ_{i} ∈ 𝒮_{1}(M)), define

D_{𝔫} = D_{ℓ_{1}} ⋯ D_{ℓ_{r}} ∈ Z[G(ℓ_{1})] ⊗ ⋯ ⊗ Z[G(ℓ_{r})] = Z[G(𝔫)].

(5.5) Lemma. For each 𝔫 ∈ 𝒮_{r}(M), the image of the point D_{𝔫}y(𝔫) ∈ A(K(x(𝔫))′) in A(K(x(𝔫))′)⊗O_{𝔭}/𝔭^{M} (which will be denoted by (D_{𝔫}y(𝔫) (mod 𝔭^{M}))) is contained in (A(K(x(𝔫))′) ⊗ O_{𝔭}/𝔭^{M})^{G(𝔫)}.

Proof. If 𝔫 = ℓ_{1} ⋯ ℓ_{r}, then we have, for each i = 1, …, r,

(σ_{ℓ_{i}} − 1) D_{𝔫} y(𝔫) = (Nℓ_{i} + 1)/u(0) D_{𝔫/ℓ_{i}} y(𝔫) − a_{ℓ_{i}} D_{𝔫/ℓ_{i}} y(𝔫/ℓ_{i}) ∈ 𝔭^{M}A(K(x(𝔫))′).

(5.6) As 𝔭^{M} is principal, the exact sequence of G_{F} -modules

0 ⟶ A[𝔭^{M}] ⟶ A(\overline{Q})⟶^{𝔭^{M}}A(\overline{Q}) ⟶ 0

induces, for each extension K′ of F, the cohomology sequence

0 ⟶ A(K′) ⊗ O_{𝔭}/𝔭^{M} ⟶^{δ}H^{1}(K′, A[𝔭^{M}]) ⟶ H^{1}(K′, A)[𝔭^{M}] ⟶ 0.

Kolyvagin’s derivative classes [c(𝔫)] (which will be constructed in 5.7-9 below) are natural lifts of the classes δ(D_{𝔫}y(𝔫) (mod 𝔭^{M})) under the restriction maps ^{(1)}

res : H^{1}(K(x), A[𝔭^{M}]) ⟶ H^{1}(K(x(𝔫))′, A[𝔭^{M}])^{G(𝔫)}  (5.6.1)

(of course, there is nothing to do if 𝔫 = (1), in which case one defines [c((1))] = δ(y((1))) = δ(ι(x) ⊗ 1)).

(5.7) In order to construct such cohomology classes [c(𝔫)] ∈ H^{1}(K(x), A[𝔭^{M}]) (where 𝔫 = ℓ_{1} ⋯ ℓ_{r} ∈ 𝒮_{r}), we fix a point z(𝔫) ∈ A(\overline{Q}) satisfying 𝔭^{M}z(𝔫) = D_{𝔫}y(𝔫) ∈ A(K(x(𝔫))′). The cohomology class δ(D_{𝔫}y(𝔫) (mod 𝔭^{M})) ∈ H^{1}(K(x(𝔫))′, A[𝔭^{M}]) is then represented by the 1-cocycle

(g ↦ (g − 1)z(𝔫)) ∈ Z^{1}(Gal(\overline{Q}/K(x(𝔫))′), A[𝔭^{M}]).

Note that the formula g ↦ (g − 1)z(𝔫) makes sense even for g ∈ Gal(\overline{Q}/K(x)); it defines a 1-cocycle in Z^{1}(Gal(\overline{Q}/K(x)), A(\overline{Q})).

Denote by

pr : Gal(\overline{Q}/K(x)) ⟶ Gal(K(x(𝔫))′/K(x)) = G(𝔫)

the natural surjection. We are going to define a 1-cocycle

d′(𝔫) ∈ Z^{1}(G(𝔫), A(K(x(𝔫))′))

such that the 1-cocycle

c(𝔫) : g ↦ (inf(d′(𝔫)))(g) + (g − 1)z(𝔫) = d′(𝔫)(pr(g)) + (g − 1)z(𝔫) ∈ Z^{1}(Gal(\overline{Q}/K(x), A(\overline{Q}))

will have values in A[𝔭^{M}]. Its cohomology class [c(𝔫)] ∈ H^{1}(K(x), A[𝔭^{M}]) will then satisfy

res([c(𝔫)]) = δ(D_{𝔫}y(𝔫) (mod 𝔭^{M})).

In order to construct d′(𝔫), we apply the following Lemma to G = G(𝔫), G_{i} = G(ℓ_{i}), σ_{i} = σ_{ℓ_{i}}, X = A(K(x(𝔫))′).

(1) As the kernel (resp., cokernel) of the map res is equal (resp., injects) to H^{i}(G(𝔫), A(K(x(𝔫))′)[𝔭^{M}]) for i = 1 (resp., i = 2) and the torsion submodule A(K[∞])_{tors} over K[∞] = ⋃ K[c] is finite (cf. [Ne-Sch, 2.2]), one sees directly, without any calculation, that such natural lifts exist if we multiply the points y(𝔫) by the square of any fixed element of O_{L} − {0} which annihilates A(K[∞])_{tors}.

(5.8) Lemma. Let G = G_{1} × ⋯ × G_{r} be a finite product of finite cyclic groups G_{i}. For each G-module X, the evaluation of 1-cocycles at fixed generators σ_{i} ∈ G_{i} induces an isomorphism of abelian groups

\begin{matrix}Z^{1}(G, X) & ⟶^{∼} \{(x_{1}, …, x_{r}) ∈ X^{r} | (∀i, j = 1, …, r) (1 + σ_{i} + ⋯ + σ_{i}^{|G_{i}|−1})x_{i} = 0, (σ_{i} − 1)x_{j} = (σ_{j} − 1)x_{i}\} \\ z & ↦ (z(σ_{1}), …, z(σ_{r})). \end{matrix}

Proof. Easy exercise.

(5.9) Proposition-Definition. If 𝔫 = ℓ_{1} ⋯ ℓ_{r} ∈ 𝒮_{r}(M) (r ≥ 1, ℓ_{i} ∈ 𝒮_{1}(M)), then:

(i) The elements

x_{i} = D_{𝔫/ℓ_{i}} (\frac{a_{ℓ_{i}}}{𝔭^{M}} y(𝔫/ℓ_{i}) − \frac{Nℓ_{i} + 1}{u(0)𝔭^{M}} y(𝔫)) ∈ A(K(x(𝔫))′) (i = 1, …, r)

satisfy

(∀i, j = 1, …, r)  Tr_{ℓ_{i}} x_{i} = 0,  (σ_{ℓ_{i}} − 1)x_{j} = (σ_{ℓ_{j}} − 1)x_{i}.

(ii) There exists a unique 1-cocycle d′(𝔫) ∈ Z^{1}(G(𝔫), A(K(x(𝔫))′) satisfying

(∀i = 1, …, r) d′(𝔫)(σ_{ℓ_{i}}) = x_{i}.

(iii) The cohomology class of d′(𝔫) lies in H^{1}(G(𝔫), A(K(x(𝔫))′))[𝔭^{M}]; more precisely, we have

(∀i = 1, …, r) 𝔭^{M}d′(𝔫)(σ_{ℓ_{i}}) = −(σ_{ℓ_{i}} − 1)D_{𝔫}y(𝔫).

(iv) Define d(𝔫) = inf(d′(𝔫)) ∈ Z^{1}(Gal(\overline{Q}/K(x)), A(\overline{Q})), i.e. d(𝔫)(g) = d′(𝔫)(pr(g)).

(v) Define d((1)) = 0 ∈ Z^{1}(Gal(\overline{Q}/K(x)), A(\overline{Q})).

Proof. (i) Firstly, the congruences 5.2.5 show that a_{ℓ_{i}}/𝔭^{M} and (Nℓ_{i} + 1)/u(0)𝔭^{M} are contained in O_{L}, so the definition of x_{i} makes sense. Secondly, the relation 4.13(i) together with 4.10(i) imply that we have

Tr_{ℓ_{i}} x_{i} = D_{𝔫/ℓ_{i}} (\frac{a_{ℓ_{i}}}{𝔭^{M}} |G(ℓ_{i})| − \frac{Nℓ_{i} + 1}{u(0)𝔭^{M}} a_{ℓ_{i}}) y(𝔫/ℓ_{i}) = 0

and, if i ≠ j,

(σ_{ℓ_{i}} − 1)x_{j} = D_{𝔫/ℓ_{i}ℓ_{j}} (\frac{Nℓ_{i} + 1}{u(0)} − Tr_{ℓ_{i}}) (\frac{a_{ℓ_{j}}}{𝔭^{M}} y(𝔫/ℓ_{j}) − \frac{Nℓ_{j} + 1}{u(0)𝔭^{M}} y(𝔫)) =

= D_{𝔫/ℓ_{i}ℓ_{j}} (−\frac{(Nℓ_{i} + 1)(Nℓ_{j} + 1)}{u(0)^{2}𝔭^{M}} y(𝔫) + \frac{(Nℓ_{i} + 1)a_{ℓ_{j}}}{u(0)𝔭^{M}} y(𝔫/ℓ_{j}) + \frac{(Nℓ_{j} + 1)a_{ℓ_{i}}}{u(0)𝔭^{M}} y(𝔫/ℓ_{i}) − \frac{a_{ℓ_{i}}a_{ℓ_{j}}}{𝔭^{M}} y(𝔫/ℓ_{i}ℓ_{j})) =

= (σ_{ℓ_{j}} − 1)x_{i}.

(ii) This follows from (i) and Lemma 5.8. (iii) We compute, using again 4.10(i) and 4.13(i), that

𝔭^{M}d′(𝔫)(σ_{ℓ_{i}}) = D_{𝔫/ℓ_{i}} (a_{ℓ_{i}} y(𝔫/ℓ_{i}) − (Nℓ_{i} + 1)/u(0) y(𝔫)) = −D_{𝔫/ℓ_{i}}(σ_{ℓ_{i}} − 1)D_{ℓ_{i}} y(𝔫) = −(σ_{ℓ_{i}} − 1)D_{𝔫}y(𝔫).

As d′(𝔫) is a 1-cocycle, it follows that 𝔭^{M}d′(𝔫) is a coboundary.

(5.10) Proposition-Definition. Let 𝔫 = ℓ_{1} ⋯ ℓ_{r} ∈ 𝒮_{r}(M) (r ≥ 0, ℓ_{i} ∈ 𝒮_{1}(M)). The 1-cocycle c(𝔫) ∈ Z^{1}(Gal(\overline{Q}/K(x)), A(\overline{Q})), defined by the formula

c(𝔫) : g ↦ d(𝔫)(g) + (g − 1)z(𝔫),

has the following properties:

(i) c(𝔫) ∈ Z^{1}(Gal(\overline{Q}/K(x)), A[𝔭^{M}]). (ii) The cohomology class [c(𝔫)] ∈ H^{1}(K(x), A[𝔭^{M}]) does not depend on the choice of z(𝔫). (iii) The image of [c(𝔫)] under the restriction map (5.6.1) is equal to

res([c(𝔫)]) = δ(D_{𝔫}y(𝔫) (mod 𝔭^{M})) ∈ H^{1}(K(x(𝔫))′, A[𝔭^{M}]).

(iv) The image of [c(𝔫)] in H^{1}(K(x), A(\overline{Q})) is equal to [d(𝔫)].

Proof. (i) As c(𝔫) is a 1-cocycle, it is sufficient to check that 𝔭^{M}c(𝔫)(g) = 0, for each element of Ker(pr) = Gal(\overline{Q}/K(x(𝔫))′) and for each g ∈ pr^{−1}(σ_{ℓ_{i}}) (i = 1, …, r). If g ∈ Ker(pr), then

𝔭^{M}c(𝔫)(g) = (g − 1)D_{𝔫}y(𝔫) = 0.

If pr(g) = σ_{ℓ_{i}}, then we have, by Proposition 5.9(iii),

𝔭^{M}c(𝔫)(g) = −(σ_{ℓ_{i}} − 1)D_{𝔫}y(𝔫) + (g − 1)D_{𝔫}y(𝔫) = 0.

(ii) Two choices of z(𝔫) differ by an element of A[𝔭^{M}]. (iii) For each element g ∈ Gal(\overline{Q}/K(x(𝔫))′), we have

(res(c(𝔫))(g) = (g − 1)z(𝔫) = δ(D_{𝔫}y(𝔫) (mod 𝔭^{M}))(g).

(iv) The cohomology class of (g − 1)z(𝔫) vanishes in H^{1}(K(x), A(\overline{Q})). (5.11) We now investigate the localizations

[c(𝔫)_{v}] ∈ H^{1}(K(x)_{v}, A[𝔭^{M}]), [d(𝔫)_{v}] ∈ H^{1}(K(x)_{v}, A(\overline{Q}))[𝔭^{M}] = H^{1}(K(x)_{v}, A)[𝔭^{M}]

of the cohomology classes [c(𝔫)] and [d(𝔫)] at various primes v of K(x).

(5.12) Proposition (Localization outside 𝔫). Let v be a non-archimedean prime of K(x) lying above a prime v_{F} of F. Denote by Ã_{v} the special fibre at v of the Néron model of A ⊗_{F} K(x) over O_{K(x)}, by π_{0}(Ã_{v}) the O_{L}-module of the connected components of Ã_{v}, and define

C_{1,v}(𝔭) = min{c ≥ 0 | 𝔭^{c} · π_{0}(Ã_{v})_{𝔭} = 0}.

If v_{F} ∤ 𝔫, then

𝔭^{C_{1,v}(𝔭)} · [d(𝔫)_{v}] = 0.

In particular, if v_{F} ∤ 𝔫 and v ∉ S, then [d(𝔫)_{v}] = 0.

Proof. If v_{F} ∤ 𝔫, then v is unramified in K(x(𝔫))′/K(x), hence the cohomology class [d(𝔫)_{v}] is inflated from an element of the cohomology group H_{ur}^{1}(K(x)_{v}, A)[𝔭^{M}], which is isomorphic to H^{1}(κ(v), π_{0}(Ã_{v}))[𝔭^{M}], where κ(v) denotes the residue field of v ([Mi 1, I.3.8]). This implies that the annihilator of π_{0}(Ã_{v})_{𝔭} kills [d(𝔫)_{v}]. If, in addition, v ∉ S, then A has good reduction at v_{F}, A ⊗_{F} K(x) has good reduction at v and π_{0}(Ã_{v}) = 0. (5.13) If ℓ ∈ 𝒮_{1}(M), fix primes w | v | λ | ℓ in the fields

K(x(ℓ)) ⊃ K(x) ⊃ K ⊃ F,

respectively (of course, λ is unique and w is uniquely determined by v, by Proposition 4.6). We denote by Fr(v) = Fr(v)_{geom} etc. the various geometric Frobenius elements.

The assumption ℓ ∈ 𝒮_{1}(M) implies that Fr(v) = Fr(λ) = Fr(ℓ)^{2} acts trivially on A[𝔭^{M}], hence the evaluation map

ev_{Fr(v)} : H_{ur}^{1}(K(x)_{v}, A[𝔭^{M}]) = Hom(⟨Fr(v)⟩, A(K(x)_{v})[𝔭^{M}]) ⟶^{∼} A(K(x)_{v})[𝔭^{M}]
f ↦ f(Fr(v))

is an isomorphism.

(5.14) Proposition. If ℓ ∈ 𝒮_{1}(M) and if v | ℓ is a prime of K(x) above ℓ, then all maps in the following diagram are isomorphisms and the diagram is commutative.

\xymatrix{A(K(x)_{v}) ⊗_{O_{L}} O_{L}/𝔭^{M} \ar[r]^{δ_{v}} \ar[d]_{_{red_{v}}} & H_{ur}^{1}(K(x)_{v}, A[𝔭^{M}]) = Hom(⟨Fr(v)⟩, A(K(x)_{v})[𝔭^{M}]) \ar[r]^{^{ev^{Fr(v)}}} & A(K(x)_{v})[𝔭^{M}] \ar[d]_{_{red_{v}}} \\ Ã_{v}(κ(v)) ⊗_{O_{L}} O_{L}/𝔭^{M} \ar[rr]^{(a_{ℓ} Fr(ℓ)−(Nℓ+1))/𝔭^{M} Nℓ=((Nℓ+1)−a_{ℓ} Fr(ℓ))/𝔭^{M}} &  & Ã_{v}(κ(v))[𝔭^{M}]}

Proof. The map δ_{v} is an isomorphism, as A has good reduction at ℓ and v ∤ p. The evaluation map ev_{Fr(v)} is an isomorphism, by 5.13. Both vertical maps are isomorphisms, as the kernel of the surjective reduction map red_{v} : A(K(x)_{v}) ⟶ Ã_{v}(κ(v)) is a pro-char(κ(v))-group and v ∤ p. It remains to show that the diagram is commutative (which will imply that the bottom horizontal map is also an isomorphism). As the composite map

f : A(K(x)_{v})[𝔭^{∞}] ↪ A(K(x)_{v}) ↠ A(K(x)_{v}) ⊗_{O_{L}} O_{L}/𝔭^{M}

is surjective, it is sufficient to compute ev_{Fr(v)} ∘ δ_{v} ∘ f(P) for P ∈ A(K(x)_{v})[𝔭^{∞}]. If we fix a point Q ∈ A(\overline{K(x)_{v}}) such that 𝔭^{M}Q = P, then

ev_{Fr(v)} ∘ δ_{v} ∘ f(P) = (Fr(v) − 1)(Q) = (Fr(ℓ)^{2} − 1)(Q) = \frac{a_{ℓ} Fr(ℓ) − (Nℓ + 1)}{Nℓ} (Q) =

= \frac{a_{ℓ} Fr(ℓ) − (Nℓ + 1)}{𝔭^{M}Nℓ} (P) = \frac{(Nℓ + 1) − a_{ℓ} Fr(ℓ)}{𝔭^{M}} (P),

as

Nℓ ≡ −1 (mod 𝔭^{M}),  Q ∈ A(K(x)_{v})[𝔭^{∞}]  and Fr(ℓ)^{−2} − a_{ℓ} Fr(ℓ)^{−1} + Nℓ = 0 on T_{𝔭}(A).

(5.15) In the situation of 5.13 we identify the O_{L}/𝔭^{M}-modules

\frac{H^{1}(K(x)_{v}, A[𝔭^{M}])}{H_{ur}^{1}(K(x)_{v}, A[𝔭^{M}])} = \frac{H^{1}(K(x)_{v}, A[𝔭^{M}])}{Im(δ_{v})} ⟶^{∼} H^{1}(K(x)_{v}, A)[𝔭^{M}].

The evaluation at the fixed generator σ_{ℓ} ∈ G(ℓ) defines an isomorphism

ev_{σ_{ℓ}} : \frac{H^{1}(K(x)_{v}, A[𝔭^{M}])}{H_{ur}^{1}(K(x)_{v}, A[𝔭^{M}])} ⟶^{∼} H^{1}(K(x)_{v}^{ur}, A[𝔭^{M}])^{Fr(v)=1} = Hom(I_{v}^{t}, A[𝔭^{M}])^{Fr(v)=1} =

= Hom(I_{v}^{t}, A[𝔭^{M}]) ⟵^{∼} Hom(G(ℓ), A(K(x)_{v})[𝔭^{M}])⟶^{ev_{σ_{ℓ}}}A(K(x)_{v})[𝔭^{M}]

(where G(ℓ) is identified with the quotient Gal(K(x(ℓ))_{w}K(x)_{v}^{ur}/K(x)_{v}^{ur}) of the tame inertia group I_{v}^{t} at v). We denote by Φ_{v} = ev_{σ_{ℓ}}^{−1} ∘ ev_{Fr(v)} the isomorphism defined by the following commutative diagram:

H_{ur}^{1}(K(x)_{v}, A[𝔭^{M}]) ⟶^{Φ_{v}}  \frac{H^{1}(K(x)_{v},A[𝔭^{M}])}{H_{ur}^{1}(K(x)_{v},A[𝔭^{M}])}
↓^{ev_{Fr(v)}}  ↓^{ev}^{σ_{ℓ}}
A(K(x)_{v})[𝔭^{M}]  = A(K(x)_{v})[𝔭^{M}]

As Nℓ ≡ −1 (mod 𝔭^{M}), the conjugation action of Fr(ℓ) on I_{v}^{t} ⊗ O_{L}/𝔭^{M} is given by multiplication by −1. In particular,

Φ_{v} ∘ Fr(ℓ) = −Fr(ℓ) ∘ Φ_{v}.  (5.15.1)

(5.16) Proposition. If 𝔫ℓ ∈ 𝒮_{r+1}(M) (𝔫 ∈ 𝒮_{r}(M), ℓ ∈ 𝒮_{1}(M), r ≥ 0) and if v | ℓ is a prime of K(x) above ℓ, then the localization

[d(𝔫ℓ)_{v}] ∈ H^{1}(K(x)_{v}, A)[𝔭^{M}] ⟵^{∼} \frac{H^{1}(K(x)_{v}, A[𝔭^{M}])}{H_{ur}^{1}(K(x)_{v}, A[𝔭^{M}])}

is equal to

[d(𝔫ℓ)_{v}] = −Φ_{v} (Fr(ℓ) [c(𝔫)_{v}]) .

Proof. The commutative diagram

H^{1}(K(x(ℓ))_{w}/K(x)_{v}, A(K(x(ℓ))_{w}))[𝔭^{M}]  ⟶^{red_{w}} H^{1}(G(ℓ), Ã_{v}(κ(w)))[𝔭^{M}]
↓  ∥
H^{1}(K(x(ℓ))_{w}K(x)_{v}^{ur}/K(x)_{v}^{ur}, A(\overline{K(x)_{v}}))[𝔭^{M}]  Hom(G(ℓ), Ã_{v}(κ(v))[𝔭^{M}])
↓  ↑^{≀red_{v}}
H^{1}(K(x)_{v}^{ur}, A(\overline{K(x)_{v}}))[𝔭^{M}]  Hom(G(ℓ), A(K(x)_{v})[𝔭^{M}])
↑^{≀}  ↓^{≀}
H^{1}(K(x)_{v}^{ur}, A[𝔭^{M}])  = Hom(I_{v}^{t}, A(K(x)_{v})[𝔭^{M}])

implies that

red_{v} ∘ ev_{σ_{ℓ}} ([d(𝔫ℓ)_{v}]) = red_{v} ([d′(𝔫ℓ)_{v}](σ_{ℓ})) = red_{v} (D_{𝔫} (\frac{a_{ℓ}}{𝔭^{M}} y(𝔫) − \frac{Nℓ + 1}{u(0)𝔭^{M}} y(𝔫ℓ))) =^{4.12(ii)}

= D_{𝔫} (\frac{a_{ℓ}}{𝔭^{M}} − \frac{(Nℓ + 1)Fr(ℓ)}{𝔭^{M}}) red_{v} y(𝔫) = (\frac{a_{ℓ} Fr(ℓ) − (Nℓ + 1)}{𝔭^{M}}) Fr(ℓ) red_{v}(D_{𝔫}y(𝔫)) =^{5.14}

= −Fr(ℓ) red_{v} (ev_{Fr(v)} ∘ δ_{v}(D_{𝔫}y(𝔫) (mod 𝔭^{M})))

(where we have used the fact that Fr(ℓ)^{2} = Fr(λ) = Fr(v) acts trivially on κ(v) and K(x)_{v}). As the reduction map red_{v} : A(K(x)_{v})[𝔭^{M}] ⟶ Ã_{v}(κ(v))[𝔭^{M}] is an isomorphism, it follows that

[d(𝔫ℓ)_{v}] = −ev_{σ_{ℓ}}^{−1} ∘ ev_{Fr(v)} (Fr(ℓ) [c(𝔫)_{v}]) = −Φ_{v} (Fr(ℓ) [c(𝔫)_{v}]) .

(5.17) If ℓ ∈ 𝒮_{1}(M), fix a prime v_{α} of K(α) above ℓ. As ℓO_{K} splits completely in K(x)/K (by Proposition 4.6(ii)), we have, for each prime v | v_{α} of K(x), natural identifications K(α)_{v_{α}} = K(x)_{v} and isomorphisms

ev_{Fr(v_{α})} : H_{ur}^{1}(K(α)_{v_{α}}, A[𝔭^{M}]) ⟶^{∼} A(K(α)_{v_{α}})[𝔭^{M}]

ev_{σ_{ℓ}} : H^{1}(K(α)_{v_{α}}, A[𝔭^{M}])/H_{ur}^{1}(K(α)_{v_{α}}, A[𝔭^{M}]) ⟶^{∼} A(K(α)_{v_{α}})[𝔭^{M}]

Φ_{v_{α}} = ev_{σ_{ℓ}}^{−1} ∘ ev_{Fr(v_{α})} : H_{ur}^{1}(K(α)_{v_{α}}, A[𝔭^{M}]) ⟶^{∼} H^{1}(K(α)_{v_{α}}, A[𝔭^{M}])/H_{ur}^{1}(K(α)_{v_{α}}, A[𝔭^{M}]).

(5.18) Proposition. If 𝔫ℓ ∈ 𝒮_{r+1}(M) (𝔫 ∈ 𝒮_{r}(M), ℓ ∈ 𝒮_{1}(M), r ≥ 0) and if v_{α} | ℓ is a prime of K(α) above ℓ, then the localization

[(cor_{K(x)/K(α)} d(𝔫ℓ))_{v_{α}}] ∈ H^{1}(K(α)_{v_{α}}, A)[𝔭^{M}] ⟵^{∼} \frac{H^{1}(K(α)_{v_{α}}, A[𝔭^{M}])}{H_{ur}^{1}(K(α)_{v_{α}}, A[𝔭^{M}])}

is equal to

[(cor_{K(x)/K(α)} d(𝔫ℓ))_{v_{α}}] = −Φ_{v} (Fr(ℓ) [(cor_{K(x)/K(α)} c(𝔫))_{v_{α}}]) .

Proof. This follows from Proposition 5.16 and the fact that

(∀X = A, A[𝔭^{M}]) (∀c ∈ H^{1}(K(x), X)) (cor_{K(x)/K(α)} (c))_{v_{α}} = ∑_{v|v_{α}} cor_{K(x)_{v}/K(α)_{vα}} (c_{v}) = ∑_{v|v_{α}} c_{v}.

(5.19) The Weil pairings. The Weil pairing

( , )_{A_{j}} : T_{p}(A_{j}) × T_{p}(Â_{j}) ⟶ Z_{p}(1)

is G_{F} -equivariant and satisfies

(∀f ∈ End_{F} (A_{j}) = O_{L_{j}}) (f(x), y)_{A_{j}} = (x, f̂(y))_{A_{j}}.

Fix a polarization φ : A_{j} ⟶ Â_{j} defined over F. As the corresponding Rosati involution acts trivially on O_{L_{j}} (by Proposition 1.18(v)), the induced skew-symmetric pairing

T_{p}(A_{j}) × T_{p}(A_{j}) ⟶ Z_{p}(1), x, y ↦ (x, φ(y))_{A_{j}}

factors through T_{p}(A_{j}) ⊗_{O_{L_{j}}} T_{p}(A_{j}) and preserves the decomposition

T_{p}(A_{j}) = ⨁_{P |p} T_{P} (A_{j}),

where P runs through all primes of L_{j} above p. Denote by

( , )_{𝔭_{j}} : T_{𝔭_{j}}(A_{j}) ⊗_{O_{𝔭_{j}}} T_{𝔭_{j}}(A_{j}) ⟶ Z_{p}(1)

its 𝔭_{j}-component, where 𝔭_{j} = 𝔭 ∩ O_{L_{j}} and O_{𝔭_{j}} = O_{L_{j},𝔭_{j}}.

Fix a generator d ∈ 𝒟_{O_{𝔭_{j}} /Z_{p}}^{−1} of the inverse different; the map

λ : O_{𝔭_{j}} ⟶^{∼} Hom_{Z_{p}}(O_{𝔭_{j}}, Z_{p}), x ↦ (y ↦ Tr_{O_{𝔭_{j}} /Z_{p}}(dxy))

is a symmetric isomorphism of O_{𝔭_{j}}-modules. The composition of the two maps

T_{𝔭_{j}}(A_{j}) ⟶ Hom_{O_{𝔭_{j}}} (T_{𝔭_{j}}(A_{j}), Hom_{Z_{p}}(O_{𝔭_{j}}, Z_{p}(1)))⟶^{Hom(id,λ^{−1})}Hom_{O_{𝔭_{j}}} (T_{𝔭_{j}}(A_{j}), O_{𝔭_{j}}(1))
x ↦ (y ↦ (a ↦ (x, ay)_{𝔭_{j}}))

defines, by adjunction, a skew-symmetric O_{𝔭_{j}}-bilinear G_{F} -equivariant pairing

( , )_{j} : T_{𝔭_{j}}(A_{j}) × T_{𝔭_{j}}(A_{j}) ⟶ O_{𝔭_{j}}(1)

characterized by the property

(∀x, y ∈ T_{𝔭_{j}}(A_{j})) (∀a ∈ O_{𝔭_{j}}) Tr_{O_{𝔭_{j}} /Z_{p}}(da(x, y)_{j}) = (x, ay)_{𝔭_{j}} = (ax, y)_{𝔭_{j}}.

On tensoring with O_{𝔭} = O_{L,𝔭}, we obtain a skew-symmetric O_{𝔭}-bilinear G_{F} -equivariant pairing on T_{𝔭}(A) = T_{𝔭_{j}}(A_{j}) ⊗_{O_{𝔭_{j}}} O_{𝔭}:

\begin{matrix}( , ) : T_{𝔭}(A) × T_{𝔭}(A) & ⟶ O_{𝔭}(1) \\ x ⊗ a, y ⊗ b & ↦ ab(x, y)_{j} (x, y ∈ T_{𝔭_{j}}(A_{j}); a, b ∈ O_{𝔭}). \end{matrix}

The adjoint map T_{𝔭}(A) ⟶ Hom_{O_{𝔭}}(T_{𝔭}(A), O_{𝔭}(1)) is injective and its cokernel is killed by deg(φ). Reducing ( , ) modulo 𝔭^{M} yields a skew-symmetric O_{𝔭}/𝔭^{M}-bilinear G_{F} -equivariant pairing

( , )_{M} : A[𝔭^{M}] × A[𝔭^{M}] ⟶ O_{𝔭}/𝔭^{M}(1),  (5.19.1)

whose kernel is killed by deg(φ). This implies that, for each O_{𝔭}-submodule Y ⊂ A[𝔭^{M}] isomorphic to O_{𝔭}/𝔭^{M−c}, we have

2 deg(φ) 𝔭^{c} Y^{⊥} ⊆ Y.  (5.19.2)

(5.20) Proposition. In the situation of 5.16, let v_{α} be the prime of K(α) induced by v and ζ_{v} ∈ K(α)_{v_{α}} = K(x)_{v} be a root of unity whose image under the reciprocity map rec_{v} : K(x)_{v}^{∗} ⟶ G_{K(x)_{v}}^{ab} induces σ_{ℓ} ∈ Gal(K(x(ℓ))_{w}/K(x)_{v}). Then we have, for each x ∈ H_{ur}^{1}(K(α)_{v_{α}}, A[𝔭^{M}]),

ζ_{v} ⊗inv_{v_{α}}(x ∪ (cor_{K(x)/K(α)} c(𝔫ℓ))_{v_{α}}) = − (ev_{Fr(v_{α})}(x), Fr(ℓ) ev_{Fr(v_{α})} (cor_{K(x)/K(α)} c(𝔫))_{v_{α}})_{M} ∈O_{𝔭}/𝔭^{M}(1),

where ∪ denotes the cup product

H^{1}(K(α)_{v_{α}}, A[𝔭^{M}]) × H^{1}(K(α)_{v_{α}}, A[𝔭^{M}])⟶^{∪}H^{2}(K(α)_{v_{α}}, O_{𝔭}/𝔭^{M}(1))⟶^{inv_{vα}} O_{𝔭}/𝔭^{M}

induced by the pairing ( , )_{M}.

Proof. Combine Proposition 5.18 with Lemma 5.21 below (using the identification K(α)_{v_{α}} = K_{λ}, where λ = ℓO_{K}).

(5.21) Lemma. Let ℓ ∈ 𝒮_{1}(M), λ = ℓO_{K}. If ζ ∈ K_{λ}^{∗} is a root of unity of order prime to Nℓ, then we have, for each x ∈ H_{ur}^{1}(K_{λ}, A[𝔭^{M}]) and y ∈ H^{1}(K_{λ}, A[𝔭^{M}]),

ζ ⊗ inv_{λ}(x ∪ y) = (x(Fr(λ)), y(rec_{λ}(ζ)))_{M} ∈ O_{𝔭}/𝔭^{M}(1),

where rec_{λ} : K_{λ}^{∗} ⟶ G_{K_{λ}}^{ab} denotes the reciprocity map (normalized in such a way that the geometric Frobenius element Fr(λ) corresponds to a uniformizer).

Proof. We can assume that ζ is a primitive root of unity of order n = Nλ−1. Denoting by δ : K_{λ}^{∗}⊗Z/nZ ⟶^{∼} H^{1}(K_{λ}, μ_{n}) the standard Kummer map, then we have

\begin{aligned}x & = δζ ⊗ a ⊗ ζ^{⊗−1} ∈ H_{ur}^{1}(K_{λ}, μ_{n}) ⊗ A[𝔭^{M}] ⊗ μ_{n}^{⊗−1} = H_{ur}^{1}(K_{λ}, A[𝔭^{M}]) \\ y & = δu ⊗ b ⊗ ζ^{⊗−1} ∈ H^{1}(K_{λ}, μ_{n}) ⊗ A[𝔭^{M}] ⊗ μ_{n}^{⊗−1} = H^{1}(K_{λ}, A[𝔭^{M}]) \end{aligned}

for some a, b ∈ A[𝔭^{M}] and u ∈ K_{λ}^{∗} (recall that G_{K_{λ}} acts trivially on A[𝔭^{M}], since ℓ ∈ 𝒮_{1}(M)). As the reciprocity map is normalized by letting the uniformizers correspond to geometric Frobenius elements,

x(Fr(λ)) = (δζ)(Fr(λ)) ⊗ a ⊗ ζ^{⊗−1} = ζ^{−1} ⊗ a ⊗ ζ^{⊗−1} = −a.

For each character χ ∈ Hom_{cont}(G_{K_{λ}}, Q/Z) = H^{1}(G_{K_{λ}}, Q/Z), we have ([Se 1, Prop. XI.2])

(∀z ∈ K_{λ}^{∗})  χ(rec_{λ}(z)) = inv_{λ}(z ∪ δχ) = −inv_{λ}(δz ∪ χ) = inv_{λ}(χ ∪ δz),  (5.21.1)

where δχ ∈ H^{2}(G_{K_{λ}}, Z) denotes the coboundary associated to the exact sequence

0 ⟶ Z ⟶ Q ⟶ Q/Z ⟶ 0.

Using (5.21.1) with z = ζ and z = u, we obtain

y(rec_{λ}(ζ)) = (δu)(rec_{λ}(ζ)) ⊗ b ⊗ ζ^{⊗−1} = −(δζ)(rec_{λ}(u)) ⊗ b ⊗ ζ^{⊗−1} (= ord_{λ}(u)b)

ζ ⊗ inv_{λ}(x ∪ y) = inv_{λ}(δζ ∪ δu) (a, b)_{M} ⊗ ζ^{⊗−1} = (δζ)(rec_{λ}(u)) (a, b)_{M} ⊗ ζ^{⊗−1} = (x(Fr(λ)), y(rec_{λ}(ζ)))_{M} .

6. Linear algebra In order to simplify the notation, we denote

H = K(α),  H_{M} = H(A[𝔭^{M+M_{0}}]),  T = T_{𝔭}(A),  V = T_{𝔭}(A) ⊗_{O_{𝔭}} L_{𝔭}

Δ = Gal(H/K), U_{M} = Gal(H_{M}/H) ⊆ Aut_{O_{𝔭}}(A[𝔭^{M+M_{0}}]) ⟶^{∼} GL_{2}(O_{𝔭}/𝔭^{M+M_{0}}).

(6.1) The constant C_{2}(𝔭)

(6.1.1) Proposition. (i) (Bogomolov [Bo]) Z_{p}(A/H) := Z_{p}^{∗} ∩ Im (G_{H} ⟶ Aut_{Z_{p}}(T_{p}(A)) is an open subgroup of Z_{p}^{∗}. (ii) (Serre [Se 3]) There exists a constant c(A/H) such that the inequality (Z_{p}^{∗} : Z_{p}(A/H)) ≤ c(A/H) holds for all prime numbers p.

(6.1.2) Proposition. Consider the restriction map

res : H^{1}(H, A[𝔭^{M}]) ⟶ H^{1}(H_{M}, A[𝔭^{M}])^{U_{M}} = Hom_{U_{M}} (G_{H_{M}}^{ab} , A[𝔭^{M}]).

There exists an integer C_{2}(𝔭) ≥ 0 which is equal to zero for all but finitely many 𝔭 and such that 𝔭^{C_{2}(𝔭)} Ker(res) = 0 for all M.

Proof. Proposition 6.1.1(i) implies that there exists u ∈ Z_{p}^{∗} − {1} such that

(∀M ≥ 0) u (mod 𝔭^{M+M_{0}}) ∈ Z(U_{M}).

It follows from “Sah’s Lemma” ([Sa], proof of Prop. 2.7(b)) that u−1 kills H^{1}(U_{M}, A[𝔭^{M}]) = Ker(res), hence 𝔭^{C_{2}(𝔭)} Ker(res) = 0 for C_{2}(𝔭) = ord_{𝔭}(u − 1). According to Proposition 6.1.1(ii), we can take u ≢ 1 (mod p) (hence C_{2}(𝔭) = 0) if p − 1 > c(A/H). (6.2) The constant C_{3}(𝔭)

(6.2.1) Proposition. The following conditions are equivalent.

(i) V is an absolutely irreducible representation of G_{H}. (ii) End_{L_{𝔭}[G_{H}]}(V) = L_{𝔭}. (iii) For every non-trivial character η : Gal(H/F) ⟶ {±1}, the L_{𝔭}[G_{F}]-modules V and V ⊗ η are not isomorphic. (iv) For every totally imaginary quadratic extension K′ of F contained in H, the abelian variety A_{j} does not acquire complex multiplication over K′.

(v) For every K′ as in (iv), every totally imaginary quadratic extension K_{j} of L_{j}, every algebraic Hecke character ψ : A_{K′}^{∗} ⟶ K_{j}^{∗} and every embedding τ : K_{j} ↪ C, the L-series L(ψ_{τ} , s) (where ψ_{τ} : A_{K′}^{∗}/K^{′∗} ⟶ C^{∗} is the idèle class character asociated to ψ at τ) is not equal to any of the L-series L(π(σ ∘ θ_{j}), s − 1/2) from Proposition 1.18.

Proof. If F′/F is any quadratic extension, denote by η_{F′/F} : G_{F} ⟶ {±1} the associated quadratic character with kernel Ker(η_{F′/F}) = G_{F′}.

(i) ⟺ (ii): Irreducibility of V implies that its restriction to G_{H} is semi-simple. The equivalence of (i) and (ii) is then well-known ([Cu-Re, Thm. 3.43]). (ii) ⟹ (iii): According to [Tay], Prop. 3.1 (and using the fact that each complex conjugation acts on V (f) by a matrix with distinct eigenvalues ±1 ∈ L_{𝔭}), V is an absolutely irreducible representation of G_{F}, hence End_{L_{𝔭}[G_{F}]}(V) = L_{𝔭}, and both conditions (ii) and (iii) are invariant under finite extensions of the field L_{𝔭}.

We can assume, therefore, that L_{𝔭} contains all roots of unity of order [H : K]. The Frobenius reciprocity yields

End_{L_{𝔭}[G_{K}]}(V) = Hom_{L_{𝔭}[G_{F}]} (V, Ind_{G_{K}}^{G_{F}} (Res_{G_{F}}^{G_{K}} (V))) = Hom_{L_{𝔭}[G_{F}]} (V, V ⊕ (V ⊗ η_{K/F})) =
= L_{𝔭} ⊕ Hom_{L_{𝔭}[G_{F}]} (V, V ⊗ η_{K/F} ) .

(6.2.1.1)

Irreducibility of V implies that any non-trivial element of Hom_{L_{𝔭}[G_{F}]}(V, V ⊗ η_{K/F}) is an isomorphism of L_{𝔭}[G_{F}]-modules V ⟶^{∼} V ⊗ η_{K/F}, which proves the equivalence (ii) ⟺ (iii) in the case H = K. Similarly,

End_{L_{𝔭}[G_{H}]}(V) = Hom_{L_{𝔭}[G_{K}]} (V, Ind_{G_{H}}^{G_{K}} (Res_{G_{K}}^{G_{H}} (V))) = Hom_{L_{𝔭}[G_{K}]}(V, ⨁_{χ∈_{Δ}} V ⊗ χ) =

= ⨁_{χ∈_{Δ}} Hom_{L_{𝔭}[G_{K}]} (V, V ⊗ χ) ,

(6.2.1.2)

where Δ̂ = Hom(Δ, L_{𝔭}^{∗}). Assume that there exists an isomorphism of L_{𝔭}[G_{F}]-modules V ⟶^{∼} V ⊗η, for a nontrivial character η : Gal(H/F) ⟶ {±1}. If η = η_{K/F}, then (6.2.1.1) implies that End_{L_{𝔭}[G_{K}]}(V) ≠ L_{𝔭}, hence End_{L_{𝔭}[G_{H}]}(V) ≠ L_{𝔭}. If η ≠ η_{K/F}, then the character χ := η|_{Δ} ∈ Δ̂ is non-trivial, hence End_{L_{𝔭}[G_{H}]}(V) ≠ L_{𝔭}, by (6.2.1.2). (iii) ⟹ (ii): Assume that End_{L_{𝔭}[G_{H}]}(V) ≠ L_{𝔭}. If End_{L_{𝔭}[G_{K}]}(V) ≠ L_{𝔭}, then V ⟶^{∼} V ⊗ η_{K/F}, by (6.2.1.1). If End_{L_{𝔭}[G_{K}]}(V) = L_{𝔭}, then V is an absolutely irreducible representation of G_{K} and, thanks to (6.2.1.2), there exists a non-trivial character χ ∈ Δ̂ and a non-trivial morphism of L_{𝔭}[G_{K}]-modules V ⟶ V ⊗ χ, which is necessarily an isomorphism. As det(V) = det(V)χ^{2}, the values of χ are contained in {±1}. It follows that χ extends to a character χ_{F} : Gal(H/F) ⟶ {±1} (χ_{F} ≠ 1, η_{K/F}, since χ ≠ 1). Applying again the Frobenius reciprocity

0 ≠ Hom_{L_{𝔭}[G_{K}]} (V ⊗ χ, V) = Hom_{L_{𝔭}[G_{F}]} (V ⊗ χ_{F} , Ind_{G_{K}}^{G_{F}} (Res_{G_{F}}^{G_{K}} (V))) =
= Hom_{L_{𝔭}[G_{F}]} (V ⊗ χ_{F} , V ⊕ (V ⊗ η_{K/F})) ,

we deduce that there exists a character η ∈ {χ_{F} , χ_{F} η_{K/F} } (hence η ≠ 1) and a morphism of L_{𝔭}[G_{F}]-modules V ⟶ V ⊗ η, which is necessarily an isomorphism. (ii) ⟹ (iv): If A_{j} acquires complex multiplication over K′ as in (iv), then Im (L_{𝔭}[G_{K′}] ⟶ End_{L_{𝔭}}(V)) is a commutative subalgebra of End_{L_{𝔭}}(V) ⟶^{∼} M_{2}(L_{𝔭}), hence End_{L_{𝔭}[G_{H}]}(V) ⊇ End_{L_{𝔭}[G_{K′}]}(V) ⊋ L_{𝔭}.

(v) ⟹ (iv): This follows from the Main Theorem of Complex Multiplication ([Sh-Ta], [Mi 2, I.5]). (iv) ⟹ (v): If the L-function L(π(σ ∘θ_{j}), s−1/2) is given by a Hecke character of K′, then Im(L_{𝔭}[G_{K′}] ⟶ End_{L_{𝔭}}(V)) is a commutative subalgebra of End_{L_{𝔭}}(V) ⟶^{∼} M_{2}(L_{𝔭}). Faltings’ isogeny theorem then implies that End_{K′}(A_{j}) ⊗ Q contains a commutative subalgebra bigger than L_{j}, hence A_{j} acquires CM over K′.

(v) ⟹ (iii): Assume that η : Gal(H/F) ⟶ {±1} is a non-trivial character such that the L_{𝔭}[G_{F}]-modules V and V ⊗η are isomorphic. Then K′ = H^{Ker(η)} is a quadratic extension of F and End_{L_{𝔭}[G_{K′}]}(V) = L_{𝔭} ⊕L_{𝔭}, hence V is a semi-simple L-rational abelian representation of G_{K′} with infinite image. This implies, according to [He, Thm. 2] (see also [Se 2, III.2.3, Thm. 2]) that K′ is totally imaginary and G_{K′} acts on V ⊗_{L_{𝔭}} \overline{L}_{𝔭} by ψ_{𝔭} ⊕ ψ_{𝔭} ∘ ρ, where ψ_{𝔭} : G_{K′} ⟶ \overline{L}_{𝔭}^{∗} is the Galois representation associated to an algebraic Hecke character ψ of K′ (cf. [Ne], proof of 12.6.5.2), hence A_{j} acquires complex multiplication over K′.

(6.2.2) Proposition. If the equivalent conditions (i)-(v) of Proposition 6.2.1 hold, then there exists an integer C_{3}(𝔭) ≥ 0, which is equal to zero for all but finitely many 𝔭 and which satisfies

Im (O_{𝔭}[G_{H}] ⟶ End_{O_{𝔭}}(T)) ⊇ 𝔭^{C_{3}(𝔭)} End_{O_{𝔭}}(T).

Proof. Im (O_{𝔭}[G_{H}] ⟶ End_{O_{𝔭}}(T)) is an O_{𝔭}-lattice in the L_{𝔭}-algebra

A = Im (L_{𝔭}[G_{H}] ⟶ End_{L_{𝔭}}(V)) ⊆ End_{L_{𝔭}}(V).

As V is a faithful simple A-module satisfying End_{A}(V) = End_{L_{𝔭}[G_{H}]}(V) = L_{𝔭}, a theorem of Burnside ([Cu-Re, Thm. 3.32]) implies that A = End_{L_{𝔭}}(V), which proves the existence of C_{3}(𝔭). It remains to show that, for all but finitely many 𝔭, the map O_{𝔭}[G_{H}] ⟶ End_{O_{𝔭}}(T) is surjective. By the Nakayama Lemma, this amounts to the surjectivity of

k[G_{H}] ⟶ End_{k}(\overline{T}) (k = O_{𝔭}/𝔭, \overline{T} = T/𝔭T),

which would follow from absolute irreducibility of Res_{G_{F}}^{G_{H}} (\overline{T}). We can assume that p ∤ [H : F] (hence p ≠ 2) and, after replacing L by a finite extension, that L contains all roots of unity of order [H : K] (hence k does, too). According to [Di, Prop. 3.1], for all but finitely many 𝔭, \overline{T} is an absolutely irreducible k[G_{F}]-module, hence End_{k[G_{F}]}(\overline{T}) = k. For such 𝔭, \overline{T} is a semi-simple k[G_{H}]-module, by Clifford’s Theorem ([Cu-Re, Thm. 11.1(i)]; this applies to an arbitrary group G and its normal subgroup H of finite index), hence

Res_{G_{F}}^{G_{H}} (\overline{T}) is absolutely irreducible ⟺ End_{k[G_{H}]}(\overline{T}) = k.

The same argument as in the proof of 6.2.1 (ii) ⟺ (iii) shows that

End_{k[G_{H}]}(\overline{T}) ≠ k ⟺ (∃η : Gal(H/F) ⟶ \{±1\}, η ≠ 1) \overline{T} ⟶^{∼} \overline{T} ⊗ η as k[G_{F}] − modules.

If this were true for infinitely many 𝔭’s, we could find a common non-trivial character η : Gal(H/F) ⟶ {±1} for which the congruences

(∀g ∈ G_{F}) Tr(g | T) ≡ Tr(g | T ⊗ η) (mod 𝔭)

held for infinitely many 𝔭, hence

(∀g ∈ G_{F}) Tr(g | V) = Tr(g | V ⊗ η),

which would imply that V ⟶^{∼} V ⊗ η (by irreducibility of V). This contradiction with 6.2.1(iii) proves that C_{3}(𝔭) = 0 for all but finitely many 𝔭. (6.3) Galois groups (6.3.1) Let W′ ⊂ H^{1}(H, A[𝔭^{M}]) be an O_{𝔭}/𝔭^{M}-submodule of finite type. Denote by W = res(W′) ⊂ Hom_{U_{M}} (G_{H_{M}}^{ab} , A[𝔭^{M}]) its image under the restriction map from Proposition 6.1.2, by W^{⊥} ⊂ G_{H_{M}}^{ab} the annihilator of W, and set H_{M}(W) := (H_{M}^{ab})^{W^{⊥}}. The natural injective map

G := Gal(H_{M}(W)/H) ↪ Hom_{O_{𝔭}}(W, A[𝔭^{M}])
g ↦ (ev_{g} : w ↦ w(g))

is a morphism of Z[U_{M}]-modules, where U_{M} = Gal(H_{M}/H) ⊂ Aut_{O_{𝔭}}(A[𝔭^{M+M_{0}}]) acts trivially on W and by conjugation on G. Denote by

X = O_{𝔭} · G = O_{𝔭} · Gal(H_{M}(W)/H) ⊂ Hom_{O_{𝔭}}(W, A[𝔭^{M}])

the O_{𝔭}/𝔭^{M}-submodule of Hom_{O_{𝔭}}(W, A[𝔭^{M}]) generated by the image of G and by

j : X ↪ Hom_{O_{𝔭}}(W, A[𝔭^{M}])

the inclusion map. By construction, j is O_{𝔭}[U_{M}]-linear and the natural map

W ⟶ Hom_{O_{𝔭}}(X, A[𝔭^{M}])

is injective.

(6.3.2) Proposition. If the equivalent conditions of Proposition 6.2.1 are satisfied, then

(∀M, W′) 𝔭^{C_{3}(𝔭)} Coker(j) = 0.

Proof. This is a special case of Proposition 6.4.3 below, for

B = Im (L_{𝔭}[G_{H}] ⟶ End_{L_{𝔭}}(V)) = End_{L_{𝔭}}(V), D = L_{𝔭},
Λ = Im (O_{𝔭}[G_{H}] ⟶ End_{O_{𝔭}}(T)) ⊇ 𝔭^{C_{3}(𝔭)} End_{O_{𝔭}}(T), R = O_{𝔭}.

(6.3.3) Corollary. If the equivalent conditions of Proposition 6.2.1 are satisfied, then

𝔭^{C_{2}(𝔭)+C_{3}(𝔭)} Coker (j′ : X⟶^{j}Hom_{O_{𝔭}}(W, A[𝔭^{M}])⟶^{Hom(res,id)}Hom_{O_{𝔭}}(W′, A[𝔭^{M}])) = 0.

Proof. It follows from Proposition 6.1.2 that Coker(Hom(res, id)) is killed by 𝔭^{C_{2}(𝔭)}.

(6.3.4) Corollary. If the equivalent conditions of Proposition 6.2.1 are satisfied and if W′ is ρ-stable (where ρ ∈ G_{F} = Gal(\overline{F} /F) denotes the complex conjugation with respect to the fixed embedding \overline{F} ↪ C extending τ_{1}), so are W, H_{M}(W) and X. The maps j, j′ are ρ-equivariant and, denoting (−)^{±} = (−)^{ρ=±1}, we have

2X^{+} ⊆ O_{𝔭}G^{+} ⊆ X^{+}
𝔭^{C_{2}(𝔭)+C_{3}(𝔭)} Coker (j′ : X^{+} ↪ Hom_{O_{𝔭}}(W′, A[𝔭^{M}])^{+}) = 0
4 · Coker (Hom_{O_{𝔭}}(W′, A[𝔭^{M}])^{+} ⟶ Hom_{O_{𝔭}}((W′)^{+}, A[𝔭^{M}]^{+}) ⊕ Hom_{O_{𝔭}}((W′)^{−}, A[𝔭^{M}]^{−})) = 0

2^{4} 𝔭^{C_{2}(𝔭)+C_{3}(𝔭)} Coker (O_{𝔭} · 2G^{+} ↪ X^{+}⟶^{j′}Hom_{O_{𝔭}}((W′)^{+}, A[𝔭^{M}]^{+}) ⊕ Hom_{O_{𝔭}}((W′)^{−}, A[𝔭^{M}]^{−})) = 0.

(6.4) In this section we prove a general abstract version of Proposition 6.3.2. (6.4.1) Assume that V is a Q_{p}-vector space of finite dimension, B ⊂ End_{Q_{p}}(V) a Q_{p}-subalgebra, Λ ⊂ B a Z_{p}-order in B and T ⊂ V a Z_{p}-lattice such that ΛT = T. The Q_{p}-algebra

D = End_{B}(V)^{op}

acts on V on the right; its action commutes with the left action of B. The ring

R = {x ∈ D | T x ⊂ T }

is a Z_{p}-order in D. (6.4.2) In the situation of 6.4.1, assume that we are given the following data:

• An integer N ≥ 1.

• A right (R/p^{N}R)-module W = W_{R}.

• A left (Λ/p^{N}Λ)-module X = _{Λ}X.

• A Z_{p}-bilinear pairing

⟨ , ⟩ : X × W ⟶ T/p^{N}T

satisfying

\begin{matrix}⟨λx, w⟩ & = λ⟨x, w⟩ \\ ⟨x, wr⟩ & = ⟨x, w⟩r \end{matrix} (∀x ∈ X, ∀w ∈ W, ∀r ∈ R, ∀λ ∈ Λ)

such that the induced homomorphisms

i = i_{W} : W_{R} ⟶ Hom_{Λ} (_{Λ}X,_{Λ}(T/p^{N}T)_{R})
j = j_{X} : _{Λ}X ⟶ Hom_{R} (W_{R},_{Λ}(T/p^{N}T)_{R})

are both injective (i is a morphism of right R-modules, while j is a morphism of left Λ-modules).

(6.4.3) Proposition. Assume that we are given the data 6.4.2 and that B is a semi-simple Q_{p}-algebra. Then there exists a maximal Z_{p}-order R_{max} ⊃ R of D; fixing R_{max}, then

Λ_{max} := {b ∈ B | bT · R_{max} ⊂ T · R_{max}}

is a maximal Z_{p}-order in B containing Λ. Let c, d ∈ Z(R_{max}) be non-zero central elements in R_{max} such that cΛ_{max} ⊂ Λ and dR_{max} ⊂ R (they always exist). Then

cd Coker(j) = 0.

Proof. We begin by reducing to the case R = R_{max}, Λ = Λ_{max}. First of all, R_{max} exists by [Cu-Re, Thm. 26.5] and Λ_{max} is a maximal Z_{p}-order in B, by [Cu-Re, Thm. 26.20, Thm. 26.23(iii)]. Consider the maps

W_{R}⟶^{i}Hom_{Λ} (_{Λ}X,_{Λ}(T/p^{N}T)_{R}) ⟶^{γ}Hom_{Λ} (_{Λ}X,_{Λ_{max}}(T R_{max}/p^{N}T R_{max})_{R_{max}});

then

W̃_{R_{max}} := (γ ∘ i)(W_{R})R_{max}

is a right R_{max}-module. Similarly, consider

ΛX⟶^{j}Hom_{R} (W_{R},_{Λ}(T/p^{N}T)_{R}) ⟶^{δ}Hom_{R_{max}} (W̃_{R_{max}},_{Λ_{max}}(T R_{max}/p^{N}T R_{max})_{R_{max}})

and put

Λ_{max} X̃ := Λ_{max}(δ ∘ j(_{Λ}X));

this is a left Λ_{max}-module and the canonical maps

ĩ : W̃_{R_{max}} ⟶ Hom_{Λ_{max}} (_{Λ_{max}} X̃,_{Λ_{max}}(T R_{max}/p^{N}T R_{max})_{R_{max}})
j̃ : _{Λ_{max}} X̃ ⟶ Hom_{R_{max}} (W̃_{R_{max}},_{Λ_{max}}(T R_{max}/p^{N}T R_{max})_{R_{max}})

are injective. Assume that the statement has been proved for the maximal orders R_{max}, Λ_{max} and c = d = 1. Then j̃ is surjective, hence

δ ∘ j(_{Λ}X) ⊃ cΛ_{max}(δ ∘ j)(_{Λ}X) = c Hom_{R_{max}} (W̃, T R_{max}/p^{N}T R_{max}) ,

i.e. c kills Coker(δ ∘ j). As (T ∩ p^{N}T R_{max})d ⊂ p^{N}T, d kills Ker(δ) = Hom_{R}(W, (T ∩ p^{N}T R_{max})/p^{N}T). The exact sequence

Ker(δ) ⟶ Coker(j) ⟶ Coker(δ ∘ j)

then implies that cd kills Coker(j). We can assume, therefore, that R = R_{max} and Λ = Λ_{max}. The next step is the reduction to the case of a simple Q_{p}-algebra B. In general there is a finite decomposition B = ∏ e_{i}B, where each e_{i}B = B_{i} is a simple Q_{p}-algebra and e_{i} are the corresponding orthogonal idempotents. Then we have V = ⨁ V_{i}, V_{i} = e_{i}V. According to [Cu-Re, Thm. 26.20], Λ = ∏ Λ_{i}, where each Λ_{i} = e_{i}Λ is a maximal Z_{p}-order in B_{i}. As ΛT ⊂ T, we have T = ⨁ T_{i} with T_{i} = e_{i}T, which implies that D = ∏ D_{i}, D_{i} = End_{B_{i}}(V_{i})^{op} and R = ∏ R_{i}, R_{i} = {x ∈ D_{i} | T_{i}x ⊂ T_{i}}. Similarly, W = ⨁ e_{i}W and X = ⨁ e_{i}X. This implies that we can assume that B is a simple Q_{p}-algebra, hence B = M_{n}(D) for a skew-field D = End_{B}(V)^{op} and V = D^{n}. In this case D has a unique maximal order R; every left or right ideal in R is bilateral and is of the form ϖ_{R}^{m}R = R ϖ_{R}^{m} (m ≥ 0), where ϖ_{R} is a fixed prime element of R ([Cu-Re, Thm. 26.23]). There is an isomorphism of right R-modules

W_{R} ⟶^{∼} ⨁_{i=1}^{k} R/ϖ_{R}^{n_{i}}R,

where ϖ_{R}^{n_{i}} | p^{N} for each i. Using the fact that multiplication by ϖ_{R}^{n_{i}}p^{−N} induces an isomorphism between T · p^{N}ϖ_{R}^{−n_{i}}/p^{N}T and T/T ϖ_{R}^{n_{i}}, we define a left Λ-module _{Λ}Y as the fibre product

\xymatrix{_{Λ}Y \ar[d] \ar[rrr] &  &  & T^{k} \ar[d] \\ _{Λ}X \ar[r]^{j} & _{Hom_{R} (W_{R},_{Λ}(T/p^{N})_{R})} \ar[r]^{∼} & _{⨁{k \atop i=1} _{Λ} (T · p^{N}ϖ_{R}^{−n_{i}}/p^{N}T )} \ar[r]^{∼} & _{⨁{k \atop i=1} _{Λ} (T/T ϖ_{R}^{n_{i}})}}

The injectivity of i_{W} implies that Y has the following property:

(∀r_{1}, …, r_{k} ∈ R s.t. ∃i r_{i} ∈ R^{∗}) (∃y_{1}, …, y_{k} ∈ Y)  ∑_{i=1}^{k} y_{i}r_{i} ∉ T ϖ_{R}  (⋆_{k})

We shall prove, by induction on k, that any Λ-submodule _{Λ}Y ⊂ _{Λ}(T^{k})_{R} satisfying (⋆_{k}) is equal to T^{k}. This will imply that j_{X} is surjective, as claimed. Let k = 1. If _{Λ}Y satisfies (⋆_{1}), then Y ⊄ T ϖ_{R}. The lattice T is free over R and, in a suitable R-basis of T, Λ = M_{r}(R)^{op}. As (Y + T ϖ_{R})/T ϖ_{R} ⊂ T/T ϖ_{R} is a non-zero Λ/ϖ_{R} = M_{r}(R/ϖ_{R})-submodule of the simple Λ/ϖ_{R}-module T/T ϖ_{R}, we have Y + T ϖ_{R} = T, hence Y = T by the Nakayama Lemma. Suppose that k > 1 and that any _{Λ}Y′ satisfying (⋆_{k−1}) is equal to T^{k−1}. If _{Λ}Y ⊂ _{Λ}(T^{k})_{R} satisfies (⋆_{k}), then its image under the projection pr_{k} : T^{k} ⟶ T on the k-th factor satisfies (⋆_{1}), hence pr_{k}(Y) = T. As Λ is a maximal order in a semi-simple Q_{p}-algebra, it is hereditary, i.e. T is a projective Λ-module [Cu-Re, Thm. 26.12(ii)]. It follows that the (surjective) projection pr_{k} : Y ⟶ T admits a Λ-linear section s : T ⟶ Y. For each i = 1, …, k − 1, the map T ⟶^{s}Y ↪ T^{k}⟶^{pr_{k}}T is an element of End_{Λ}(T) = R^{op}, hence is given by right multiplication by some a_{i} ∈ R. It follows that

Y = {(xa_{1} + y_{1}, …, xa_{k−1} + y_{k−1}, x) | x ∈ T, y_{1}, …, y_{k−1} ∈ Y ∩ ker(pr_{k})}.  (6.4.3.1)

If r_{1}, …, r_{k−1} ∈ R and (∃i) r_{i} ∈ R^{∗}, then there exist y_{1}, …, y_{k−1} ∈ Y ∩ ker(pr_{k}) such that

∑_{i=1}^{k−1} y_{i}r_{i} = ∑_{i=1}^{k−1}(xa_{i} + y_{i})r_{i} − x ∑_{i=1}^{k−1} a_{i}r_{i} ∉ T ϖ_{R},

since Y satisfies (⋆_{k}). This implies that Y ∩ker(pr_{k}) satisfies (⋆_{k−1}), hence Y ∩ker(pr_{k}) = T^{k−1} by inductive hypothesis. It follows from (6.4.3.1) that Y = T^{k}, which concludes the proof of the Proposition.

(6.5) Galois groups and Frobenius elements

(6.5.1) In the situation of 6.3.1, assume that W′ is ρ-stable. Given an arbitrary element

g ∈ G^{+} = Gal(H_{M}(W)/H_{M})^{+},

then h = g^{2} = (ρ + 1)g ∈ 2G^{+}. The Čebotarev density theorem implies that there exist infinitely many non-archimedean primes ℒ (W) of H_{M}(W) satisfying the following properties:

(6.5.1.1) ℒ (W) is unramified in H_{M}(W)/F. (6.5.1.2) ℒ (W) is prime to (pu(0))c(x)N(I_{0}) and the prime ℓ of F induced by ℒ (W) does not lie in S. (6.5.1.3) Fr_{H_{M} (W)/F} (ℒ (W)) = ρg ∈ Gal(H_{M}(W)/F).

Denote by ℓ, λ, λ_{H} and ℒ, respectively, the primes of F, K, H and H_{M} induced by ℒ (W).

(6.5.2) Lemma. (i) ℓ ∈ 𝒮_{1}(M). (ii) The prime λ = ℓO_{K} splits completely in H_{M}/K. (iii) Fr_{H_{M} (W)/K}(ℒ (W)) = (ρg)^{2} = ρgρg = ^{ρ}g g = (ρ + 1)g = g^{2} = h ∈ 2G^{+}. (iv) The decomposition group of ℒ in H_{M}/F is equal to Gal(H_{M}/F)_{ℒ} = {1, ρ}. In particular, ρ(ℒ) = ℒ.

Proof. The statements (i) and (ii) follow from (6.5.1.2) and the equalities Fr_{H_{M} /F} (ℒ) = ρg|_{H_{M}} = ρ, Fr_{H_{M} /K}(ℒ) = Fr_{H_{M} /F} (ℒ)^{2} = ρ^{2} = 1. The statement (iii) is clear and (iv) follows from the fact that Gal(H_{M}/F)_{ℒ} is generated by Fr_{H_{M} /F} (ℒ) = ρ. (6.5.3) In the situation of 6.5.1, the element Fr(ℒ) := Fr_{H_{M} (W)/H_{M}} (ℒ (W)) ∈ G depends only on ℒ; more precisely, Fr(ℒ) = g^{2} = h ∈ 2G^{+}, by Lemma 6.5.2(ii). Its image via the map

j′ : G^{+} ↪ X^{+}⟶^{j}Hom_{O_{𝔭}}(W, A[𝔭^{M}])^{+} ⟶ Hom_{O_{𝔭}}(W′, A[𝔭^{M}])^{+}

is given by the evaluation map

j(Fr(ℒ) )(w′) = res(w′)(Fr(ℒ) )  (w′ ∈ W′)  (6.5.3.1)

(6.5.4) The prime ℒ determines identifications

A(H_{λ_{H}})[𝔭^{M}] = A((H_{M})_{ℒ} )[𝔭^{M}] = A(H_{M})[𝔭^{M}] = A(\overline{F})[𝔭^{M}] (=: A[𝔭^{M}]).

The image of W′ ⊂ H^{1}(H, A[𝔭^{M}]) under the localization map

res_{λ_{H}} : H^{1}(H, A[𝔭^{M}]) ⟶ H^{1}(H_{λ_{H}} , A[𝔭^{M}])

is contained in H_{ur}^{1}(H_{λ_{H}} , A[𝔭^{M}]) and the composite map

W′⟶^{res_{λ_{H}}}H_{ur}^{1}(H_{λ_{H}} , A[𝔭^{M}])⟶^{ev_{Fr(λ_{H})}}A[𝔭^{M}]

coincides with the map (6.5.3.1)

j(Fr(ℒ) ) : w′ ↦ res(w′)(Fr(ℒ) ).

We shall use repeatedly the fact that this map is ρ-equivariant (= is contained in Hom_{O_{𝔭}}(W′, A[𝔭^{M}])^{+}). In order to stress the dependence on ℒ, we shall write

w′_{ℒ} := res_{λ_{H}} (w′) (w′ ∈ W′).

(6.6) Linear and quadratic forms

(6.6.1) Lemma. Let

f(x) = ∑_{i=1}^{n} f_{i} x_{i},  g(x) = ∑_{1≤i≤j≤n} g_{ij} x_{i}x_{j}  (f_{i}, g_{ij} ∈ O_{𝔭}/𝔭^{N}, x = ∑_{i=1}^{n} x_{i}e_{i} ∈ ⨁_{i=1}^{n} O_{𝔭} e_{i} = O_{𝔭}^{⊕n})

be a linear and a quadratic form, respectively, in n ≥ 1 variables, with coefficients in O_{𝔭}/𝔭^{N}. Writing Z_{p}^{⊕n} = ⨁_{i=1}^{n} Z_{p} e_{i} ⊂ O_{𝔭}^{⊕n}, we have, for each m ∈ {0, …, N}:

(i) f(Z_{p}^{⊕n}) ⊆ 𝔭^{m}(O_{𝔭}/𝔭^{N}) ⟺ (∀i = 1, …, n) f_{i} ∈ 𝔭^{m}(O_{𝔭}/𝔭^{N}) ⟺ f(O_{𝔭}^{⊕n}) ⊆ 𝔭^{m}(O_{𝔭}/𝔭^{N}). (ii) g(Z_{p}^{⊕n}) ⊆ 𝔭^{m}(O_{𝔭}/𝔭^{N}) ⟺ (∀1 ≤ i ≤ j ≤ n) g_{ij} ∈ 𝔭^{m}(O_{𝔭}/𝔭^{N}) ⟺ g(O_{𝔭}^{⊕n}) ⊆ 𝔭^{m}(O_{𝔭}/𝔭^{N}). (iii) If U ⊊ Z_{p}^{⊕n} is a proper Z_{p}-submodule and g(Z_{p}^{⊕n}) ⊄ 𝔭^{m}(O_{𝔭}/𝔭^{N}) (1 ≤ m < N − ord_{𝔭}(2)), then

(∃x ∈ Z_{p}^{⊕n}, x ∉ U) g(x) ∉ 2 𝔭^{m}(O_{𝔭}/𝔭^{N}) = 𝔭^{m+ord_{𝔭}(2)}(O_{𝔭}/𝔭^{N}).

Proof. The statements (i) and (ii) follow from the equalities f_{i} = f(e_{i}), g_{ii} = g(e_{i}), g_{ij} = g(e_{i} + e_{j}) − g(e_{i}) − g(e_{j}) (i < j). (iii) There exists an integer r ≥ 1 and a Z_{p}-basis e′_{1}, …, e′_{n} of Z_{p}^{⊕n} such that

U ⊆ ⨁_{i=1}^{r} pZ_{p} e′_{i} ⊕ ⨁_{j=r+1}^{n} Z_{p} e′_{j}.

Writing x = ∑_{i=1}^{n} x_{i}e_{i} = ∑_{i=1}^{n} x′_{i}e′_{i}, we have

g(x) = ∑_{1≤i≤j≤n} g′_{ij} x′_{i}x′_{j},  g′_{ii} = g(e′_{i}),  g′_{ij} = g(e′_{i} + e′_{j}) − g(e′_{i}) − g(e′_{j}) (i < j).

It follows from (ii) that

(∃k ≤ l) g′_{kl} ∉ 𝔭^{m}(O_{𝔭}/𝔭^{N}).

We distinguish three cases:

(1) k ≤ l ≤ r: if k = l, then x := e′_{k} ∉ U and g(x) = g′_{kk} ∉ 𝔭^{m}(O_{𝔭}/𝔭^{N}). If k < l, then g(e′_{k}+e′_{l})−g(e′_{k})−g(e′_{l}) ∉ 𝔭^{m}(O_{𝔭}/𝔭^{N}), hence there exists x ∈ {e′_{k}, e′_{l}, e′_{k} + e′_{l}} (⟹ x ∉ U) such that g(x) ∉ 𝔭^{m}(O_{𝔭}/𝔭^{N}).

(2) k ≤ r < l and (∀i ≤ j ≤ r) g′_{ij} ∈ 𝔭^{m}(O_{𝔭}/𝔭^{N}): the congruence

(∀λ ∈ Z_{p}) g(e′_{k} + λe′_{l}) = g′_{kk} + λg′_{kl} + λ^{2}g′_{ll} ≡ λg′_{kl} + λ^{2}g′_{ll} (mod 𝔭^{m}(O_{𝔭}/𝔭^{N}))

implies that there exists λ ∈ {1, 2} such that x := e′_{k} + λe′_{l} ∉ U satisfies g(x) ∉ 2 𝔭^{m}(O_{𝔭}/𝔭^{N}).

(3) (∀i ≤ r) (∀j ≥ i) g′_{ij} ∈ 𝔭^{m}(O_{𝔭}/𝔭^{N}): in this case k > r and we have

(∀y ∈ ⨁_{j=r+1}^{n} O_{𝔭} e′_{j}) g(e′_{1} + y) ≡ g(y) (mod 𝔭^{m}(O_{𝔭}/𝔭^{N})).

If k = l, then x := e′_{1} + e′_{k} ∉ U satisfies g(x) ≡ g′_{kk} ≢ 0 (mod 𝔭^{m}(O_{𝔭}/𝔭^{N})). If k < l, then

g(e′_{1} + e′_{k} + e′_{l}) − g(e′_{1} + e′_{k}) − g(e′_{1} + e′_{l}) ≡ g(e′_{k} + e′_{l}) − g(e′_{k}) − g(e′_{l}) ≡ g′_{kl} ≢ 0 (mod 𝔭^{m}(O_{𝔭}/𝔭^{N})),

hence there exists x ∈ {e′_{1} + e′_{k} + e′_{l}, e′_{1} + e′_{k}, e′_{1} + e′_{l}} (⟹ x ∉ U) such that g(x) ∉ 𝔭^{m}(O_{𝔭}/𝔭^{N}). (6.6.2) Proposition. Let a, b, N ≥ 0 be integers such that a, 2b+ord_{𝔭}(2) < N. Let Z ⊂ (O_{𝔭}/𝔭^{N})^{⊕5} be a Z_{p}- submodule satisfying O_{𝔭}·Z ⊇ 𝔭^{a}(O_{𝔭}/𝔭^{N})⊕(𝔭^{b}(O_{𝔭}/𝔭^{N}))^{⊕4}. Then there exists an element z = (z_{0}, …, z_{4}) ∈ Z (z_{i} ∈ O_{𝔭}/𝔭^{N}) such that

ℓ_{O_{𝔭}} ((O_{𝔭}/𝔭^{N})/(O_{𝔭}/𝔭^{N})z_{0}) ≤ a
ℓ_{O_{𝔭}} ((O_{𝔭}/𝔭^{N})^{⊕2}/(O_{𝔭}/𝔭^{N})(z_{1}, z_{2}) + (O_{𝔭}/𝔭^{N})(z_{3}, z_{4})) ≤ 2b + ord_{𝔭}(2)

(where ℓ_{O_{𝔭}}(C) denotes the length of a finite O_{𝔭}-module C), hence

(O_{𝔭}/𝔭^{N})z_{0} ⊇ 𝔭^{a}(O_{𝔭}/𝔭^{N}), (O_{𝔭}/𝔭^{N})(z_{1}, z_{2}) + (O_{𝔭}/𝔭^{N})(z_{3}, z_{4}) ⊇ 2 𝔭^{2b}(O_{𝔭}/𝔭^{N})^{⊕2}.

Proof. Fix a set of Z_{p}-generators z^{(i)} = (z_{0}^{(i)}, …, z_{4}^{(i)}) ∈ Z (1 ≤ i ≤ n) of Z and set

f(x) = ∑_{i=1}^{n} x_{i}z_{0}^{(i)},  g(x) = | \begin{matrix}∑_{i=1}^{n} x_{i}z_{1}^{(i)} & ∑_{i=1}^{n} x_{i}z_{2}^{(i)} \\ ∑_{i=1}^{n} x_{i}z_{3}^{(i)} & ∑_{i=1}^{n} x_{i}z_{4}^{(i)} \end{matrix} |  (x = ∑_{i=1}^{n} x_{i} e_{i} ∈ ⨁_{i=1}^{n} O_{𝔭} e_{i} = O_{𝔭}^{⊕n}).

The assumptions imply that f(O_{𝔭}^{⊕n}) ⊄ 𝔭^{a+1}(O_{𝔭}/𝔭^{N}), hence

U := {x ∈ Z_{p}^{⊕n} | f(x) ∈ 𝔭^{a+1}(O_{𝔭}/𝔭^{N})}

is a proper (U ⊊ Z_{p}^{⊕n}) Z_{p}-submodule of Z_{p}^{⊕n}, by Lemma 6.6.1(i). The assumptions also imply that g(O_{𝔭}/𝔭^{N}) ⊄ 𝔭^{2b+1}(O_{𝔭}/𝔭^{N}), hence

(∃x = ∑_{i=1}^{n} x_{i}e_{i} ∈ Z_{p}^{⊕n}, x ∉ U) g(x) ∉ 𝔭^{2b+1+ord_{𝔭}(2)}(O_{𝔭}/𝔭^{N}),

by Lemma 6.6.1(iii). The element z = ∑_{i=1}^{n} x_{i} z^{(i)} ∈ Z then has the desired properties.

7. The main result We continue to use the notation from Sect. 3-6. (7.1) Selmer groups (7.1.1) Let F′/F be a finite extension and Σ an arbitrary set of primes of F′. We denote

S_{Σ}(A/F′, 𝔭^{M}) := Ker(H^{1}(F′, A[𝔭^{M}]) ⟶ ⨁_{v∉Σ} H^{1}(F′_{v}, A[𝔭^{M}])/Im(δ_{v})),

where

δ_{v} : A(F′_{v}) ⊗ O_{𝔭}/𝔭^{M} ↪ H^{1}(F′_{v}, A[𝔭^{M}])

denotes the coboundary map arising from the cohomology exact sequence associated to

0 ⟶ A[𝔭^{M}] ⟶ A(\overline{F})⟶^{𝔭^{M}}A(\overline{F}) ⟶ 0.

If v ∤ p∞ and if A has good reduction at the prime of F induced by v, then Im(δ_{v}) = H_{ur}^{1}(F′_{v}, A[𝔭^{M}]). This implies that, if Σ is finite, so is S_{Σ}(A/F′, 𝔭^{M}). (7.1.2) The classical Selmer group

S(A/F′, 𝔭^{M}) := S_{∅}(A/F′, 𝔭^{M})

sits in the standard exact sequence

0 ⟶ A(F′) ⊗ O_{𝔭}/𝔭^{M} ⟶^{δ}S(A/F′, 𝔭^{M}) ⟶ � (A/F′)[𝔭^{M}] ⟶ 0.

The inductive limit \underrightarrow{lim}_{M} S(A/F′, 𝔭^{M}) is an O_{𝔭}-module of co-finite type.

(7.1.3) If F′′/F′ is a finite Galois extension, then the restriction map induces a canonical homomorphism

S(A/F′, 𝔭^{M}) ⟶ S(A/F′′, 𝔭^{M})^{Gal(F′′/F′)},

whose kernel and cokernel are killed by [F′′ : F′]. (7.1.4) The Tate local duality [Mi 1, I.3.4-5] implies that, for each prime v of F′, Im(δ_{v}) is an isotropic subspace of H^{1}(F′_{v}, A[𝔭^{M}]) with respect to the cup product

H^{1}(F′_{v}, A[𝔭^{M}]) × H^{1}(F′_{v}, A[𝔭^{M}])⟶^{∪}H^{2}(F′_{v}, O_{𝔭}/𝔭^{M}(1))⟶^{inv_{v}}O_{𝔭}/𝔭^{M}

induced by the pairing ( , )_{M} from (5.19.1). (7.1.5) If Σ is a finite set of primes of F′, s ∈ S(A/F′, 𝔭^{M}) and c ∈ S_{Σ}(A/F′, 𝔭^{M}), then the reciprocity law

(∀a ∈ H^{2}(G_{F′}, O_{𝔭}/𝔭^{M}(1)))  ∑_{v} inv_{v}(a_{v}) = 0 ∈ O_{𝔭}/𝔭^{M}

applied to a = s ∪ c yields (thanks to 7.1.4)

∑_{v∈Σ} inv_{v}(s_{v} ∪ c_{v}) = 0 ∈ O_{𝔭}/𝔭^{M}.

(7.2) Annihilation relations (7.2.1) Kolyvagin’s classes. As in Sect. 6, we denote H = K(α). For each 𝔫 = ℓ_{1} ⋯ ℓ_{r} ∈ 𝒮_{r}(M) (r ≥ 0, ℓ_{i} ∈ 𝒮_{1}(M)), we define

κ_{𝔫} = ϖ^{C_{1}(𝔭)} e_{β} (cor_{K(x)/H}(c(𝔫))) ∈ H^{1}(H, A[𝔭^{M}])^{(β)},

where C_{1}(𝔭) = max_{v}C_{1,v}(𝔭) (the constants C_{1,v}(𝔭) were defined in 5.2). For r = 0, the cohomology class κ_{1} := κ_{(1)} is equal to

κ_{1} = ϖ^{C_{1}(𝔭)} δ(e_{β}(y)) ∈ S(A/H, 𝔭^{M})^{(β)},

where y = cor_{K(x)/H}(y_{j}) ∈ A(H). For r ≥ 1, Proposition 5.12 implies that

κ_{𝔫} ∈ S_{{v|𝔫}}(A/H, 𝔭^{M})^{(β)}.

(7.2.2) Assume that 𝔫 = ℓ_{1} ⋯ ℓ_{r} ∈ 𝒮_{r}(M) (r ≥ 1, ℓ_{i} ∈ 𝒮_{1}(M)). For each i ∈ {1, …, r}, fix a prime ℒ_{i} | ℓ_{i} of H_{M} such that Fr_{H_{M} /F} (ℒ_{i}) = ρ (as an element of Gal(H_{M}/F), not only as a conjugacy class) and denote by v_{i} the prime of H induced by ℒ_{i}. As in 6.5.4, ℒ_{i} determines identifications

A(H_{v_{i}})[𝔭^{M}] = A((H_{M})_{ℒ_{i}} )[𝔭^{M}] = A(\overline{F})[𝔭^{M}] (=: A[𝔭^{M}]),

and the localization map

W′_{i} := S_{{v|(𝔫/ℓ_{i})}}(A/H, 𝔭^{M})⟶^{res_{v_{i}}}H_{ur}^{1}(H_{v_{i}}, A[𝔭^{M}])⟶^{ev_{Fr(v_{i})}}A[𝔭^{M}]

coincides with the evaluation map w′ ↦ w′_{ℒ_{i}} := (res(w′))(Fr(ℒ_{i}) ).

(7.2.3) Proposition. In the situation of 7.2.2, assume that the generators σ_{ℓ_{i}} ∈ G(ℓ_{i}) have been chosen in the following compatible manner: there exists a global root of unity ζ ∈ μ_{p^{∞}}(H_{M}) such that, for each i = 1, …, r, the image of σ_{ℓ_{i}} in G(ℓ_{i}) ⊗ O_{𝔭}/𝔭^{M} coincides with the image of ζ under the composite map

μ_{p^{∞}}(H_{M}) ⊗ O_{𝔭}/𝔭^{M} ⟶ μ_{p^{∞}}((H_{M})_{ℒ_{i}} ) ⊗ O_{𝔭}/𝔭^{M} ⟵^{∼} μ_{p^{∞}}(H_{v_{i}}) ⊗ O_{𝔭}/𝔭^{M} ⟶^{rec_{v_{i}} ⊗id}G(ℓ_{i}) ⊗ O_{𝔭}/𝔭^{M}

(such a compatible choice always exists). Then we have, for each s ∈ S(A/H, 𝔭^{M})^{(β)}, an equality

[H : K] ∑_{i=1}^{r} (s_{ℒ_{i}} , (κ_{𝔫/ℓ_{i}})_{ℒ_{i}} )_{M} = 0

in μ_{p^{∞}}(H_{M}) ⊗ O_{𝔭}/𝔭^{M} = O_{𝔭}/𝔭^{M}(1).

Proof. Applying 7.1.5 to F′ = H, Σ = {v | 𝔫}, ρ(s) and c = κ_{𝔫}, we obtain

∑_{i=1}^{r} ∑_{σ∈Δ} inv_{σ(v_{i})} (res_{σ(v_{i})}(ρ(s)) ∪ res_{σ(v_{i})}(κ_{𝔫})) = 0 ∈ O_{𝔭}/𝔭^{M},  (7.2.3.1)

where Δ = Gal(H/K). As

(∀σ ∈ Δ)  σ(ρ(s)) = β^{−1}(σ)ρ(s),  σ(κ_{𝔫}) = β(σ)κ_{𝔫},

we deduce from (7.2.3.1) the equality

[H : K] ∑_{i=1}^{r} inv_{v_{i}} (res_{v_{i}}(ρ(s)) ∪ res_{v_{i}}(κ_{𝔫})) = 0 ∈ O_{𝔭}/𝔭^{M}.  (7.2.3.2)

Combined with the formula from Proposition 5.20 (with ζ_{v} = ζ), (7.2.3.2) yields

[H : K] ∑_{i=1}^{r} (ρ(s)_{ℒ_{i}} , Fr(ℓ_{i})(κ_{𝔫/ℓ_{i}})_{ℒ_{i}} )_{M} = 0 ∈ O_{𝔭}/𝔭^{M}(1).

Finally, it follows from

ρ(s)_{ℒ_{i}} = ρ(s_{ℒ_{i}} ) = Fr(ℓ_{i})s_{ℒ_{i}} ∈ A[𝔭^{M}]

that

((ρ(s)_{ℒ_{i}} , Fr(ℓ_{i})(κ_{𝔫/ℓ_{i}})_{ℒ_{i}} )_{M} = (Fr(ℓ_{i})s_{ℒ_{i}} , Fr(ℓ_{i})(κ_{𝔫/ℓ_{i}})_{ℒ_{i}} )_{M} = Fr(ℓ_{i}) (s_{ℒ_{i}} , (κ_{𝔫/ℓ_{i}})_{ℒ_{i}} )_{M} =
= − (s_{ℒ_{i}} , (κ_{𝔫/ℓ_{i}})_{ℒ_{i}} )_{M} .

(7.2.4) (i) If we do not choose the generators σ_{ℓ_{i}} compatibly, then there exist ζ ∈ μ_{p^{∞}}(H_{M}) and u_{1}, …, u_{r} ∈ Z_{p}^{∗} such that the image of σ_{ℓ_{i}} in G(ℓ_{i}) ⊗ O_{𝔭}/𝔭^{M} is equal to the image of ζ ⊗ u_{i} (i = 1, …, r). The statement of Proposition 7.2.3 then becomes

[H : K] ∑_{i=1}^{r} (s_{ℒ_{i}} , (κ_{𝔫/ℓ_{i}})_{ℒ_{i}} )_{M} ⊗ u_{i} = 0 ∈ μ_{p^{∞}}(H_{M}) ⊗ O_{𝔭}/𝔭^{M} = O_{𝔭}/𝔭^{M}(1).

(ii) We shall apply Proposition 7.2.3 only in situations when it is known a priori that

(∀i = 2, …, r) (s_{ℒ_{i}} , (κ_{𝔫/ℓ_{i}})_{ℒ_{i}} )_{M} = 0.

In such a case one does not have to worry about the compatibility of the σ_{ℓ_{i}}’s.

(7.3) Theorem. Assume that the condition (⋆) from Theorem 3.2 holds. If e_{β}(y) ∉ A(H)_{tors}, then there exists an integer C(𝔭) ≥ 0 which is equal to zero for all but finitely many 𝔭 and such that

(∀M >> 0) 𝔭^{C(𝔭)} (S(A/H, 𝔭^{M})^{(β)}/O_{𝔭} · κ_{1}) = 0.

(7.4) Thanks to 7.1.2, Theorem 7.3 implies Theorem 3.2 (the statements about the α^{−1}-components follow by applying ρ to the α-components). The proof of Theorem 7.3 will occupy the rest of Sect. 7. Recall that the constants C_{1}(𝔭), C_{2}(𝔭) and C_{3}(𝔭) were introduced in 7.2.1, 6.1 and 6.2, respectively. We set

C_{5}(𝔭) := ord_{𝔭}([H : K]), C_{6}(𝔭) := ord_{𝔭}(deg(φ)),

where φ : A_{j} ⟶ Â_{j} is the isogeny from 5.19. If β^{2} ≠ 1, then another constant C_{4}(𝔭) will be defined in 7.6.1 below. The assumption e_{β}(y) ∉ A(H)_{tors} implies that the constant

C_{0}(𝔭) := max{c ∈ Z_{≥0} | e_{β}(y) ∈ A(H)_{tors} + 𝔭^{c}A(H)}

is defined (and C_{0}(𝔭) = 0 for all but finitely many 𝔭). In order to simplify the notation, we write C_{i} = C_{i}(𝔭) (i = 0, …, 6). We also denote, for each O_{𝔭}/𝔭^{M}-module Y and an element y ∈ Y,

exp(y) := min{c ∈ Z_{≥0} | 𝔭^{c} y = 0}, exp(Y) := max{exp(y) | y ∈ Y }.

Using this notation, we have, for M >> 0,

exp(κ_{1}) ≥ M − C_{0} − C_{1}.  (7.4.1)

(7.5) Proof of Theorem 7.3 in the case β^{2} = 1. In this case S := S(A/H, 𝔭^{M})^{(β)} is ρ-stable; we denote S^{±} := S^{ρ=±1}. As

2κ_{1} = (1 + ρ)κ_{1} + (1 − ρ)κ_{1}, (1 ± ρ)κ_{1} ∈ S^{±},

it follows that

(∃ε ∈ {±1}) exp((1 + ερ)κ_{1}) ≥ M − C_{0} − C_{1} − ord_{𝔭}(2).

Fix such an ε; then x_{ε} := (1 + ερ)κ_{1} ∈ S^{ε}. (7.5.1) Bounding the exponent of S^{−ε}. Fix s ∈ S^{−ε} with maximal exp(s) and set x_{−ε} = s. Choosing the first Kolyvagin prime. We apply the discussion from 6.3 and 6.5 to W′ = S. Set

U_{±ε} := {h ∈ 2G^{+} | exp(j′(h))(x_{±ε}) < exp(x_{±ε}) − (C_{2} + C_{3} + 4 ord_{𝔭}(2))}.

It follows from Corollary 6.3.4 that U_{±ε} ⊊ 2G^{+} are proper subgroups of 2G^{+}, hence there exists h = g^{2} ∈ 2G^{+} − (U_{ε} ∪ U_{−ε}). Applying the discussion in 6.5.1 to the element h = g^{2}, choose a prime ℒ (W) satisfying (6.5.1.1-3). The induced primes ℒ | λ_{H} | λ | ℓ of H_{M}, H, K, F, respectively, satisfy ℓ ∈ 𝒮_{1}(M) and ρ(ℒ) = ℒ. The definition of U_{±ε} implies that

exp ((x_{±ε})_{ℒ} ) ≥ exp (x_{±ε}) − (C_{2} + C_{3} + 4 ord_{𝔭}(2)),

hence

exp (((1 + ερ)κ_{1})_{ℒ} ) ≥ M − ∑_{i=0}^{3} C_{i} − 5 ord_{𝔭}(2)  (7.5.1.1)
exp (s_{ℒ} ) ≥ exp (S^{−ε}) − (C_{2} + C_{3} + 4 ord_{𝔭}(2)).  (7.5.1.2)

The first annihilation relation. Applying Proposition 7.2.3 with 𝔫 = ℓ to s and ρ(s), we obtain (using the ρ-equivariance of the map w′ ↦ w′_{ℒ} ) that

[H : K] (s_{ℒ} , ((1 + ερ)κ_{1})_{ℒ} )_{M} = 0 ∈ O_{𝔭}/𝔭^{M}(1).  (7.5.1.3)

Combining (7.5.1.3) with (7.5.1.1) and (5.19.2), we obtain

2^{6} 𝔭^{C_{0}+C_{1}+C_{2}+C_{5}+C_{6}}s_{ℒ} ∈ O_{𝔭} · ((1 + ερ)κ_{1})_{ℒ} ⊂ A[𝔭^{M}]^{ρ=ε}.

On the other hand, s_{ℒ} ∈ A[𝔭^{M}]^{ρ=−ε} (since s ∈ S^{−ε}) and 2(A[𝔭^{M}]^{ρ=ε} ∩ A[𝔭^{M}]^{ρ=−ε}) = 0, hence

2^{7} 𝔭^{C_{0}+C_{1}+C_{2}+C_{5}+C_{6}}s_{ℒ} = 0.

Combined with (7.5.1.2), we obtain

𝔭^{M_{1}} S^{−ε} = 0,  M_{1} = C_{0} + C_{1} + 2C_{2} + 2C_{3} + C_{5} + C_{6} + 11 ord_{𝔭}(2).  (7.5.1.4)

In particular, 𝔭^{M_{1}}(1 − ερ)κ_{1} = 0, hence 2 𝔭^{M_{1}}κ_{1} = 𝔭^{M_{1}}(1 + ερ)κ_{1}. (7.5.2) Bounding the exponent of S^{ε}/O_{𝔭} · (1 + ερ)κ_{1}. Denote

S̃ = Ker (res_{λ_{H}} : S ⟶ H^{1}(H_{λ_{H}} , A[𝔭^{M}]))

and fix s̃ ∈ S̃^{ε} = S̃^{ρ=ε} with maximal exp(s̃). It follows from Proposition 5.18 combined with (5.15.1) that

exp((1 − ερ)κ_{ℓ}) ≥ exp (f (((1 − ερ)κ_{ℓ})_{ℒ} )) = exp (((1 + ερ)κ_{1})_{ℒ} ) ≥ M − ∑_{i=0}^{3} C_{i} − 5 ord_{𝔭}(2),

where f denotes the map

f : H^{1}(H_{λ_{H}} , A[𝔭^{M}]) ⟶ H^{1}(H_{λ_{H}} , A[𝔭^{M}])/H_{ur}^{1}(H_{λ_{H}} , A[𝔭^{M}])

(cf. 7.6.4 below). Choosing the second Kolyvagin prime. We apply the discussion from 6.3 and 6.5 to the submodule W′ = S_{{v|ℓ}}(A/H, 𝔭^{M})^{(β)}. Set x′_{ε} = s̃ ∈ (W′)^{ε} and x′_{−ε} = (1 − ερ)κ_{ℓ} ∈ (W′)^{−ε}. The argument as in 7.5.1 shows that there exists a prime ℓ′ ∈ 𝒮_{1}(M) (ℓ′ ≠ ℓ) and a prime ℒ′ | ℓ′ of H_{M} such that ρ(ℒ′) = ℒ′ and

exp (((1 − ερ)κ_{ℓ})_{ℒ′} ) ≥ M − ∑_{i=0}^{3} C_{i} − 5 ord_{𝔭}(2) − (C_{2} + C_{3} + 4 ord_{𝔭}(2))  (7.5.2.1)
exp (s_{ℒ′} ) ≥ exp \binom{}{S^{ε}} − (C_{2} + C_{3} + 4 ord_{𝔭}(2)).  (7.5.2.2)

The second annihilation relation. Applying Proposition 7.2.3 with 𝔫 = ℓℓ′ to s̃ and ρ(s̃), we obtain (using the ρ-equivariance of the map w′ ↦ w′_{ℒ′} and the assumption s̃_{ℒ} = 0) that

[H : K] (s_{ℒ′} , ((1 − ερ)κ_{ℓ})_{ℒ′} )_{M} = 0 ∈ O_{𝔭}/𝔭^{M}(1).  (7.5.2.3)

Combining (7.5.2.3) with (7.5.2.1) and (5.19.2), we deduce that

2^{10} 𝔭^{C_{0}+C_{1}+2C_{2}+2C_{3}+C_{5}+C_{6}}s_{ℒ′} ∈ O_{𝔭} · ((1 − ερ)κ_{ℓ})_{ℒ′} ⊂ A[𝔭^{M}]^{ρ=−ε}.

On the other hand, s̃ ∈ S̃^{ε} implies that s_{ℒ′} ⊂ A[𝔭^{M}]^{ρ=ε}, hence

2^{11} 𝔭^{C_{0}+C_{1}+2C_{2}+2C_{3}+C_{5}+C_{6}}s_{ℒ′} = 0.

Applying (7.5.2.2), we obtain

2^{15} 𝔭^{C_{0}+C_{1}+3C_{2}+3C_{3}+C_{5}+C_{6}} S̃^{ε} = 0.  (7.5.2.4)

(7.5.3) Bounding the exponent of S/O_{𝔭} κ_{1}. As 2H_{ur}^{1}(H_{λ_{H}} , A[𝔭^{M}])^{ε} is a cyclic O_{𝔭}-module, it follows from (7.5.1.1) that

2^{5} 𝔭^{C_{0}+C_{1}+C_{2}+C_{3}} (S^{ε}/ (S̃^{ε} + O_{𝔭} · (1 + ερ)κ_{1})) = 0.

Putting this together with (7.5.1.4) and (7.5.2.4), we deduce

2^{21} 𝔭^{2C_{0}+2C_{1}+4C_{2}+4C_{3}+C_{5}+C_{6}} (S/O_{𝔭} κ_{1}) = 0,

which finishes the proof of Theorem 7.3 in the case β^{2} = 1. (7.6) Proof of Theorem 7.3 in the case β^{2} ≠ 1. We write \overline{β} := β^{−1} ≠ β, S := S(A/H, 𝔭^{M}) (for M >> 0) and, as in 7.5, C_{i} := C_{i}(𝔭) (i = 0, …, 6).

(7.6.1) Lemma-Definition. Define

C_{4}(𝔭) = \{ \begin{matrix}0, & p ∤ order of β^{2} \\ ord_{𝔭}(p)/(p − 1), & p | order of β^{2}. \end{matrix}

(i) C_{4}(𝔭) ∈ Z_{≥0}. (ii) C_{4}(𝔭) ≤ C_{5}(𝔭)/(p − 1). (iii) For each O_{𝔭}[Δ]-module Y (where Δ = Gal(H/K)), 𝔭^{C_{4}(𝔭)} (Y^{(β)} ∩ Y^{(\overline{β})}) = 0.

Proof. (i) If p divides the order of β^{2}, then L_{𝔭} contains the values of β^{2}, hence L_{𝔭} ⊃ Q_{p}(μ_{p}).

(ii) This is clear. (iii) (∃σ ∈ Δ) β(σ) ≠ \overline{β}(σ); then Y^{(β)} ∩ Y^{(\overline{β})} is killed by β^{2}(σ) − 1 ≠ 0, but

ord_{𝔭}(β^{2}(σ) − 1) = \{ \begin{matrix}0, & β^{2}(σ) ∉ μ_{p^{∞}} \\ ≤ ord_{𝔭}(p)/(p − 1), & β^{2}(σ) ∈ μ_{p^{∞}}. \end{matrix}

(7.6.2) Definition. (i) Fix t_{±} ∈ T^{±} = T^{ρ=±1} such that T^{±} = O_{𝔭} t_{±} and let t_{±,M} be the image of t_{±} in (T/𝔭^{M}T)^{±} = A[𝔭^{M}]^{±}. (ii) exp(t_{±,M}) ≥ M − ord_{𝔭}(2); fix u_{±} ∈ (O_{𝔭}/𝔭^{M})t_{±,M} with exp(u_{±}) = M − ord_{𝔭}(2). (iii) Let W′ be as in 6.3.1 and let f ∈ Hom_{O_{𝔭}}(W′, A[𝔭^{M}])^{+}. Then 2f((W′)^{±}) ⊂ (O_{𝔭}/𝔭^{M−ord_{𝔭}(2)})u_{±}; we define the morphisms

f_{±}(W′)^{±} ⟶ O_{𝔭}/𝔭^{M−ord_{𝔭}(2)},  (∀w ∈ (W′)^{±})  f_{±}(w)u_{±} = 2f(w).

(iv) For each w ∈ W′, 2w = (1 + ρ)w + (1 − ρ)w, hence

4f(w) = f_{+}((1 + ρ)w)u_{+} + f_{−}((1 − ρ)w)u_{−}.

(7.6.3) Choosing the first Kolyvagin prime. We have

κ_{1} ∈ S^{(β)},  ρ(κ_{1}) ∈ S^{(\overline{β})},  exp(κ_{1}) = exp(ρ(κ_{1})) ≥ M − C_{0} − C_{1}.

As 𝔭^{C_{4}} (S^{(β)} ∩ S^{(\overline{β})}) = 0, it follows that the elements (1 ± ρ)κ_{1} ∈ S^{±} = S^{ρ=±1} satisfy

exp((1 ± ρ)κ_{1}) ≥ M − C_{0} − C_{1} − C_{4} − ord_{𝔭}(2).  (7.6.3.1)

Applying Corollary 6.3.4 to W′ = S, we obtain, as in 7.5.1, primes ℓ ∈ 𝒮_{1}(M) and ℒ | ℓ in H_{M} such that ρ(ℒ) = ℒ and

exp (((1 ± ρ)κ_{1}))_{ℒ} ) ≥ exp((1 ± ρ)κ_{1}) − (C_{2} + C_{3} + 4 ord_{𝔭}(2)) ≥ M − M_{2},  (7.6.3.2)

where

M_{2} = ∑_{i=0}^{4} C_{i} + 5 ord_{𝔭}(2).

As 2κ_{1} = (1 + ρ)κ_{1} + (1 − ρ)κ_{1} and 2(A[𝔭^{M}]^{+} ∩ A[𝔭^{M}]^{−}) = 0, it follows that

exp ((κ_{1})_{ℒ} ) , exp ((ρ(κ_{1}))_{ℒ} ) ≥ M − M_{2} − 2 ord_{𝔭}(2).  (7.6.3.3)

The first annihilation relation. Let s ∈ S^{(β)}. Applying Proposition 7.2.3 with 𝔫 = ℓ, we obtain

[H : K] (s_{ℒ} , (κ_{1})_{ℒ} )_{M} = 0 ∈ O_{𝔭}/𝔭^{M}(1),

hence, using (5.19.2) and (7.6.3.3),

2^{3} 𝔭^{M_{2}+C_{5}+C_{6}} s_{ℒ} = 2^{8} 𝔭^{∑_{i=0}^{6} C_{i}} s_{ℒ} ∈ O_{𝔭} (κ_{1})_{ℒ} .  (7.6.3.4)

Set

S̃ = Ker(S⟶^{(res_{v})} ⨁_{v|ℓ} H^{1}(H_{v}, A[𝔭^{M}]));

then

S̃^{(β)} = Ker (res_{λ_{H}} : S^{(β)} ⟶ H^{1}(H_{λ_{H}} , A[𝔭^{M}])) ,

and the inclusion (7.6.3.4) can be reformulated as follows:

2^{3} 𝔭^{M_{2}+C_{5}+C_{6}} (S^{(β)}/ (S̃^{(β)} + O_{𝔭} κ_{1})) = 2^{8} 𝔭^{∑_{i=0}^{6} C_{i}} (S^{(β)}/ (S̃^{(β)} + O_{𝔭} κ_{1})) = 0.  (7.6.3.5)

(7.6.4) Choosing the second Kolyvagin prime. In the exact sequence of O_{𝔭}/𝔭^{M}-modules

0 ⟶ H_{ur}^{1}(H_{λ_{H}} , A[𝔭^{M}]) ⟶ H^{1}(H_{λ_{H}} , A[𝔭^{M}])⟶^{f} \frac{H^{1}(H_{λ_{H}} , A[𝔭^{M}])}{H_{ur}^{1}(H_{λ_{H}} , A[𝔭^{M}])} ⟶ 0,

both flank terms are isomorphic to A[𝔭^{M}] ⟶^{∼} (O_{𝔭}/𝔭^{M})^{⊕2}, hence the sequence splits and the middle term is isomorphic, as an O_{𝔭}/𝔭^{M}-module, to (O_{𝔭}/𝔭^{M})^{⊕4}. We know that

(ρ(κ_{1}))_{ℒ} = ρ((κ_{1})_{ℒ} ) ∈ H_{ur}^{1}(H_{λ_{H}} , A[𝔭^{M}]) ⟶^{∼} A[𝔭^{M}], ((1 ± ρ)κ_{1})_{ℒ} ∈ A[𝔭^{M}]^{±}

and

exp (((1 ± ρ)κ_{1})_{ℒ} ) ≥ M − M_{2}.

On the other hand, (5.15.1) and Proposition 5.18 imply that

f (((1 ± ρ)κ_{ℓ})_{ℒ} ) = −Φ_{λ_{H}} (((1 ∓ ρ)κ_{1})_{ℒ} ) ,

hence

exp (f (((1 ± ρ)κ_{ℓ})_{ℒ} )) ≥ M − M_{2}.  (7.6.4.1)

Set

W′ = S_{{v|ℓ}}(A/H, 𝔭^{M}),  W̃′ = Ker(W′⟶^{(res_{v})} ⨁_{v|ℓ} H^{1}(H_{v}, A[𝔭^{M}]));

then

(W̃′)^{(β)} = S̃^{(β)},  (W̃′)^{(\overline{β})} = S̃^{(\overline{β})},  κ_{1}, κ_{ℓ} ∈ (W′)^{(β)},  ρ(κ_{1}), ρ(κ_{ℓ}) ∈ (W′)^{(\overline{β})}.

Denote by U ⊆ W′ the O_{𝔭}/𝔭^{M}-submodule generated by (1 + ρ)κ_{1}, (1 − ρ)κ_{1}, (1 + ρ)κ_{ℓ} and (1 − ρ)κ_{ℓ}. It follows from (7.6.3.2) and (7.6.4.1) that

res_{λ_{H}} (U) ⊇ 𝔭^{M_{2}} H^{1}(H_{λ_{H}} , A[𝔭^{M}]),  (7.6.4.2)

hence

𝔭^{M_{2}}(U ∩ W̃′) = 0  (7.6.4.3)

and U contains an O_{𝔭}/𝔭^{M}-submodule isomorphic to (𝔭^{M_{2}}(O_{𝔭}/𝔭^{M}))^{⊕4}.

Fix an element s̃ ∈ (W̃′)^{(β)} = S̃^{(β)} with maximal exp(s̃); then the argument used in the proof of (7.6.3.1) shows that

exp((1 + ρ)s̃) ≥ exp(s̃) − C_{4} − ord_{𝔭}(2).  (7.6.4.4)

Define a homomorphism of O_{𝔭}/𝔭^{M}-modules

z : Hom_{O_{𝔭}}(W′, A[𝔭^{M}])^{+} ⟶ (O_{𝔭}/𝔭^{N})^{⊕5} (N = M − ord_{𝔭}(2))

by the formula

z(f) = (f_{+}((1 + ρ)s̃), f_{+}((1 + ρ)κ_{1}), f_{−}((1 − ρ)κ_{1}), f_{+}((1 + ρ)κ_{ℓ}), f_{−}((1 − ρ)κ_{ℓ})))

and set

Z = (z ∘ j′)(2G^{+}) = (z ∘ j′)(2 Gal(H_{M}(W)/H_{M})^{+}),

which is a Z_{p}-submodule of (O_{𝔭}/𝔭^{N})^{⊕5}. It follows from (7.6.3.2), (7.6.4.1), (7.6.4.3) and (7.6.4.4) that

Im(z) ⊇ 2 𝔭^{a′}(O_{𝔭}/𝔭^{N}) ⊕ (2 𝔭^{M_{2}}(O_{𝔭}/𝔭^{N}))^{⊕4},

where

N − a′ ≥ exp(s̃) − (M_{2} + C_{4} + ord_{𝔭}(2)).

Corollary 6.3.4 implies that

O_{𝔭} · Z ⊇ 𝔭^{a}(O_{𝔭}/𝔭^{N}) ⊕ 𝔭^{b}(O_{𝔭}/𝔭^{N}))^{⊕4},

where

N − a ≥ exp(s̃) − (M_{2} + C_{2} + C_{3} + C_{4} + 6 ord_{𝔭}(2)), b = M_{2} + C_{2} + C_{3} + 5 ord_{𝔭}(2).

According to Proposition 6.6.2, there exists an element h = g^{2} ∈ 2G^{+} such that the corresponding vector

(z_{0}, …, z_{4}) = (z ∘ j′)(h) ∈ (O_{𝔭}/𝔭^{N})^{⊕5}

satisfies

z_{0} ∉ 𝔭^{a+1}(O_{𝔭}/𝔭^{N}),  | \begin{matrix}z_{1} & z_{2} \\ z_{3} & z_{4} \end{matrix} | ∉ 2 𝔭^{2b+1}(O_{𝔭}/𝔭^{N}).  (7.6.4.5)

Applying the discussion from 6.5.1, we choose ℒ′(W) satisfying (6.5.1.1-3) and not dividing ℓ. We denote by ℒ′, λ′_{H}, ℓ′ ∈ 𝒮_{1} (ℓ′ ≠ ℓ) the induced primes of H_{M}, H and F, respectively. By construction, we have

2 ((1 + ρ)s̃)_{ℒ′} = z_{0}u_{+}, \begin{aligned}2 ((1 + ρ)κ_{1})_{ℒ′} & = z_{1}u_{+}, 2 ((1 − ρ)κ_{1})_{ℒ′} = z_{2}u_{−}, \\ 2 ((1 + ρ)κ_{ℓ})_{ℒ′} & = z_{3}u_{+}, 2 ((1 − ρ)κ_{ℓ})_{ℒ′} = z_{4}u_{−}, \end{aligned}

hence

4(κ_{1})_{ℒ′} = z_{1}u_{+} + z_{2}u_{−}, 4(κ_{ℓ})_{ℒ′} = z_{3}u_{+} + z_{4}u_{−}.

It follows from (7.6.4.5) that

exp (((1 + ρ)s̃)_{ℒ′} ) ≥ M − a ≥ exp(s̃) − (M_{2} + C_{2} + C_{3} + C_{4} + 5 ord_{𝔭}(2)) = exp(s̃) − (b + C_{4}) (7.6.4.6)

and

res_{λ′_{H}} (O_{𝔭} κ_{1} + O_{𝔭} κ_{ℓ}) ⊇ 𝔭^{2b}H_{ur}^{1}(H_{λ′_{H}} , A[𝔭^{M}]) ⟶^{∼} 𝔭^{2b}A[𝔭^{M}].  (7.6.4.7)

The second annihilation relation. Applying Proposition 7.2.3 with 𝔫 = ℓℓ′ to the above element s = s̃ ∈ S̃^{(β)}, we obtain, as in (7.5.2.3),

[H : K] (s_{ℒ′} , (κ_{ℓ})_{ℒ′} )_{M} = 0 ∈ O_{𝔭}/𝔭^{M}(1).

Applying Proposition 7.2.3 with 𝔫 = ℓ′ to s = s̃ (i.e., the first annihilation relation with ℓ′ instead of ℓ), we obtain

[H : K] (s_{ℒ′} , (κ_{1})_{ℒ′} )_{M} = 0 ∈ O_{𝔭}/𝔭^{M}(1).

Combining these two relations with (7.6.4.7), we deduce that the element 𝔭^{2b+C_{5}} s_{ℒ′} ∈ A[𝔭^{M}] lies in the kernel of the pairing ( , )_{M}, hence

𝔭^{2b+C_{5}+C_{6}} s_{ℒ′} = 0.

It follows from (7.6.4.6) that

exp \binom{}{S^{(β)}} = exp(s̃) ≤ exp (((1 + ρ)s̃)_{ℒ′} ) + (b + C_{4}) ≤ (2b + C_{5} + C_{6}) + (b + C_{4}) =
= 3b + C_{4} + C_{5} + C_{6} = 3C_{0} + 3C_{1} + 6C_{2} + 6C_{3} + 4C_{4} + C_{5} + C_{6} + 30 ord_{𝔭}(2).

Combined with (7.6.3.5), this relation yields

2^{38} 𝔭^{4C_{0}+4C_{1}+7C_{2}+7C_{3}+5C_{4}+2C_{5}+2C_{6}} (S^{(β)}/O_{𝔭} κ_{1}) = 0,

which concludes the proof of Theorem 7.3 (hence also the proof of Theorem 3.2) in the case β^{2} ≠ 1.

References

[Be] M. Bertolini, Selmer groups and Heegner points in anticyclotomic Z_{p}-extensions, Compositio Math. 99

(1995), 153–182. [Be-Da] M. Bertolini, H. Darmon, Kolyvagin’s descent and Mordell-Weil groups over ring class fields, J. Reine

Angew. Math. 412 (1990), 63–74.

[Bi] B. Birch, Heegner points of elliptic curves, In: Symposia Mathematica, Vol. XV (Convegno di Strutture in Corpi Algebrici, INDAM, Rome, 1973), pp. 441–445. Academic Press, London, 1975.

[Bo] F.A. Bogomolov, Sur l’algébricité des représentations l-adiques, C. R. Acad. Sci. Paris Sér. A-B 290 (1980), no. 15, A701–A703.

[Bu] K. Buzzard, Integral models of certain Shimura curves, Duke Math. J. 87 (1997), 591–612. [Ca 1] H. Carayol, Sur la mauvaise réduction des courbes de Shimura, Compositio Math. 59 (1986), 151–230. [Ca 2] H. Carayol, Sur les représentations ℓ-adiques attachées aux formes modulaires de Hilbert, Ann. Sci.

E.N.S. 19 (1986), 409–469.

[Cas] W. Casselman, On abelian varieties with many endomorphisms and a conjecture of Shimura’s, Invent. Math. 12 (1971), 225–236.

[Con] B. Conrad, Gross-Zagier revisited, in: Heegner points and Rankin L-series, (H. Darmon, S.-W. Zhang, eds.), MSRI Publ. 49, Cambridge Univ. Press, Cambridge, 2004, pp. 67–163.

[Co] C. Cornut, Réduction de Familles de Points CMs, Thesis, Strasbourg, 2000. [Co-Va 1] C. Cornut, V. Vatsal, Nontriviality of Rankin-Selberg L-functions and CM points, to appear in: L-

functions and Galois representations (Durham 2004).

[Co-Va 2] C. Cornut, V. Vatsal, CM points and quaternion algebras, Documenta Math. 10 (2005), 263-309. [Cu-Re] C.W. Curtis, I. Reiner, Methods of Representation Theory, Vol. I, Wiley, New York, 1990.

[De] P. Deligne, Travaux de Shimura, Séminaire Bourbaki, exposé 389, 1970/71, Lect. Notes in Math. 244,

Springer, Berlin, 1971, pp. 123–165. [De-Ra] P. Deligne, M. Rapoport, Les schémas de modules de courbes elliptiques, in: Modular functions of one

variable II (Antwerp, 1972), Lect. Notes in Math. 349, Springer, Berlin, 1973, pp. 143–316.

[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, Two theorems on modular curves (Russian), Funkcional. Anal. i Priložen. 7 (1973), 83–84. English translation: Functional Anal. Appl. 7 (1973), 155–156.

[Fa] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), 349–366. Erratum: Invent. Math. 75 (1984), 381.

[Ge] S. Gelbart, Lectures on the Arthur-Selberg trace formula, Univ. Lect. Series 9, Amer. Math. Soc., Providence, 1996.

[Gr 1] B.H. Gross, Kolyvagin’s work on modular elliptic curves, in: L-functions and arithmetic (Durham, 1989; J. Coates, M.J. Taylor, eds.), London Math. Soc. Lect. Note Ser. 153, Cambridge Univ. Press, Cambridge, 1991, pp. 235–256.

[Gr 2] B.H. Gross, Heegner points and representation theory, in: Heegner points and Rankin L-series, (H. Dar-

mon, S.-W. Zhang, eds.), MSRI Publ. 49, Cambridge Univ. Press, Cambridge, 2004, pp. 37–65.

[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, pp. 107–126. [Ho 1] B. Howard, The Heegner point Kolyvagin system, Compos. Math. 140 (2004), 1439–1472. [Ho 2] B. Howard, Iwasawa Theory of Heegner points on abelian varieties of GL_{2}-type, Duke Math. J. 124

(2004), 1–45. [Ja-La] H. Jacquet, R. Langlands, Automorphic forms on GL(2), Lect. Notes in Math. 114, Springer, Berlin-

New York, 1970.

[Ka-Ma] N. Katz, B. Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies 108, Princeton

Univ. Press, Princeton, 1985.

[Ko] V. A. Kolyvagin, Euler systems, in: The Grothendieck Festschrift II, Progress in Mathematics 87,

Birkhäuser, Boston, Basel, Berlin, 1990, pp. 435–483.

[Ko-Lo 1] 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), 171–196. English translation: Leningrad Math. J. 1 (1990), 1229–1253.

[Ko-Lo 2] V.A. Kolyvagin, D.Yu. Logachev, Finiteness of � over totally real fields (Russian), Izv. Akad. Nauk.

SSSR, Ser. Math. 55 (1991), 851–876. English translation: Math. USSR-Izv. 39 (1992), 829–853. [Mi 1] J.S. Milne, Arithmetic duality theorems, Persp. in Math. 1, Academic Press, Boston, 1986. [Mi 2] 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,

[Mi 3] J.S. Milne, The points on a Shimura variety modulo a prime of good reduction, in: The zeta functions of Picard modular surfaces, (R. Langlands, D. Ramakrishnan, eds.), Univ. Montreal, Montreal, 1992, pp. 151–253.

[Miy] T. Miyake, Modular forms, Springer, Berlin, 1989.

[Mu] D. Mumford, Abelian varieties, Oxford Univ. Press, London, 1970.

[Ne] J. Nekovář, Selmer complexes, to appear in Astérisque. Available at

http://www.math.jussieu.fr/∼nekovar/pu/

[Ne-Sch] J. Nekovář, N. Schappacher, On the asymptotic behaviour of Heegner points, Turkish J. of Math. 23

(1999), 549–556.

[Oh] M. Ohta, On l-adic representations attached to automorphic forms, Japan. J. Math. 8 (1982), 1–47.

[Ri] K. Ribet, Galois action on division points of Abelian varieties with real multiplications, Amer. J. Math. 98 (1976), 751–804.

[Sa] C.-H. Sah, Automorphisms of finite groups, J. Algebra 10 (1968), 47–68. [Se 1] J.-P. Serre, Corps locaux, Hermann, Paris, 1962. [Se 2] J.-P. Serre, Abelian l-adic representations and elliptic curves, W. A. Benjamin, New York-Amsterdam,

1968. [Se 3] J.-P. Serre, Lettre à Marie-France Vignéras du 10/2/1986, Collected Papers IV, 38–55. [Sh 1] G. Shimura, On canonical models of arithmetic quotients of bounded symmetric domains, Ann. of Math.

91 (1970), 144–222. [Sh 2] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton Univ. Press,

Princeton, 1971. [Sh-Ta] G. Shimura, Y. Taniyama, Complex multiplication of abelian varieties and its applications to number

theory, Publ. Math. Soc. Japan 6, Tokyo, 1961.

[Tay] 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, Boston, 1995, pp. 185–191.

[Ti] Y. Tian, Euler systems of CM points on Shimura curves, Thesis, Columbia University, 2003. [Ti-Zh] Y. Tian, S.-W. Zhang, in preparation.

[Wa] L.C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics 83, Second ed., Springer, New York, 1997.

[Zh 1] S.-W. Zhang, Heights of Heegner points on Shimura curves, Ann. of Math. (2) 153 (2001), 27–147. [Zh 2] S.-W. Zhang, Gross-Zagier formula for GL_{2}, Asian J. Math. 5 (2001), 183–290.

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

www: http://www.math.jussieu.fr/∼nekovar/