Non-triviality of CM points in ring class field towers

Esther Aflalo, Jan Nekovář

Introduction

In [Co-Va 1], Cornut and Vatsal proved a generalisation of Mazur’s conjecture on higher Heegner points, valid for CM points on Shimura curves with unramified central character over totally real number fields. In the present work, which grew out of the thesis of the first author at University Paris 6, we extend the results of [Co-Va 1] to more general families of CM points on slightly less general Shimura curves (those with trivial central character).

In more concrete terms, let X be such a Shimura curve over a totally real number field F and let A_{0} be a simple quotient of the Jacobian of X. Fix a quadratic CM extension K of F and two relatively prime ideals c and b ≠ (1) of O_{F}. Let K[cb^{∞}] be the union of all ring class fields of K (in the sense of 1.1.2 below) of conductors cb^{m} (m ≥ 1); fix a character χ_{0} of Gal(K[cb^{∞}]/K)_{tors}. If {x_{m}}_{m∈N} is a sequence of CM points on X lying in the same b-isogeny class (in the sense of 2.2.4 below) and such that b^{m} divides the conductor c(x_{m}) of x_{m} for all m ≥ 1, then our main result (Theorem 4.3.1) states that, asuming a certain natural necessary compatibility condition for A_{0} and χ_{0}, for each sufficiently large m >> 0 there exists a character χ_{m} of Gal(K[cb^{∞}]/K) of finite order which extends χ_{0}, whose conductor differs from c(x_{m}) by a divisor of c, and for which the \overline{χ}_{m}-component of the image of x_{m} in A_{0} is not torsion.

The proof is based on the techniques of Cornut and Vatsal, who proved in [Co-Va 1, Thm. 4.2] this result (for slightly more general Shimura curves) in the special case c = (1), b = P prime: the Galois action on the CM points in the given b-isogeny class is decomposed into a “chaotic” and a “geometric” part. The chaotic action is treated using a known case of the André-Oort conjecture proved by Edixhoven and Yafaev [Ed-Ya, Thm. 1.2]. The corresponding argument in [Co-Va 2, 3.21] was somewhat sketchy; the appendix [Co 2] to the present article provides the required translation of the general results of [Ed-Ya] to the special case considered here and in [Co-Va 1,2] (it is very likely that the chaotic action could have also been treated using a p-adic version of Ratner’s theorem, as in [Co-Va 1,2], but we did not pursue this line of argument). The geometric Galois action is analysed - and this our principal innovation - using representation theory of GL(2) and its inner forms over p-adic fields. This approach leads naturally to the questions considered by Gross and Prasad [Gr-Pr]. In particular, the compatibility condition for A_{0} and χ_{0} alluded to above has a natural formulation in terms of “test vectors” in the sense of [Gr-Pr]. The existence of test vectors is controlled by the values of certain local ε-factors ([Tu], [Wa], [Sa]); this is exploited in [Ne 3], where the second author deduces a fairly general result on the parity-induced growth of Selmer groups of Hilbert modular forms over ring class fields.

Let us describe the contents of the present article in more detail. In §1 we collect background material on ring class fields and local ε-factors. In §2, after preliminaries on Shimura curves (§2.1) and CM points (§2.2) we analyse first the chaotic (§2.3) and then the geometric Galois action (§2.4); this leads to a weak nonvanishing result (Theorem 2.5.1) which generalises [Co-Va 1, Thm. 4.10]. The purpose of the representationtheoretical interlude in §3 is two-fold: firstly, to exhibit explicit test vectors (and thus generalise [Gr-Pr, Prop. 2.3]), which implies an explicit variant (Theorem 2.5.2) of Theorem 2.5.1; secondly, to prepare ground for §4, in which we deduce our main result (Theorem 4.3.1) from Theorem 2.5.1 and the distribution relations for CM points that were studied in [Co-Va 1, §6].

The authors are grateful to C. Cornut for helpful discussions and for the appendix [Co 2].

1. Galois groups

1.1. Ring class fields

1.1.1. Throughout §1, F is a totally real number field of degree d and K a totally imaginary quadratic extension of F. Let η = η_{K/F} : A_{F}^{∗} /F^{∗}N_{K/F} (A_{K}^{∗}) ⟶ {±1} be the quadratic character associated to the extension K/F. We denote by rec_{K} : K̂^{∗} = (K ⊗ Ẑ)^{∗} ⟶ Gal(K^{ab}/K) the reciprocity map normalised by letting uniformisers correspond to geometric Frobenius elements. For a subextension L/K of K^{ab}/K we denote by rec_{L/K} the morphism K̂^{∗} ⟶ Gal(K^{ab}/K) ↠ Gal(L/K). 1.1.2. For a non-zero ideal c ⊂ O_{F}, the ring class field of K of conductor c is the finite abelian extension K[c] of K satisfying

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

where O_{c} = O_{F} + cO_{K} is the O_{F} -order in O_{K} of conductor c, Ô_{c} = O_{c} ⊗ Ẑ and Ô_{c}^{∗} = (Ô_{c})^{∗}. The group

U(c) = Ô_{K}^{∗}/Ô_{c}^{∗} ⟶^{∼} (O_{K}/cO_{K})^{∗}/(O_{F} /cO_{F})^{∗}

maps naturally to K̂^{∗}/K^{∗}F̂^{∗}Ô_{c}^{∗}, giving rise to an exact sequence

(1.1.2.1)  0 ⟶ Z(c) ⟶ U(c) ⟶ Gal(K[c]/K) ⟶ Gal(K[1]/K) ⟶ 0,

in which Z(c) is a subquotient of the finite abelian group (K^{∗} ∩ F̂^{∗}Ô_{K}^{∗})/F^{∗}. Put K[∞] = ⋃_{c} K[c]. 1.1.3. If b ⊂ O_{F} is a non-zero ideal satisfying bO_{K} ∤ lcm {(u − 1) | u ∈ (O_{K}^{∗})_{tors}, u ≠ 1}, then Z(bc) = Z(b) by ([Ne 2, Cor. 2.11]), hence

Ô_{b}^{∗}/Ô_{bc}^{∗} ⟶^{∼} Gal(K[bc]/K[b]).

1.1.4. For any subgroup X ⊂ K̂^{∗}, put X^{(c)} = {x ∈ X | (∀w|c) x_{w} = 1}. 1.1.5. For any finite prime v of F, put K_{v} = K ⊗_{F} F_{v} and O_{K,v} = O_{K} ⊗_{O_{F}} O_{F,v}. Let (K_{v}^{∗})^{∘} ⊂ K_{v}^{∗} be the preimage of (K_{v}^{∗}/O_{K,v}^{∗})^{Gal(K/F)} ⊂ K_{v}^{∗}/O_{K,v}^{∗}. As

K_{v}^{∗}/O_{K,v}^{∗}F_{v}^{∗} ⟶^{∼} \{ \begin{matrix}0, & v is inert in K/F \\ Z/2Z, & v is ramified in K/F \\ Z, & v splits in K/F, \end{matrix}

it follows that the quotient (K_{v}^{∗})^{∘}/F_{v}^{∗} is compact and

K_{v}^{∗}/(K_{v}^{∗})^{∘} ⟶^{∼} \{ \begin{matrix}Z, & v splits in K/F \\ 0, & otherwise, \end{matrix} (K_{v}^{∗})^{∘}/O_{K,v}^{∗}F_{v}^{∗} ⟶^{∼} \{ \begin{matrix}Z/2Z, & v is ramified in K/F \\ 0, & otherwise. \end{matrix}

Put K̂^{∗∘} = {x ∈ K̂^{∗} | (∀v) x_{v} ∈ (K_{v}^{∗})^{∘}}. 1.1.6. Let b, c ⊂ O_{F} be ideals satisfying (b, c) = (1). Denote by U(c, b) (resp., by U′(c, b)) the image of Ô_{K}^{∗(b)} (resp., of (K̂^{∗∘})^{(b)}) in K̂^{∗}/F̂^{∗}Ô_{bc}^{∗}. There are canonical isomorphisms

U(c, b) ⟶^{∼} Ô_{K}^{∗(b)}/Ô_{bc}^{∗(b)} ⟶^{∼} Ô_{K}^{∗}/Ô_{c}^{∗} = U(c) = ∏_{v|c} U(c)_{v},  U′(c, b) ⟶^{∼} ∏_{{v|cd \atop v∤b}} U′(c, b)_{v}  (d = d_{K/F}),

where U(c)_{v} (resp., U′(c, b)_{v}) is the quotient of O_{K,v}^{∗}/O_{F,v}^{∗} (resp., of (K_{v}^{∗})^{∘}/F_{v}^{∗}) by (O_{F,v} + cO_{K,v})^{∗}/O_{F,v}^{∗}. It follows from 1.1.5 that, for each prime v | cd_{K/F} not dividing b,

(1.1.6.1)  U′(c, b)_{v}/U(c)_{v} ⟶^{∼} \{ \begin{matrix}Z/2Z, & v | d_{K/F}, v ∤ b \\ 0, & otherwise, \end{matrix}

which implies that there is an exact sequence

(1.1.6.2)  0 ⟶ U(c) ⟶ U′(c, b) ⟶ (Z/2Z)^{{v|d_{K/F} , v∤b}} ⟶ 0.

1.1.7. For any subextension L/K of K[∞]/K, denote by Gal(L/K)^{b−rat} ⊃ Gal(L/K)^{∘,b−rat} the respective images of K̂^{∗(b)} ⊃ (K̂^{∗∘})^{(b)} in Gal(L/K) under the map rec_{L/K}. In particular, rec_{K[bc]/K} induces a surjective morphism

(1.1.7.1)  U′(c, b) ↠ Gal(K[bc]/K)^{∘,b−rat}.

In the terminology of [Co-Va 2], the elements of Gal(L/K)^{b−rat} are “b-rational” (note, however, that our ring class fields K[c] are slightly smaller than the corresponding fields in [Co-Va 1,2]).

1.2. Ring class towers

1.2.1. Data. Throughout this section, P_{1}, …, P_{s} (s ≥ 1) are distinct prime ideals of O_{F} and c ⊂ O_{F} is an ideal relatively prime to P_{1} ⋯ P_{s}. We denote by p_{i} the rational prime below P_{i}.

1.2.2. Notation. Let N_{∞} = N ∪ {∞} and N_{>0} = N − {0}. For a multiindex n = (n_{1}, …, n_{s}) ∈ N_{∞}^{s}, we abbreviate P^{n} := P_{1}^{n_{1}} ⋯ P_{s}^{n_{s}} and we use the following notation:

G^{(c)}(n) = Gal(K[cP^{n}]/K),  G(n) = G^{(1)}(n)  (n ∈ N^{s}),
G^{(c)} = G^{(c)}(∞) = Gal(K[cP^{∞}]/K) = \underleftarrow{lim}_{n} G^{(c)}(n),  K[cP^{∞}] = ⋃_{n∈N^{s}} K[cP^{n}],  G = G^{(1)}.

1.2.3. Proposition-Definition. (1) The kernels and cokernels of the maps

U(cP^{n}) = U(c) ⊕ ⨁_{i=1}^{s} U(P_{i}^{n_{i}}) ⟶ G^{(c)}(n) (n ∈ N^{s})

(defined in 1.1.2) are finite and their orders are bounded.

(2) The torsion subgroups G_{0}^{(c)} := (G^{(c)})_{tors} and G_{0} := G_{0}^{(1)} = G_{tors} are finite and

G^{(c)}/G_{0}^{(c)} ⟶^{∼} G/G_{0} ⟶^{∼} ⨁_{i=1}^{s} Z_{p_{i}}^{[F_{P_{i}}:Q_{p_{i}}]}.

(3) We define G_{1}^{(c)} := G_{0}^{(c)} ∩ (G^{(c)})^{P_{1}···P_{s}−rat} and G_{1} := G_{0} ∩ G^{P_{1}···P_{s}−rat}.

Proof. (1) Apply (1.1.2.1). (2) This follows from (1) and the structure of \underleftarrow{lim}_{m} U(P_{i}^{m}).

1.2.4. Proposition. For all but finitely many n ∈ N^{s}, the map from 1.2.3(1) gives rise to an exact sequence

0 ⟶ U(c) ⟶ G^{(c)}(n) ⟶ G(n) ⟶ 0.

Proof. Apply 1.1.3 with b = P^{n}.

1.2.5. Corollary. There is a commutative diagram with exact rows

0 ⟶ U(c) ⟶ G_{0}^{(c)} ⟶ G_{0} ⟶ 0
∥  ∩ ↓  ∩ ↓
0 ⟶ U(c) ⟶ G^{(c)} ⟶ G ⟶ 0.

1.2.6. Denote by d the prime-to-(P_{1} ⋯ P_{s}) part of d_{K/F}. It follows from 1.1.6 that

∀n ∈ N_{>0}^{s}  U′(c, P^{n}) = U′(c, P_{1} ⋯ P_{s}),  U′(c, P^{n})/U(c) ⟶^{∼} (Z/2Z)^{{v|d}} ⟶^{∼} U′(1, P^{n}).

1.2.7. Proposition. (1) The reciprocity map rec_{K} induces isomorphisms

K̂^{∗}/K^{∗}F̂^{∗}Ô_{c}^{∗(P_{1}···P_{s})} ⟶^{∼} G^{(c)},  K̂^{∗(P_{1}···P_{s})}/F̂^{∗(P_{1}···P_{s})}Ô_{c}^{∗(P_{1}···P_{s})} ⟶^{∼} (G^{(c)})^{P_{1}···P_{s}−rat}.

(2) The surjective maps (1.1.7.1)

U′(1, P_{1} ⋯ P_{s}) = U′(1, P^{n}) ↠ G(n)^{∘,P_{1}···P_{s}−rat} (n ∈ N_{>0}^{s})

give rise to an isomorphism

(Z/2Z)^{{v|d_{K/F} , v∤P_{1}···P_{s}}} ⟶^{∼} U′(1, P_{1} ⋯ P_{s}) ⟶^{∼} G_{1}.

(3) The surjective maps (1.1.7.1)

U′(c, P_{1} ⋯ P_{s}) = U′(c, P^{n}) ↠ G^{(c)}(n)^{∘,P_{1}···P_{s}−rat} (n ∈ N_{>0}^{s})

give rise to an isomorphism of exact sequences

0 ⟶ U(c) ⟶ U′(c, P_{1} ⋯ P_{s}) ⟶ U′(1, P_{1} ⋯ P_{s}) ⟶ 0
∥  ↓^{≀}  ↓^{≀}
0 ⟶ U(c) ⟶  G_{1}^{(c)}  ⟶  G_{1}  ⟶ 0.

Proof. (1) The first isomorphism is obtained as in ([Co-Va 1], Lemma 2.1); the second follows from K̂^{∗(P_{1}···P_{s})} ∩ K^{∗}F̂^{∗}Ô_{c}^{∗(P_{1}···P_{s})} = F̂^{∗(P_{1}···P_{s})}Ô_{c}^{∗(P_{1}···P_{s})}.

(2) (cf. [Co-Va 1, Lemma 2.3]) Applying (1) to c = (1), we obtain

G^{P_{1}···P_{s}−rat} ⟶^{∼} K̂^{∗(P_{1}···P_{s})}/F̂^{∗(P_{1}···P_{s})}Ô_{K}^{∗(P_{1}···P_{s})} ⟶^{∼} I_{K}^{(P_{1}···P_{s})}/I_{F}^{(P_{1}···P_{s})},

where we have denoted by I_{E}^{(a)} (E = F, K; a ⊂ O_{F}) the group of fractional ideals of E prime to a. It follows that

G_{1} = (G^{P_{1}···P_{s}−rat})_{tors} ⟶^{∼} (I_{K}^{(P_{1}···P_{s})}/I_{F}^{(P_{1}···P_{s})})_{tors} = (Z/2Z)^{{v|d_{K/F} , v∤P_{1}···P_{s}}},

and this isomorphism is induced by the map (1.1.7.1).

(2) It follows from Corollary 1.2.5 and 1.2.6 that the reciprocity map defines a map between exact sequences

0 ⟶ U(c) ⟶ U′(c, P_{1} ⋯ P_{s}) ⟶ U′(1, P_{1} ⋯ P_{s}) ⟶ 0
∥  ↓  ↓
0 ⟶ U(c) ⟶  G_{1}^{(c)}  ⟶  G_{1}

By (2), the third vertical arrow is an isomorphism; this implies that the map G_{1}^{(c)} ⟶ G_{1} is surjective and that the middle vertical arrow is also an isomorphism (by the Snake Lemma).

1.2.8. We define the conductor c(χ) of a character χ : G_{1}^{(c)} ⟶ C^{∗} to be the smallest divisor c′ of c (with respect to divisibility) such that χ factors through G_{1}^{(c′)}. Equivalently, it is the smallest divisor c′ of c such that the restriction of χ to U(c) factors through U(c′) (by Proposition 1.2.7(3) applied to c and c′).

1.3. Local ε-factors

1.3.1. Let v be a finite prime of F and B_{v} a quaternion algebra over F_{v} equipped with an F_{v}-embedding t_{v} : K_{v} ↪ B_{v}. Denote the invariant of B_{v} by

inv_{v}(B_{v}) = \{ \begin{matrix}1, & B_{v} = M_{2}(F_{v}) \\ −1, & B_{v} ≠ M_{2}(F_{v}). \end{matrix}

Let π_{v} be an irreducible smooth representation of B_{v}^{∗}/F_{v}^{∗} (on a C-vector space) and χ_{v} : K_{v}^{∗}/F_{v}^{∗} ⟶ C^{∗} a continuous (quasi)-character. The ε-factor

ε(π_{v} × χ_{v}, \frac{1}{2}) = ε(π_{v} × χ_{v}, ψ_{v}, \frac{1}{2}) ∈ \{±1\}

is independent of the choice of an additive character ψ_{v} of F_{v} and satisfies

(1.3.1.1)  ε(π_{v} × χ_{v}^{−1}, \frac{1}{2}) = ε(π_{v} × χ_{v}, \frac{1}{2})^{−1} = ε(π_{v} × χ_{v}, \frac{1}{2}).

Its value admits the following representation-theoretical characterisation.

1.3.2. Theorem ([Tu], [Wa, Thm. 2], [Sa]). If B_{v} = M_{2}(F_{v}), assume that dim(π_{v}) = ∞.

(1) If η_{v}(−1)ε(π_{v}×χ_{v}, \frac{1}{2}) = inv_{v}(B_{v}), then there exists a non-zero t_{v}(K_{v}^{∗})-invariant linear form π_{v}⊗χ_{v} ⟶ C (unique up to a scalar multiple). Equivalently, there exists a non-zero t_{v}(K_{v}^{∗})-invariant linear map (unique up to a scalar multiple) ℓ_{v} : π_{v} ⟶ C(χ_{v}^{−1}), i.e., a linear map ℓ_{v} : π_{v} ⟶ C satisfying

∀a ∈ K_{v}^{∗} ∀u ∈ π_{v} ℓ_{v} (t_{v}(a)u) = χ_{v}(a)^{−1} ℓ_{v}(u).

A “test vector” (for ℓ_{v}) is a vector u ∈ π_{v} such that ℓ_{v}(u) ≠ 0.

(2) If η_{v}(−1)ε(π_{v} × χ_{v}, \frac{1}{2}) ≠ inv_{v}(B_{v}), then no such ℓ_{v} exists.

1.3.3. (1) If v splits in K/F or if (B_{v} = M_{2}(F_{v}) and π_{v} is a principal series representation), then ([Ne 1, Prop. 12.6.2.4]):

η_{v}(−1)ε(π_{v} × χ_{v}, \frac{1}{2}) = 1 = inv_{v}(B_{v}).

(2) If v does not split in K/F, then K_{v}^{∗}/F_{v}^{∗} is compact, which implies that π_{v} is a direct sum of finitedimensional (smooth) representations of t_{v}(K_{v}^{∗}). This means that the existence of a non-zero t_{v}(K_{v}^{∗})-invariant linear form π_{v} ⊗ χ_{v} ⟶ C is equivalent to the existence of a non-zero t_{v}(K_{v}^{∗})-invariant vector in π_{v} ⊗ χ_{v}.

(3) If B_{v} = M_{2}(F_{v}) and π_{v} = St ⊗ μ (where St is the Steinberg representation and μ : F_{v}^{∗} ⟶ {±1}), then ([Ne 1, Prop. 12.6.2.4]):

η_{v}(−1)ε(π_{v} × χ_{v}, \frac{1}{2}) = \{ \begin{matrix}−1, & v does not split in K/F and χ_{v} = μ ∘ N_{K_{v}/F_{v}} \\ 1, & otherwise. \end{matrix}

(4) If B_{v} ≠ M_{2}(F_{v}), denote by JL(π_{v}) the representation of P GL_{2}(F_{v}) associated to π_{v} by the Jacquet-Langlands correspondence; then ε(π_{v} × χ_{v}, \frac{1}{2}) = ε(JL(π_{v}) × χ_{v}, \frac{1}{2}).

2. Shimura curves and CM points

2.1. Shimura curves

We recall the basic definitions in the language of [Co-Va 1] (see also [Ne 2]). 2.1.1. Let F be as in 1.1.1. Let B be a quaternion algebra over F such that B_{τ_{j}} := B ⊗_{F,τ_{j}} R is isomorphic to M_{2}(R) (resp., to the algebra of Hamilton quaternions) for one infinite prime τ_{1} of F (resp., for the remaining infinite primes τ_{2}, …, τ_{d} of F). If F = Q, we also allow B = M_{2}(Q).

2.1.2. Put B̂ = B ⊗ Ẑ and, for each prime v of B, B_{v} = B ⊗_{F} F_{v}. For any open compact subgroup H ⊂ B̂^{∗} (for example, if R ⊂ B is an O_{F} -order, then R̂^{∗} is an open compact subgroup of B̂^{∗}), let N_{H} be the (non-singular) Shimura curve over F whose associated Riemann surface is given by

N_{H}^{an} = (N_{H} ⊗_{F,τ_{1}} C)(C) = B^{∗}\((C − R) × B̂^{∗}/HF̂^{∗}),

where B^{∗} ⊂ B_{τ_{1}}^{∗} acts on C − R via a fixed isomorphism B_{τ_{1}} ⟶^{∼} M_{2}(R) and the standard action of GL_{2}(R) by Möbius transformations. The point of N_{H}^{an} represented by z ∈ C − R and b ∈ B̂^{∗} will be denoted by [z, b]_{H} = [z, b]. If H′ ⊂ H is an open subgroup of H, then there is a natural projection map

(2.1.2.1)  pr : N_{H′} ⟶ N_{H},  [z, b]_{H′} ↦ [z, b]_{H},

which is finite and flat. 2.1.3. If B ≠ M_{2}(Q), then the curve N_{H} is projective; set N_{H}^{∗} = N_{H}. If B = M_{2}(Q), denote by N_{H}^{∗} the standard non-singular compactification of N_{H} obtained by adjoining the cusps. In either case, the smooth projective curve N_{H}^{∗} over F is connected, but not necessarily geometrically connected. Its field of constants is (non-canonically) isomorphic to the subfield F_{H} ⊂ F^{ab} satisfying

rec_{F} : F̂^{∗}/F_{+}^{∗}F̂^{∗2}nr(H) ⟶^{∼} Gal(F_{H}/F),

where F_{+}^{∗} is the group of totally positive elements of F^{∗} and nr is the reduced norm on B. As in [Co-Va 1, 3.3], we define the Jacobian of N_{H}^{∗} as

J(N_{H}^{∗}) = Pic^{∘}(N_{H}^{∗}/F) = R_{F_{H}/F} (Pic^{∘}(N_{H}^{∗}/F_{H}))

and we let

ι_{H} : N_{H}^{∗} ⟶ J(N_{H}^{∗})

be the F -morphism given by a suitable multiple of the Hodge class (resp., of the cusp at infinity), as in [Co-Va 1, 3.5] (see also [Ne 2, 1.19]). 2.1.4. The curves N_{H}^{∗} (for variable H) form a projective system with respect to the projections (2.1.2.1) (resp., their natural extensions to N_{H}^{∗}, if B = M_{2}(Q)). Note that N_{H}^{∗} depends only on HÔ_{F}^{∗}, which always contains a subgroup of the form R̂^{∗}, for a suitable O_{F} -order R ⊂ B. As a result, the set of curves {N ∗ ; _{R^{∗}}}_{R} is cofinal in the projective system {N_{H}^{∗}}_{H}.

2.1.5. For each g ∈ B̂^{∗}, right multiplication by g induces a holomorphic isomorphism

[·g] : N_{H}^{an} ⟶^{∼} N_{g^{−1}Hg}^{an}, [z, b]_{H} ↦ [z, bg]_{g^{−1}Hg},

which comes from an F -isomorphism

[·g] : N_{H}^{∗} ⟶^{∼} N_{g^{−1}Hg}^{∗}.

2.1.6. As explained in [Co-Va 1, 3.6] (see also [Ne 2, 1.6-1.7]), the space of holomorphic 1-forms

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

is naturally identified with a certain space S_{F^{∗}} ; 2 of automorphic forms of weight (2, 0, …, 0) on B_{A}^{∗} with trivial central character. All we need to know about this space are the following three facts. 2.1.6.1. For each g ∈ B̂^{∗}, the left action

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

corresponds, in the limit, to the standard action of g on S_{2}.

2.1.6.2. As a (smooth) representation of B̂^{∗}, the space S_{F^{∗}} ; 2 decomposes into a direct sum

S_{F^{∗}} ; 2 = ⨁_{π} π^{∞},

where π = π_{∞} ⊗π^{∞} = ⨂′_{v} π_{v} runs through all irreducible cuspidal representations of B_{A}^{∗} with trivial central character whose archimedean component π_{∞} is isomorphic to the weight two holomorphic discrete series at τ_{1} and to the trivial representation at the remaining infinite primes τ_{2}, …, τ_{d}. Moreover, each representation π^{∞} = ⊗′_{v∤∞}π_{v} occurs in S_{F^{∗}} ; 2 with multiplicity one.

2.1.6.3. For each open compact subgroup H of B̂^{∗}, Γ(N_{H}^{∗an}, Ω^{an}) is identified with the subspace S_{2}^{_{F^{∗}}H} of H-fixed vectors in S_{F^{∗}} ; 2.

2.1.7. As recalled in ([Ne 2, 1.18]), for each open compact subgroup H of B̂^{∗} there exists an isogeny (defined over F)

(2.1.7.1)  J(N_{H}^{∗}) ⟶ ∏_{j∈J} A_{j}^{a_{j}}  (a_{j} ≥ 1),

where each A_{j} is an F -simple abelian variety over F for which End_{F} (A_{j}) = O_{L_{j}} is the ring of integers in a totally real number field L_{j} of degree [L_{j} : Q] = dim(A_{j}). In particular, Γ(A_{j}, Ω_{A_{j}/F}^{1} ) is a free (L_{j} ⊗_{Q} F)- module of rank one, which implies that the term corresponding to A_{j} in

(2.1.7.2)  S_{2}^{_{F^{∗}}H} = Γ(J(N_{H}^{∗})^{an}, Ω^{an}) = Γ(J(N_{H}^{∗}), Ω_{J(N_{H}^{∗})/F}^{1} ) ⊗_{F,τ_{1}} C ⟶^{∼} ⨁_{j∈J} Γ(A_{j}, Ω_{A_{j}/F}^{1} )^{a_{j}} ⊗_{F,τ_{1}} C

is a free module of rank a_{j} over

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

More precisely, there exists a finite set S of finite primes of F such that H = H_{S}H^{S}, where H_{S} is an open compact subgroup of ∏_{v∈S} B_{v}^{∗} and H^{S} = ∏_{v∉S} H_{v}, where each H_{v} is a maximal compact subgroup of B_{v}^{∗}. As recalled in [Ne 2, 1.17], the spherical Hecke algebra T_{H}^{S} = Z[H^{S}\(B̂^{∗})^{S}/H^{S}] acts on the j-th factor of (2.1.7.2) via a morphism φ_{j} : T_{H}^{S} ⟶ O_{L_{j}} and the natural action of O_{L_{j}} on A_{j}. Strong multiplicity one theorem for B_{A}^{∗} implies that, for each embedding σ : L_{j} ↪ R, there exists a unique automorphic representation π = π(σ) of the type described in 2.1.6.2 for which T_{H}^{S} acts on π(σ)^{H} ≠ 0 by σ ∘ φ_{j}. In particular,

∀σ : L_{j} ↪ Q dim_{C} (π(σ)^{∞})^{H} = a_{j}.

In more arithmetic terms, the L-function of A_{j} is automorphic in the following sense ([Ne 2, 1.18]):

∀σ : L_{j} ↪ Q L(σA_{j}/F, s) = L(π(σ), s − \frac{1}{2})

(Euler factor by Euler factor).

2.2. CM points

2.2.1. Let F and B be as in 2.1.1. Let K be a totally imaginary quadratic extension of F such that all finite primes of F at which B ramifies are either inert or ramified in K/F. This implies that there exists an embedding t : K ↪ B over F, which is unique up to B^{∗}-conjugacy, by the Skolem-Noether theorem. Fix such an embedding t and denote by

t_{v} : K_{v} ↪ B_{v},  t̂: K̂ ↪ B̂

the corresponding local and finite-adelic embeddings.

2.2.2. 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). The set of CM-points by K on the curve N_{H} is defined as

CM(N_{H}, K) = {[z, b]_{H} ∈ N_{H}(C) | b ∈ B̂^{∗}}.

This set does not depend on the choice of t and is contained in N_{H}(K[∞]), thanks to “Shimura’s reciprocity law”

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

In particular, a CM point x = [z, b]_{H} is defined over K[c], provided t̂(Ô_{c}^{∗}) ⊆ bHb^{−1}.

The map b ↦ [z, b]_{H} defines a bijection t(K^{∗})\B̂^{∗}/HF̂^{∗} ⟶^{∼} CM(N_{H}, K).

2.2.3. From now on (with the exception of §2.6) we assume, for simplicity, that H = R̂^{∗} for some O_{F} -order R ⊂ B (in view of 2.1.4, this is not a serious restriction). For each CM point x = [z, b]_{R^{∗}} ∈ CM(N_{R^{∗}}, K), there is a unique non-zero ideal c(x) ⊂ O_{F} (called the conductor of x) such that t^{−1}(bRb^{−1}) = O_{c(x)} = O_{F} + c(x)O_{K}. As t̂^{−1}(bR̂^{∗}b^{−1}) = Ô_{c(x)}^{∗}, it follows from (2.2.2.1) that x is defined over K[c(x)].

2.2.4. Given distinct prime ideals P_{1}, …, P_{s} (s ≥ 1) of O_{F}, a (P_{1} ⋯ P_{s})-isogeny class of CM points is a subset of CM(N_{H}, K) of the form

{[z, g_{1} ⋯ g_{s}b]_{H} | g_{i} ∈ B_{P_{i}}^{∗}}

for fixed b ∈ B̂^{∗}. Note that, if x, x′ ∈ CM(N_{H}, K) lie in the same (P_{1} ⋯ P_{s})-isogeny class, then

(2.2.4.1)  ∀v ∤ P_{1} ⋯ P_{s}  ord_{v}(c(x)) = ord_{v}(c(x′)).

2.3. CM points in ring class towers: the chaotic Galois action

2.3.1. Proposition ([Co-Va 2, Prop. 3. 18] if s = 1). Let P_{1}, …, P_{s} be distinct prime ideals of O_{F} such that B_{P_{i}} = M_{2}(F_{P_{i}}) for all i = 1, …, s. Fix σ ∈ Gal(K[∞]/K) and put, for any x ∈ CM(N_{H}, K), δ_{σ}(x) = (x, σx) ∈ (N_{H} ⊗_{F,τ_{1}} C)(C)^{2}. The following properties of σ are equivalent:

(1) σ ∈ Gal(K[∞]/K)^{P_{1}···P_{s}−rat} (= the image of K̂^{∗(P_{1}···P_{s})} under rec_{K[∞]/K}).

(2) For any (P_{1} ⋯ P_{s})-isogeny class C ⊂ CM(N_{H}, K), the Zariski closure of δ_{σ}(C) in (N_{H} ⊗_{F,τ_{1}} C)^{2} has dimension 1.

(3) For any sequence of CM points {x_{n}}_{n∈N} contained in some (P_{1} ⋯ P_{s})-isogeny class C ⊂ CM(N_{H}, K) and satisfying (P_{1} ⋯ P_{s})^{n} | c(x_{n}) for all n ∈ N, the Zariski closure of {δ_{σ}(x_{n})}_{n∈N} in (N_{H} ⊗_{F,τ_{1}} C)^{2} has dimension 1.

Proof. The proof of ([Co-Va 2, Prop. 3. 18]) applies with trivial modifications (one has to consider the distances on the trees associated to B_{P_{i}}^{∗}/F_{P_{i}}^{∗} for all i = 1, …, s).

2.3.2. Corollary ([Co-Va 2, Rmk. 3. 21] if s = 1). Given elements σ_{1}, …, σ_{r} ∈ Gal(K[∞]/K) that are distinct modulo Gal(K[∞]/K)^{P_{1}···P_{s}−rat}, then for any sequence {x_{n}}_{n∈N} satisfying the assumptions of 2.3.1(3), the Zariski closure of {(σ_{1}(x_{n}), …, σ_{r}(x_{n}))}_{n∈N} in (N_{H} ⊗_{F,τ_{1}} C)^{r} contains a connected component of (N_{H} ⊗_{F,τ_{1}} C)^{r}.

Proof. The statement being trivial for r = 1, assume r > 1. Let Y be a connected component of the Zariski closure of {(σ_{1}(x_{n}), …, σ_{r}(x_{n}))}_{n∈N}. According to a special case of the André-Oort conjecture proved in ([Ed-Ya, Thm. 1.2]), Y is a subvariety of Hodge type of (N_{H} ⊗_{F,τ_{1}} C)^{r}. For 1 ≤ i < j ≤ r, denote by p_{ij} : (N_{H} ⊗_{F,τ_{1}} C)^{r} ⟶ (N_{H} ⊗_{F,τ_{1}} C)^{2} the projection on the i-th and the j-th factors. Proposition 2.3.1 implies that p_{ij}(Y) contains a connected component of (N_{H} ⊗_{F,τ_{1}} C)^{2}. The description of the subvarieties of Hodge type of (N_{H} ⊗_{F,τ_{1}} C)^{r} given in [Co 2, Prop. 2.1] (which generalises the special case B = M_{2}(Q) treated in [Ed, Prop. 2.1]) then yields the desired result.

2.4. CM points in ring class towers: the geometric Galois action

2.4.1. We continue to assume that H = R̂^{∗} for some O_{F} -order R ⊂ B (which implies that H = ∏_{v} H_{v}, H_{v} = R_{v}^{∗}). Assume that we are given the following data:

(D1) Distinct prime ideals P_{1}, …, P_{s} (s ≥ 1) of O_{F} such that B_{P_{i}} = M_{2}(F_{P_{i}}) for all i = 1, …, s. We denote by p_{i} the rational prime below P_{i}. (D2) An ideal c ⊂ O_{F} relatively prime to P_{1} ⋯ P_{s}. (D3) A (P_{1} ⋯ P_{s})-isogeny class C ⊂ CM(N_{H}, K) such that for at least one x ∈ C ( ⟺ for all x ∈ C) c(x) | c(P_{1} ⋯ P_{s})^{∞}. (D4) A simple quotient J(N_{H}^{∗}) ↠ A_{0} defined over F. As recalled in 2.1.7, there exists a totally real number field L_{0} such that [L_{0} : Q] = dim(A_{0}) and (possibly after replacing A_{0} by an isogeneous abelian variety) End_{F} (A_{0}) = O_{L_{0}}. Denote by

α : N_{H}^{∗} ⟶^{ι_{H}} J(N_{H}^{∗}) ↠ A_{0}

the composition with the map ι_{H} from 2.1.3 (α is not constant and is defined over F).

(D5) A character χ_{0} : G_{0}^{(c)} ⟶ O_{L}^{∗}, where L is a number field containing L_{0}. Put

e_{\overline{χ}_{0}} = ∑_{g∈G_{0}^{(c)}} χ_{0}(g) g ∈ O_{L}[G_{0}^{(c)}].

2.4.2. Our goal is to show (under suitable assumptions) that, for sufficiently many CM points x ∈ C,

e_{\overline{χ}_{0}}(1 ⊗ α(x)) = ∑_{g∈G_{0}^{(c)}} χ_{0}(g) ⊗ α(g(x)) ∈ O_{L} ⊗_{O_{L_{0}}} A_{0}(K[cP^{∞}])

is not torsion. As in [Co-Va 1, 4.5-4.6], we decompose the action of G_{0}^{(c)} into a “geometric” action of G_{1}^{(c)} and a “chaotic” action of a set of representatives of G_{0}^{(c)}/G_{1}^{(c)}. 2.4.3. Put S = {v | cd_{K/F} , v ∤ P_{1} ⋯ P_{s}} (which should not be confused with the set that was denoted by the same letter in 2.1.7). As observed in Proposition 1.2.7, the reciprocity map induces an isomorphism

U′ ⟶^{∼} G_{1}^{(c)}, where

U′ = U′(c, P_{1} ⋯ P_{s}) = ⨁_{v∈S} U′_{v}, U′_{v} = U′(c, P_{1} ⋯ P_{s})_{v} = (K_{v}^{∗})^{∘}/F_{v}^{∗}(O_{F,v} + cO_{K,v})^{∗}.

This means that we can consider the restriction of χ_{0} to G_{1}^{(c)} as a character χ_{1} : U′ ⟶ O_{L}^{∗}, and then write it as a product of its local components χ_{1,v} : U′_{v} ⟶ O_{L}^{∗} (v ∈ S). We shall abuse the notation and denote by χ_{1,v} also the corresponding character

χ_{1,v} : (K_{v}^{∗})^{∘}/F_{v}^{∗} ↠ U′_{v} ⟶ O_{L}^{∗} (v ∈ S).

2.4.4. The decomposition χ_{1} = ∏_{v∈S} χ_{1,v} implies that

(2.4.4.1)  e_{\overline{χ}_{1}} := ∑_{a∈U′} χ_{1}(a) a = ∏_{v∈S} ( ∑_{a_{v}∈U′_{v}} χ_{1,v}(a_{v}) a_{v}) = ∏_{v∈S} e_{\overline{χ}_{1,v}} ∈ O_{L}[U′].

We are going to write the action of e_{\overline{χ}_{1}} on all CM points in C in geometric terms. Fix g ∈ B̂^{∗} such that C = {[z, b_{1} ⋯ b_{s}g]_{H} | b_{i} ∈ B_{P_{i}}^{∗}} and define new embeddings

\begin{aligned}j_{v} & = Ad(g_{v})^{−1} ∘ t_{v} : K_{v} ↪ B_{v},  j_{v}(a) = g_{v}^{−1}t_{v}(a)g_{v}  (v ∈ S) \\ j & = (j_{v})_{v∈S} : ∏_{v∈S} K_{v} ↪ ∏_{v∈S} B_{v}. \end{aligned}

Shimura’s reciprocity law (2.2.2.1) implies that

(2.4.4.2)

∀x = [z, b]_{H} ∈ C ∀a ∈ ∏_{v∈S} K_{v}^{∗} ∀a′ ∈ K̂^{∗} rec_{K}(a)rec_{K}(a′)(x) = [z, t̂(a)t̂(a′)b]_{H} = [z, t̂(a′)bj(a)]_{H},

and that, for each v ∈ S, the order j_{v}^{−1}(R_{v}) = O_{F_{v}} + c(x)O_{K,v} (with ord_{v}(c(x)) ≤ ord_{v}(c)) is independent of x ∈ C. In particular, for each a ∈ (K_{v}^{∗})^{∘} (v ∈ S), the order Adj_{v}(a)R_{v} = j_{v}(a)R_{v}j_{v}(a)^{−1} ⊂ B_{v} depends only on the image of a in (K_{v}^{∗})^{∘}/F_{v}^{∗}(O_{F_{v}} + cO_{K,v})^{∗} = U′_{v}. 2.4.5. Change of the Shimura curve. Define a new O_{F} -order R_{1} ⊂ B by its localisations

R_{1,v} = \{ \begin{matrix}R_{v}, & v ∉ S \\ ⋂_{a∈U′_{v}} Adj_{v}(a)R_{v}, & v ∈ S \end{matrix}

and consider the Shimura curve N_{H_{1}}^{∗} for H_{1} = R̂_{1}^{∗}. 2.4.6. Degeneracy maps. For each a ∈ U′, choose its lift ã ∈ ∏_{v∈S}(K_{v}^{∗})^{∘}; the morphism

f_{a} : N_{H_{1}}^{∗} ⟶^{[·j(_{a})]} N_{j(_{a})^{−1}H_{1}j(_{a})}^{∗} ⟶^{pr} N_{H}^{∗}

is defined over F and depends only on a, not on the chosen lift ã. Put

(2.4.6.1)  β_{a} = α ∘ f_{a} : N_{H_{1}}^{∗} ⟶^{f_{a}} N_{H}^{∗} ⟶^{α} A_{0}.

The formula (2.4.4.2) implies that

∀x = [z, b]_{H} ∈ C ∀g ∈ Gal(K[∞]/K) e_{\overline{χ}_{1}}(1 ⊗ α(g(x))) = ∑_{a∈U′} χ_{1}(a) ⊗ β_{a}(g(x_{1})) ∈ O_{L} ⊗_{O_{L_{0}}} A_{0}(K[cP^{∞}]),

where x_{1} = [z, b]_{H_{1}} ∈ CM(N_{H_{1}}, K). Equivalently, consider the abelian variety A = O_{L} ⊗_{O_{L_{0}}} A_{0} (see [Con, §7]) and define a morphism (over F)

(2.4.6.2)  α_{1} : N_{H_{1}}^{∗} ⟶ A,  α_{1} = ∑_{a∈U′} χ_{1}(a) β_{a};

then

(2.4.6.3)  ∀x = [z, b]_{H} ∈ C ∀g ∈ Gal(K[∞]/K)  e_{\overline{χ}_{1}}(1 ⊗ α(g(x))) = α_{1}(g([z, b]_{H_{1}})).

Functoriality of the Hodge class [Co-Va 1, §3.5] implies that, if we multiply the Hodge classes of H and H_{1} by the same integer in order to define ι_{H} and ι_{H_{1}}, then α_{1} is also given by

(2.4.6.4)  α_{1} : N_{H_{1}}^{∗} ⟶^{ι_{H_{1}}} J(N_{H_{1}}^{∗})⟶^{((f_{a})_{∗})} J(N_{H}^{∗})^{U′} \underrightarrow{∑ χ_{1}(a)(−)} O_{L} ⊗_{O_{L_{0}}} J(N_{H}^{∗}) ⟶^{id⊗α} A.

2.4.7. Proposition ([Co-Va 1, §4.6] if s = 1). Assume that the morphism α_{1} from (2.4.6.2) is not constant. If {x_{n}}_{n∈N} ⊆ C is a sequence of CM points contained in the (P_{1} ⋯ P_{s})-isogeny class C such that (P_{1} ⋯ P_{s})^{n} | c(x_{n}) for all n ∈ N, then the point

e_{\overline{χ}_{0}}(1 ⊗ α(x_{n})) = ∑_{g∈G_{0}^{(c)}} χ_{0}(g) ⊗ α(g(x_{n})) ∈ O_{L} ⊗_{O_{L_{0}}} A_{0}(K[cP^{∞}]) = A(K[cP^{∞}])

is non-torsion for each sufficiently large n >> 0.

Proof. We briefly repeat the argument from [Co-Va 1]. Fix b ∈ B̂^{∗} such that x_{0} = [z, b]_{H}. For each n ∈ N, choose an element u_{n} ∈ ∏_{i=1}^{s} B_{P_{i}} such that x_{n} = [z, u_{n}b]_{H} and put x_{n,1} = [z, u_{n}b]_{H_{1}}; the sequence {x_{n,1}}_{n∈N} is then contained in the (P_{1} ⋯ P_{s})-isogeny class C_{1} ⊂ CM(N_{H_{1}}, K) of x_{0,1}. Fix a set of representatives R ⊂ G_{0}^{(c)} of G_{0}^{(c)}/G_{1}^{(c)} and define

\begin{aligned}α′ : & (N_{H_{1}}^{∗})^{R} & ⟶^{α_{1}^{R}} & A^{R} & ⟶^{Σ} & A \\ & & & (a_{g})_{g∈R} & ↦ & ∑_{g∈R} χ_{0}(g)a_{g}, \end{aligned}

which is an algebraic morphism defined over F. We also define a map

\begin{aligned}Δ : & C_{1} & ⟶ & (N_{H_{1}}^{∗} ⊗_{F,τ_{1}} C)(C)^{R} \\ & x_{1} & ↦ & (g(x_{1}))_{g∈R}. \end{aligned}

The formula (2.4.6.3) implies that

∀n ∈ N e_{\overline{χ}_{0}}(1 ⊗ α(x_{n})) = α′ ∘ Δ(x_{n,1}).

Assume that α′ ∘Δ(x_{n,1}) ∈ A_{tors} for infinitely many n. As A(K[∞])_{tors} is finite [Co-Va 1, 4.2], it follows that there is a subsequence {x_{n(k),1}} of {x_{n,1}} for which α′ ∘ Δ({x_{n(k),1}}) = {y} is a single point (y ∈ A(K[∞])), which implies that the Zariski closure Z of {Δ(x_{n(k),1})} in (N_{H_{1}}^{∗} ⊗_{F,τ_{1}} C)(C)^{R} satisfies α′(Z) = {y}. On the other hand, Corollary 2.3.2 applies to R = {σ_{1}, …, σ_{r}} and {x_{n(k),1}}, hence for each g ∈ R there exists a connected component Y_{g} of (N_{H_{1}}^{∗} ⊗_{F,τ_{1}} C)(C) such that ∏_{g∈R} Y_{g} ⊆ Z; thus α′ ⊗ id is constant on ∏_{g∈R} Y_{g}, which implies that α_{1} ⊗ id is constant on each Y_{g}. As α_{1} is defined over F and N_{H_{1}}^{∗} is connected, α_{1} is constant, which contradicts our assumptions. 2.4.8. Action on differentials. The morphism α_{1} is non-constant if and only if the induced map on differentials

α_{1}^{∗} : Γ(A, Ω_{A/F}^{1} ) ⟶ Γ(N_{H_{1}}^{∗}, Ω_{N_{H_{1}}^{∗} /F}^{1} )

is non-zero. Let us express this map in terms of the representation-theoretical description of differentials given in 2.1.6-2.1.7. Firstly, applying (−) ⊗_{F,τ_{1}} C to the algebraic differentials in the sequence

β_{a}^{∗} : Γ(A_{0}, Ω_{A_{0}/F}^{1} ) ⟶^{α^{∗}} Γ(N_{H}^{∗}, Ω_{N_{H}^{∗}/F}^{1} ) ⟶^{f_{a}^{∗}} Γ(N_{H_{1}}^{∗}, Ω_{N_{H_{1}}^{∗} /F}^{1} ),

we obtain the following commutative diagram:

β_{a}^{∗} ⊗ id :  Γ(A_{0}^{an}, Ω^{an})  ↪ Γ(N_{H}^{∗an}, Ω^{an}) ⟶ Γ(N_{H_{1}}^{∗an}, Ω^{an})
∩ ↓  ∥  ∥
⨁_{σ_{0}:L_{0}↪R}(π(σ_{0})^{∞})^{H} ↪  S_{2}^{_{F^{∗}}H}  ⟶^{j(a)}  S_{2}^{_{F^{∗}}H_{1}},

from which we deduce another commutative diagram

Γ(A^{an}, Ω^{an}) = L ⊗_{L_{0}} Γ(A_{0}^{an}, Ω^{an}) ⟶^{(α_{1}⊗id)^{∗}}  Γ(N_{H_{1}}^{∗an}, Ω^{an})
∩ ↓  ∥
⨁_{σ:L↪C}(π(σ|_{L_{0}})^{∞})^{H}  ⟶  S_{2}^{_{F^{∗}}H_{1}}
ω ∈ (π(σ|_{L_{0}})^{∞})^{H} = (^{σ}π^{∞})^{H}  ↦  (∑_{a∈U′} ^{σ}χ_{1}(a)j(a)) ω,

where we have put (for each embedding σ : L ↪ C) ^{σ}χ_{1} = σ ∘ χ_{1} : G_{1}^{(c)} ⟶ C^{∗} and ^{σ}π = π(σ|_{L_{0}}); the left vertical arrow is given on L ⊗_{L_{0}} (Γ(A_{0}^{an}, Ω^{an}) ∩ π(σ_{0})) by

λ ⊗ ω_{0} ↦ (σ ↦ \{ \begin{matrix}σ(λ)ω_{0}, & if σ|_{L_{0}} = σ_{0} \\ 0, & otherwise \end{matrix} ).

In other words, we have

(2.4.8.1)

∀σ : L ↪ C ∀ω ∈ Γ(A^{an}, Ω^{an})∩(^{σ}π^{∞})^{H} (α_{1} ⊗id)^{∗}(ω) = ( ∑_{a∈U′} ^{σ}χ_{1}(a)j(a))ω = ∏_{v∈S} ( ∑_{a∈U′_{v}} ^{σ}χ_{1,v}(a)j_{v}(a))ω.

This formula can be reinterpreted as follows: for each σ : L ↪ C, the space ^{σ}π decomposes under the action of the compact abelian group j(∏_{v∈S}(K_{v}^{∗})^{∘}/F_{v}^{∗}) into a direct sum of one-dimensional representations. The action on the subspace (^{σ}π^{∞})^{H} = ⊗′_{v∤∞}^{σ}π_{v}^{H_{v}} (recall that H = R̂^{∗} = ∏_{v} R_{v}^{∗} = ∏_{v} H_{v}) factors through U′, and (2.4.8.1) simply says that

(2.4.8.2)  |U′|^{−1}(α_{1} ⊗ id)^{∗}(ω) = |U′|^{−1} ^{σ}e_{\overline{χ}_{1}}(ω) = |U′|^{−1}( ∏_{v∈S} ^{σ}e_{\overline{χ}_{1,v}})(ω)

is equal to the projection of ω on the ^{σ}χ_{1}^{−1}-eigenspace with respect to this action. A fancy reformulation of this simple observation is given in Proposition 2.4.10 below.

2.4.9. Proposition. If ∀v ∈ S dim(^{σ}π_{v}^{H_{v}}) = 1 (for one, hence for all σ : L ↪ C), then: the map α_{1} is not constant ⟺ there exists σ : L ↪ C such that ∀v ∈ S ^{σ}e_{\overline{χ}_{1,v}}(^{σ}π_{v}^{H_{v}}) ≠ 0.

Proof. The assumptions imply that, for each σ : L ↪ C, any non-zero element ω ∈ (^{σ}π^{∞})^{H} = ⨂′_{v∤∞} ^{σ}π_{v}^{H_{v}} is of the form ω = ⨂_{v∈S} ω_{v} ⊗ ω^{S} (Cω_{v} = ^{σ}π_{v}^{H_{v}}, ω^{S} ≠ 0); then

(α_{1} ⊗ id)^{∗}(ω) = ⨂_{v∈S}^{σ}e_{\overline{χ}_{1,v}}(ω_{v}) ⊗ ω^{S}

is non-zero ⟺ ∀v ∈ S ^{σ}e_{\overline{χ}_{1,v}}(^{σ}π_{v}^{H_{v}}) ≠ 0.

2.4.10. Proposition. (1) The map α_{1} is not constant ⟺ there exist σ : L ↪ C, ω ∈ Γ(A^{an}, Ω^{an}) ∩^{σ}π^{H} and a j(∏_{v∈S}(K_{v}^{∗})^{∘}/F_{v}^{∗})-invariant linear map^{σ}ℓ : ^{σ}π ⟶ C(^{σ}χ_{1}^{−1}) (i.e., such that^{σ}ℓ(j(a)u) = ^{σ}χ_{1}(a)^{−1} ^{σ}ℓ(u)) satisfying ^{σ}ℓ(ω) ≠ 0.

(2) Assume that there exist σ : L ↪ C and 0 ≠ ω = ⨂_{v∈S} ω_{v} ⊗ ω^{S} ∈ Γ(A^{an}, Ω^{an}) ∩^{σ}π^{H} (ω_{v} ∈ ^{σ}π_{v}^{H_{v}}) and, for each v ∈ S, a j_{v}((K_{v}^{∗})^{∘}/F_{v}^{∗})-invariant linear map ^{σ}ℓ_{v} : ^{σ}π_{v} ⟶ C(^{σ}χ_{1,v}^{−1}) such that ^{σ}ℓ_{v}(ω_{v}) ≠ 0. Then α_{1} is not constant.

(3) Assume that, for each v ∈ S, dim(^{σ}π_{v}^{H_{v}}) = 1 (for one, hence for all σ : L ↪ C). Then: the map α_{1} is not constant ⟺ there exist σ : L ↪ C and, for each v ∈ S, a j_{v}((K_{v}^{∗})^{∘}/F_{v}^{∗})-invariant linear map σℓ_{v} : ^{σ}π_{v} ⟶ C(^{σ}χ_{1,v}^{−1}) such that ^{σ}ℓ_{v}(^{σ}π_{v}^{H_{v}}) ≠ 0.

Proof. (1) If ^{σ}ℓ exists, then ω′ := (α_{1} ⊗ id)^{∗}(ω) is non-zero (and hence α_{1} is not constant), since

σℓ(ω′) = ∑_{a∈U′} ^{σ}χ_{1}(a)^{σ}ℓ(j(a)ω) = |U′| ·^{σ}ℓ(ω) ≠ 0.

Conversely, if α_{1} is not constant, then there exist σ and ω as above such that ω′ := (α_{1} ⊗ id)^{∗}(ω) ≠ 0; we let ^{σ}ℓ be the j(∏_{v∈S}(K_{v}^{∗})^{∘}/F_{v}^{∗})-equivariant projection of ^{σ}π onto the line Cω′ (on which j(∏_{v∈S}(K_{v}^{∗})^{∘}/F_{v}^{∗}) acts by the character ^{σ}χ_{1}^{−1}).

(2) Fix a linear form ℓ^{S} : ⨂′_{v∉S,v∤∞} ^{σ}π_{v} ⟶ C such that ℓ^{S}(ω^{S}) ≠ 0 and apply (1) to the linear map σℓ = ⨂_{v∈S} ^{σ}ℓ_{v} ⊗ ℓ^{S}.

(3) The assumptions imply that, for each σ : L ↪ C, any element ω ∈ (^{σ}π^{∞})^{H} is of the form ω = ⨂_{v∈S} ω_{v} ⊗ ω^{S} (ω_{v} ∈ ^{σ}π_{v}^{H_{v}}); the two implications then follow from (1) and (2), respectively. 2.4.11. Invariant linear maps. Let v ∈ S. If v does not split in K/F, then (K_{v}^{∗})^{∘} = K_{v}^{∗} and χ_{1,v} is a character of K_{v}^{∗}/F_{v}^{∗}. The value of ε(^{σ}π_{v} ×^{σ}χ_{1,v}, \frac{1}{2}) ∈ \{±1\} does not depend on σ; denote it by ε(π_{v} × χ_{v}, \frac{1}{2}). As recalled in 1.3.2, a non-zero j_{v}(K_{v}^{∗}/F_{v}^{∗})-invariant linear map ^{σ}ℓ_{v} : ^{σ}π_{v} ⟶ C(^{σ}χ_{1,v}^{−1}) exists if and only if the following condition is satisfied:

(⋆_{v})  η_{v}(−1)ε(π_{v} × χ_{v}, \frac{1}{2}) = inv_{v}(B_{v}).

If this is the case, then ^{σ}ℓ_{v} is unique up to a scalar multiple.

If, on the other hand, v splits in K/F, then B = M_{2}(F_{v}) and there are many different j_{v}((K_{v}^{∗})^{∘}/F_{v}^{∗})- invariant linear maps ^{σ}π_{v} ⟶ C(^{σ}χ_{1,v}^{−1}). For example, if χ̃_{1,v} : K_{v}^{∗}/F_{v}^{∗} ⟶ O_{L}^{∗} is any character extending χ_{1,v} (there are plenty of them, since K_{v}^{∗}/(K_{v}^{∗})^{∘}F_{v}^{∗} ⟶^{∼} Z), then ε(π_{v} × χ_{v}, \frac{1}{2}) := ε(^{σ}π_{v} ×^{σ}χ̃_{1,v}, \frac{1}{2}) ∈ \{±1\} (where σχ̃_{1,v} = σ ∘ χ̃_{1,v} : K_{v}^{∗}/F_{v}^{∗} ⟶ C^{∗}) does not depend on σ, nor on the chosen extension χ̃_{1,v} of χ_{1,v}, and the condition (⋆_{v}) automatically holds (by 1.3.3(1)). As a result, there exists a non-zero j_{v}(K_{v}^{∗}/F_{v}^{∗})-invariant linear map (unique up to a scalar multiple) ^{σ}ℓ_{v} : ^{σ}π_{v} ⟶ C(^{σ}χ_{1,v}^{−1}), which is then j_{v}((K_{v}^{∗})^{∘}/F_{v}^{∗})-invariant as a map ^{σ}π_{v} ⟶ C(^{σ}χ_{1,v}^{−1}). However, there are many j_{v}((K_{v}^{∗})^{∘}/F_{v}^{∗})-invariant maps ^{σ}π_{v} ⟶ C(^{σ}χ_{1,v}^{−1}) which cannot be obtained in this way.

In particular, if (⋆_{v}) holds for all v ∈ S, then there is a non-zero j(∏_{v∈S}(K_{v}^{∗})^{∘}/F_{v}^{∗})-invariant linear map

⨂_{v∈S}^{σ}ℓ_{v} : ^{σ}π_{S} = ⨂_{v∈S}^{σ}π_{v} ⟶ C(^{σ}χ_{1}^{−1}).

Tensoring it with any non-zero linear form ℓ^{S} : ⨂′_{v∉S,v∤∞} ^{σ}π_{v} ⟶ C, we obtain a non-zero j(∏_{v∈S}(K_{v}^{∗})^{∘}/F_{v}^{∗})- invariant linear map

σℓ = ⨂_{v∈S}^{σ}ℓ_{v} ⊗ ℓ^{S} : ^{σ}π ⟶ C(^{σ}χ_{1}^{−1}).

2.4.12. Proposition. (1) If α_{1} is not constant, then (⋆_{v}) holds for all v ∈ S.

(2) If χ : G^{(c)} ⟶ O_{L}^{∗} is a character of finite order whose restriction to G_{1}^{(c)} is equal to χ_{1}, then (⋆_{v}) holds for all finite primes v of F not dividing cd_{K/F} P_{1} ⋯ P_{s}.

(3) If (⋆_{v}) holds for all v ∈ S and χ : G^{(c)} ⟶ O_{L}^{∗} is as in (2), then the global ε-factor ε(π × χ, \frac{1}{2}) = ∏_{v} ε(π_{v} × χ_{v}, \frac{1}{2}) is equal to

ε(π × χ, \frac{1}{2}) = − ∏_{v|P_{1}···P_{s}} η_{v}(−1)ε(π_{v} × χ_{v}, \frac{1}{2}).

If all primes P_{1}, …, P_{s} split in K/F or if (P_{1} ⋯ P_{s})^{n} | c(χ) for large enough n, then ε(π × χ, \frac{1}{2}) = −1.

Proof. (1) As α_{1} is not constant, there exist σ : L ↪ C and ω ∈ Γ(A^{an}, Ω^{an}) ∩ ^{σ}π such that ω′ := (α_{1} ⊗ id)^{∗}(ω) ≠ 0. As

∀a ∈ ∏_{v∈S}(K_{v}^{∗})^{∘}/F_{v}^{∗} ^{σ}π(j(a)) ω′ = ^{σ}χ_{1}(a)^{−1} ω′,

it follows that, for each v ∈ S, there exists a non-zero element of ^{σ}π_{v} ⊗^{σ}χ_{1,v} invariant under j_{v}((K_{v}^{∗})^{∘}/F_{v}^{∗}). If v splits in K/F, then the condition (⋆_{v}) is automatically satisfied, by 1.3.3(1). If v does not split, then (K_{v}^{∗})^{∘} = K_{v}^{∗} and the statement follows from 1.3.3(2) and 1.3.2.

(2) As before, there is nothing to prove if v splits in K/F. If v does not split, then our assumptions imply that v is inert and χ_{v} is unramified, hence χ_{v} is trivial on F_{v}^{∗}O_{K,v}^{∗} = K_{v}^{∗}. Similarly, the fact that ∀x ∈ C v ∤ c(x)

implies that t_{v}(O_{K,v}^{∗}) ⊆ R_{v}^{∗}; thus t_{v}(K_{v}^{∗}) = t_{v}(F_{v}^{∗}O_{K,v}^{∗}) acts trivially on ^{σ}π_{v}^{R_{v}^{∗}} ≠ 0, hence there exists a t_{v}(K_{v}^{∗})-invariant vector in ^{σ}π_{v} ⊗^{σ}χ_{v}; we conclude again by 1.3.3(2) and 1.3.2.

(3) For each prime v of F, put a_{v} = inv_{v}(B_{v})η_{v}(−1)ε_{v} ∈ \{±1\}, where ε_{v} = ε(π_{v} × χ_{v}, \frac{1}{2}); then the global ε-factor is equal to ε = ∏_{v} a_{v}. If v ∤ ∞P_{1} ⋯ P_{s}, then a_{v} = 1, by (2) (resp., by assumption) if v ∉ S (resp., if v ∈ S). If v = τ_{j} | ∞, then ε_{v} = 1, η_{v}(−1) = −1 and inv_{v}(B_{v}) = −1 (resp., inv_{v}(B_{v}) = 1) if j ≠ 1 (resp., if j = 1); thus

ε = ∏_{v|∞P_{1}···P_{s}} a_{v} = − ∏_{v|P_{1}···P_{s}} a_{v} = − ∏_{v|P_{1}···P_{s}} η_{v}(−1)ε_{v}

(the last equality follows from the assumption (D1)). If v = P_{i} splits in K/F, then η_{v}(−1)ε_{v} = 1, by 1.3.3(1). Finally, if v = P_{i} does not split and χ_{v} is sufficiently ramified, then η_{v}(−1)ε_{v} = 1, by [J-L, Prop. 3.8] and [J, Thm. 20.6]. 2.4.13. We are now going to describe several special cases in which Proposition 2.4.10(3) (equivalently, Proposition 2.4.9) applies. We drop σ from the notation and consider the following, purely local, situation: let v be a finite prime of F, B_{v} a quaternion algebra over F_{v}, K_{v} a quadratic extension of F_{v} (the split case K_{v} = F_{v} × F_{v} is allowed), j_{v} : K_{v} ↪ B_{v} an F_{v}-embedding, χ_{v} : (K_{v}^{∗})^{∘}/F_{v}^{∗} ⟶ C^{∗} a character (of finite order), π an irreducible smooth representation of B_{v}^{∗}/F_{v}^{∗} (of infinite dimension if B_{v} = M_{2}(F_{v})) and R_{v} ⊂ B_{v} an O_{F,v}-order such that π_{v}^{R_{v}^{∗}} ≠ 0.

We are interested in the existence of a non-zero j_{v}((K_{v}^{∗})^{∘}/F_{v}^{∗})-invariant linear map ℓ_{v} : π_{v} ⟶ C(χ_{v}^{−1}), i.e., a linear form ℓ_{v} : π_{v} ⟶ C satisfying

(2.4.13.1)  ∀a ∈ (K_{v}^{∗})^{∘}/F_{v}^{∗} ∀u ∈ π_{v}  ℓ_{v}(j_{v}(a)u) = χ_{v}(a)^{−1} u,

for which

(2.4.13.2)  ℓ_{v}(π_{v}^{R_{v}^{∗}}) ≠ 0,

particularly in the following case (studied in [Gr-Pr], for example):

(2.4.13.3)  dim(π_{v}^{R_{v}^{∗}}) = 1.

The requirements (2.4.13.1-2) are compatible only if χ_{v}(j_{v}^{−1}(R_{v}^{∗})) = 1, which is equivalent to

(2.4.13.4)  c(χ_{v}) | c(j_{v}^{−1}(R_{v})),

where the conductor of any O_{F,v}-order O of K_{v} is the ideal c(O) ⊂ O_{F,v} given by O = O_{F,v} + c(O)O_{K,v}, and the conductor c(χ_{v}) of χ_{v} is the conductor of the biggest O_{F,v}-order O ⊂ K_{v} such that O^{∗} ⊆ Ker(χ_{v}).

In the situation considered in 2.4.4 we have, for each v ∈ S, χ_{v} = χ_{1,v}, hence c(χ_{v}) is equal to the v-part of c(χ_{1}), and c(j_{v}^{−1}(R_{v})) is equal to the common v-part of c(x) for all x ∈ C. The divisibility (2.4.13.4) then simply says that

(2.4.13.5)  ∀v ∈ S ∀x ∈ C  ord_{v}(c(χ_{1})) ≤ ord_{v}(c(x)).

We can reformulate (2.4.13.1-2) in more down-to-earth terms, by returning to the formula (2.4.8.1): the assumption (2.4.13.4) implies that χ_{v} factors through the finite abelian group

U′_{v} = (K_{v}^{∗})^{∘}/F_{v}^{∗}(O_{F,v} + c(j_{v}^{−1}(R_{v}))O_{K,v})^{∗};

similarly, the action of j_{v}(K_{v}^{∗})^{∘}/F_{v}^{∗}) on π_{v}^{R_{v}^{∗}} factors through U′_{v}. Denoting

e_{\overline{χ}_{v}} = ∑_{a∈U′_{v}} χ_{v}(a) a ∈ C[U′_{v}],

then the two conditions (2.4.13.1-2) are equivalent to the condition

(2.4.13.6)  e_{\overline{χ}_{v}}(π_{v}^{R_{v}^{∗}}) ≠ 0

from Proposition 2.4.9, as shown in the proof of Proposition 2.4.10. 2.4.14. We retain the notations of 2.4.13. We list a few cases in which the properties (2.4.13.1-3) are known to be satisfied. 2.4.14.1. If B_{v} = M_{2}(F_{v}) and dim(π_{v}) = ∞, denote by o(π_{v}) the conductor exponent of π_{v} (i.e., the valuation of the conductor of π_{v}). If R_{v} ⊂ B_{v} is an Eichler order of B_{v} of level n ≥ 0, then

dim(π_{v}^{R_{v}^{∗}}) = max(n − o(π_{v}) + 1, 0),

by a representation-theoretical reformulation of the Atkin-Lehner theory, due to Casselman [Ca, §1] (cf. 3.1.7 below). In particular, (2.4.13.3) holds if n = o(π_{v}). 2.4.14.2. If B_{v} ≠ M_{2}(F_{v}), then dim(π_{v}) < ∞; put o(π_{v}) := o(JL(π_{v})). We have

dim(π_{v}) = 1 ⟺ π_{v} = μ ∘ nr : B_{v}^{∗} ⟶^{nr} F_{v}^{∗} ⟶^{μ} {±1} ⟺ JL(π_{v}) = St ⊗ μ.

2.4.14.3. Assume that o(π_{v}) = 0 and that R_{v} is a maximal order. In this case B_{v} = M_{2}(F_{v}), π_{v} is an unramified principal series representation and (2.4.13.3) holds.

A result of Gross and Prasad ([Gr-Pr, Prop. 2.3]) implies that a linear form satisfying (2.4.13.1-2) exists, provided c(χ_{v}) = c(j_{v}^{−1}(R_{v})).

In Proposition 3.4.3 below we generalise this result as follows: write the eigenvalue of the Hecke operator T_{v} on π_{v}^{R_{v}^{∗}} as (Nv)^{1/2}(λ_{v} + λ_{v}^{−1}) (λ_{v} ∈ C). Assume that c(χ_{v}) = v^{m}, c(j_{v}^{−1}(R_{v})) = v^{n}, 1 ≤ m ≤ n. If λ_{v}^{2} = 1 or λ_{v}^{2(n−m+1)} ≠ 1 (which is automatic if m = n, the case covered by [Gr-Pr]), then (2.4.13.6) holds. 2.4.14.4. Assume that o(π_{v}) = 1, B_{v} ≠ M_{2}(F_{v}) and that R_{v} is any order in B_{v}. In this case v does not split in K/F, dim(π_{v}) = 1, π_{v} = μ ∘ nr, μ : F_{v}^{∗} ⟶ {±1} is unramified and (2.4.13.3) holds. It follows from 1.3.2-3 that a linear form ℓ_{v} satisfying (2.4.13.1) exists (and is then trivially unique up to a scalar multiple) if and only if

(2.4.14.4.1) χ_{v} = μ∘N_{K_{v}/F_{v}} ⟺ \{ \begin{matrix}v is inert in K/F, χ_{v} = 1 \\ v is ramified in K/F, χ_{v} is unramified, χ_{v}(w) = μ(v) (vO_{K} = w^{2}). \end{matrix}

If (2.4.14.4.1) holds, then the form ℓ_{v} satisfies (2.4.13.2), for trivial reasons, and (2.4.13.6) holds. 2.4.14.5. Assume that o(π_{v}) = 1, B_{v} = M_{2}(F_{v}) and that R_{v} is an Eichler order of level one. In this case π_{v} = St ⊗ μ, μ : F_{v}^{∗} ⟶ {±1} is unramified and (2.4.13.3) holds. It follows again from 1.3.2-3 that a linear form ℓ_{v} satisfying (2.4.13.1) exists (and is unique up to a scalar multiple if v does not split in K/F) if and only if either

(2.4.14.5.1)  v splits in K/F,

or

(2.4.14.5.2)

v does not split in K/F, χ_{v} ≠ μ ∘ N_{K_{v}/F_{v}} ⟺ \{ \begin{matrix}v is inert in K/F, χ_{v} ≠ 1 \\ v is ramified in K/F, χ_{v} is ramified \\ v is ramified in K/F, χ_{v} is unramified, χ_{v}(w) = −μ(v). \end{matrix}

(vO_{K} = w^{2}). It is shown in Proposition 3.4.5 below that if (2.4.14.5.1) or (2.4.14.5.2) holds, then (2.4.13.6) holds in this case, too.

2.5. Weak non-vanishing results

2.5.1. Theorem. Assume that we are given the data (D1) - (D5) from 2.4.1. Assume also that there exist σ : L ↪ C, ω ∈ Γ(A^{an}, Ω^{an}) ∩^{σ}π^{H} and a j(∏_{v∈S}(K_{v}^{∗})^{∘}/F_{v}^{∗})-invariant linear map ^{σ}ℓ : ^{σ}π ⟶ C(^{σ}χ_{1}^{−1}) (i.e., such that ^{σ}ℓ(j(a)u) = ^{σ}χ_{1}(a)^{−1} ^{σ}ℓ(u)) satisfying ^{σ}ℓ(ω) ≠ 0. If {x_{n}}_{n∈N} ⊆ C is a sequence of CM points contained in the (P_{1} ⋯ P_{s})-isogeny class C such that (P_{1} ⋯ P_{s})^{n} | c(x_{n}) for all n ∈ N, then

e_{\overline{χ}_{0}}(1 ⊗ α(x_{n})) = ∑_{g∈G_{0}^{(c)}} χ_{0}(g) ⊗ α(g(x_{n})) ∈ O_{L} ⊗_{O_{L_{0}}} A_{0}(K[cP^{∞}]) = A(K[cP^{∞}])

is non-torsion for all sufficiently large n >> 0.

Proof. Combine Proposition 2.4.7 and 2.4.10(1).

2.5.2. Theorem (an explicit special case of 2.5.1). Assume that we are given the data (D1) - (D5) from 2.4.1. Assume also that, for each v ∈ S, the following conditions (i)–(iv) hold:

(i) o(π_{v}) ≤ 1. (ii) If o(π_{v}) = 0, then R_{v} is a maximal order in B_{v} = M_{2}(F_{v}); if o(π_{v}) = 1 and B_{v} = M_{2}(F_{v}), then R_{v} is an Eichler order of level one in B_{v}. (iii) (⋆_{v}) holds. (iv) If o(π_{v}) = 0, write the eigenvalue of T_{v} on A_{0} as (Nv)^{1/2}(λ_{v} + λ_{v}^{−1}). Put n_{v} = ord_{v}(c(x)) (for any x ∈ C) and m_{v} = ord_{v}(c(χ_{0})). Assume that 1 ≤ m_{v} ≤ n_{v} and [λ_{v}^{2} = 1 or λ_{v}^{2(n_{v}−m_{v}+1)} ≠ 1] (which is automatic if m_{v} = n_{v} ≥ 1). Then: if {x_{n}}_{n∈N} ⊆ C is a sequence of CM points contained in the (P_{1} ⋯ P_{s})-isogeny class C such that (P_{1} ⋯ P_{s})^{n} | c(x_{n}) for all n ∈ N, then

e_{\overline{χ}_{0}}(1 ⊗ α(x_{n})) = ∑_{g∈G_{0}^{(c)}} χ_{0}(g) ⊗ α(g(x_{n})) ∈ O_{L} ⊗_{O_{L_{0}}} A_{0}(K[cP^{∞}]) = A(K[cP^{∞}])

is non-torsion for all sufficiently large n >> 0.

Proof. Combine Proposition 2.4.7, 2.4.9 and 2.4.14.3-5.

2.6. Non-vanishing and ε-factors

In this section H ⊂ B̂^{∗} denotes any open compact subgroup (not necessarily of the form R̂^{∗}).

2.6.1. Let α : N_{H}^{∗} ⟶^{ι_{H}} J(N_{H}^{∗}) ↠ A_{0} be as in (D4). Assume that we are given a CM point x = [z, b]_{H} ∈ CM(N_{H}, K) and a character χ : G_{x} = Gal(K(x)/K) ⟶ O_{L}^{∗} (L ⊃ L_{0}), where K(x) = (K^{ab})^{rec_{K}(_{t^{−1}}(bH_{F^{∗}}b^{−1}))} ⊂ K[∞] is the field of definition of x. As in 2.4.1-6, set e\overline{χ} = ∑_{g∈G_{x}} χ(g) g ∈ O_{L}[G_{x}] and A = O_{L} ⊗_{O_{L_{0}}} A_{0}. The character

K̂^{∗} ⟶^{rec_{K}} Gal(K^{ab}/K) ↠ G_{x} ⟶^{χ} O_{L}^{∗}

will also be denoted by χ. For each embedding σ : L ↪ C, let ^{σ}π and ^{σ}χ = σ ∘ χ : G_{x} ⟶ C^{∗} be as in 2.4.8. For each prime v of F, the local ε-factor ε(^{σ}π_{v} ×^{σ}χ_{v}, \frac{1}{2}) ∈ \{±1\} does not depend on σ; denote it by ε(π_{v} × χ_{v}, \frac{1}{2}) (cf. 2.4.11).

The following Proposition seems to be well-known to the experts ([Gr], [Zh]), but we have not been able to find a precise reference.

2.6.2. Proposition. If the point e\overline{χ}(1 ⊗ α(x)) ∈ A(K(x)) is not torsion, then:

(1) ∀v ∤ ∞ η_{v}(−1) ε(π_{v} × χ_{v}, \frac{1}{2}) = inv_{v}(B_{v}). In particular, B is uniquely determined by K, χ and the isogeny class of A_{0} over F.

(2) The global ε-factor is equal to ε(π × χ, \frac{1}{2}) = −1.

Proof. (1) We slightly modify the arguments from 2.4.4-12. Let S′ be a finite set of non-archimedean primes of F containing all primes at which B, π or K(x)/F ramify and such that the composite morphism

r = (r_{v})_{v∈S′} : ∏_{v∈S′} K_{v}^{∗} ⟶ K̂^{∗} ⟶^{rec_{K}} Gal(K^{ab}/K) ↠ G_{x} (r_{v} : K_{v}^{∗} ⟶ G_{x})

is surjective. As in 2.4.4, if we define

j_{v} = Ad(b_{v})^{−1} ∘ t_{v} : K_{v} ↪ B_{v} (v ∈ S′), j = (j_{v})_{v∈S′} : ∏_{v∈S′} K_{v} ↪ ∏_{v∈S′} B_{v},

then

∀a ∈ ∏_{v∈S′} K_{v}^{∗} rec_{K}(a)[z, b]_{H} = [z, t̂(a)b]_{H} = [z, bj(a)]_{H}.

Denote by S′_{1} (resp., S′_{2}) the set of those primes v ∈ S′ that do (resp., do not) split in K/F. For each v ∈ S′, C_{v} = O_{K,v}^{∗} ∩ Ker(r_{v}) is an open subgroup of O_{K,v}^{∗} and the quotient V_{v}^{∘} := (K_{v}^{∗})^{∘}/F_{v}^{∗}C_{v} is finite. If v ∈ S′_{2}, then V_{v}^{∘} = V_{v} := K_{v}^{∗}/F_{v}^{∗}C_{v}. If v ∈ S′_{1}, fix a splitting of the surjection K_{v}^{∗} ↠ K_{v}^{∗}/(K_{v}^{∗})^{∘} = Δ_{v} ⟶^{∼} Z; then K_{v}^{∗} = (K_{v}^{∗})^{∘} × Δ_{v} and Ker(r_{v}|_{Δ_{v}}) = n_{v}Δ_{v} for some n_{v} ≥ 1. Fix a set of representatives Δ′_{v} ⊂ Δ_{v} of Δ_{v}/n_{v}Δ_{v} and set V_{v} = V_{v}^{∘}Δ′_{v} ⊂ K_{v}^{∗}/F_{v}^{∗}C_{v}. This defines a finite subset V = ∏_{v∈S′} V_{v} ⊂ ∏_{v∈S′} K_{v}^{∗}/F_{v}^{∗}C_{v} stable under multiplication by the abelian group V^{∘} = ∏_{v∈S′} V_{v}^{∘} and such that the composite map V ↪ ∏_{v∈S′} K_{v}^{∗}/F_{v}^{∗}C_{v} ⟶^{r} G_{x} is surjective, with each fibre of cardinality |V |/|G_{x}|. It follows that

\frac{|V |}{|G_{x}|} e\overline{χ}(1 ⊗ α(x)) = ∑_{a∈V} χ(a) α([z, bj(a)]_{H}).

As in 2.4.5-6, fix an open compact subgroup H_{1} ⊂ B̂^{∗} such that

H_{1}F̂^{∗} ⊆ ⋂_{a∈V} Ad(j(a))HF̂^{∗}

and define, for a ∈ V,

β_{a} : N_{H_{1}}^{∗} ⟶^{[·j(a)]} N_{j(a)^{−1}H_{1}j(a)}^{∗} ⟶^{pr} N_{H}^{∗} ⟶^{α} A_{0}, α_{1} = ∑_{a∈V} χ(a) β_{a} : N_{H_{1}}^{∗} ⟶ A;

then

\frac{|V |}{|G_{x}|} e\overline{χ}(1 ⊗ α(x)) = α_{1}(x_{1}), x_{1} = [z, b]_{H_{1}}.

We claim that the map

α_{1}^{∗} : Γ(A, Ω_{A/F}) ⟶ Γ(N_{H_{1}}^{∗}, Ω_{N_{H_{1}}^{∗} /F})

is non-zero. If this were not the case, the morhism α_{1} (which is defined over F) would be constant, as the curve N_{H_{1}}^{∗} is irreducible. However, α_{1} commutes with the Hecke operator T (v) for any non-archimedean prime v of F at which B and ^{σ}π are unramified, which implies that the closed point α_{1}(N_{H_{1}}^{∗}) of A is killed by deg(T (v))−λ(v) = Nv+1−λ(v) ∈ O_{L_{0}}, where σ(λ(v)) ∈ C is the eigenvalue of T (v) on ^{σ}π. As Nv+1−λ(v) is non-zero by Weil’s bounds, it follows that α_{1}(N_{H_{1}}^{∗}) is torsion. This contradiction with our assumption that α_{1}(x_{1}) is not torsion implies that α_{1}^{∗} ≠ 0, as claimed. As in 2.4.8, we obtain that there is an embedding σ : L ↪ C and a differential form ω ∈ Γ(A^{an}, Ω^{an}) ∩ (^{σ}π^{∞})^{H} for which

ω_{1} := (α_{1} ⊗ id)^{∗}(ω) = ∑_{a∈V} ^{σ}χ(a) j(a)ω = ∏_{v∈S′} ( ∑_{a_{v}∈V_{v}} ^{σ}χ_{v}(a_{v}) j_{v}(a_{v})) ω ≠ 0

(above, χ_{v} = χ ∘ r_{v} : K_{v}^{∗} ⟶ O_{L}^{∗}). As V is stable by multiplication by V^{∘}, it follows that

∀a ∈ ∏_{v∈S′}(K_{v}^{∗})^{∘} j(a)ω_{1} = ^{σ}χ^{−1}(a) ω_{1}.

Writing

ω_{1} = ω_{1,S′} ⊗ ⨂_{v∉S′}(a spherical vector in ^{σ}π_{v}), ω_{1,S′} ∈ ⨂_{v∈S′} ^{σ}π_{v},

we deduce that ω_{1,S′} is a non-zero j (∏_{v∈S′}(K_{v}^{∗})^{∘})-invariant vector in ⨂_{v∈S′}(^{σ}π_{v} ⊗ ^{σ}χ_{v}). As each ^{σ}π_{v} is a direct sum of one-dimensional representations of (K_{v}^{∗})^{∘} (on each of which (K_{v}^{∗})^{∘} acts through a finite quotient), it follows (as in the proof of Proposition 2.4.10(1)) that there exists a non-zero j (∏_{v∈S′}(K_{v}^{∗})^{∘})- invariant linear map ^{σ}ℓ_{S′} : ⨂_{v∈S′} ^{σ}π_{v} ⟶ C(^{σ}χ^{−1}) for which ^{σ}ℓ_{S′}(ω_{1,S′}) ≠ 0. In particular, for each v ∈ S′_{2} there exists a non-zero j_{v}(K_{v}^{∗})-invariant linear map ^{σ}π_{v} ⟶ C(^{σ}χ_{v}^{−1}), hence

η_{v}(−1) ε(π_{v} × χ_{v}, \frac{1}{2}) = inv_{v}(B_{v}),

by 1.3.2. For each non-archimedean prime v ∉ S′_{2} it follows from 1.3.3(1) and the definition of S′ that

ε(π_{v} × χ_{v}, \frac{1}{2}) = η_{v}(−1) = inv_{v}(B_{v}) = 1.

(2) If v is an archimedean prime of F, then

ε(π_{v} × χ_{v}, \frac{1}{2}) = 1,  η_{v}(−1) = −1,  inv_{v}(B_{v}) = \{ \begin{matrix}−1, & v ≠ τ_{1} \\ 1, & v = τ_{1}. \end{matrix}

Combined with (1), this implies that

ε(π × χ, \frac{1}{2}) = − ∏_{v} η_{v}(−1) inv_{v}(B_{v}) = −1.

2.6.3. (1) In particular, the assumptions of ([Ne 2], Thm. 3.2) imply that the corresponding global ε-factor is equal to −1 (note that, if A_{j} = A_{0} acquires CM over K, then this ε-factor is very often equal to +1).

(2) With minor modifications, Proposition 2.6.2 also holds for a simple quotient J(M_{H}^{∗}) ↠ A_{0} of the Jacobian of the Shimura curve M_{H}^{∗} ([Co-Va 1], [Ne 2]) which does not factor through J(N_{H}^{∗}). In this case L_{0} is a CM field, ^{σ}π has a non-trivial central character (of finite order, with trivial archimedean part) ^{σ}φ = σ ∘ φ, where

φ : A_{F}^{∗} /F^{∗} ⟶ O_{L_{0}}^{∗}. A CM point x = [z, b]_{H} ∈ CM(M_{H}, K) is defined over K(x) = (K^{ab})^{rec_{K}(_{t^{−1}}(bH_{b^{−1}}))},

which is not necessarily contained in K[∞]. Fix χ : K̂^{∗} ⟶ G_{x} = Gal(K(x)/K) ⟶ O_{L}^{∗} (L ⊃ L_{0}) and assume that e\overline{χ}(1 ⊗ α(x)) ∈ A(K(x)) is not torsion. The proof of Proposition 2.6.2(1) yields a non-zero j (∏_{v∈S′}(K_{v}^{∗})^{∘})-invariant vector in ⨂_{v∈S′}(^{σ}π_{v} ⊗ ^{σ}χ_{v}), which implies that φ · χ|_{F^{∗}} = 1. Appealing to the results of ([Tu], [Wa, Thm. 2], [Sa]) in the case of a non-trivial central character, we deduce, as in the proof of Proposition 2.6.2, that

∀v ∤ ∞ ε(π_{v} × χ_{v}, \frac{1}{2}) = (φη)_{v}(−1) inv_{v}(B_{v}), ε(π × χ, \frac{1}{2}) = −1.

3. Representation theory

3.1. The Bruhat-Tits tree 3.1.1. For any set X, denote by C[X] = {∑_{x∈X} n_{x}x | n_{x} ∈ C, the sum is finite} the C-vector space on X. If G is a group acting on X (on the left), then C[X] is a left G-module. A map f : X ⟶ Y with finite fibres induces linear maps

\begin{matrix}f^{∗} : C[Y] & ⟶ C[X] \\ ∑_{y∈Y} n_{y}y & ↦ ∑_{x∈X} n_{f(x)}x, \end{matrix}  \begin{matrix}f_{∗} : C[X] & ⟶ C[Y] \\ ∑_{x∈X} n_{x}x & ↦ ∑_{y∈Y} ( ∑_{f(x)=y} n_{x})y \end{matrix}

satisfying

(3.1.1.1)  C[X] = Im(f^{∗}) ⊕ Ker(f_{∗}).

3.1.2. Let O be a complete discrete valuation ring with a uniformiser ϖ and a finite residue field O/ϖO with q elements; denote by E the fraction field of O and by v : E^{∗} ⟶ Z the valuation. Let \underline{G} = GL(2) ⊃

\underline{B} = \{ ( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} ) \} ⊃ \underline{Z} = \{ ∗ ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) \} (as group schemes over O) and put, for n ∈ N,

G = \underline{G}(E),  B = \underline{B}(E),  Z = \underline{Z}(E),  K = \underline{G}(O),  K(n) = Ker(\underline{G}(O) ⟶ \underline{G}(O/ϖ^{n}O)),
K_{0}(n) = the inverse image of \underline{B}(O/ϖ^{n}O) ⊂ \underline{G}(O/ϖ^{n}O) in \underline{G}(O).

3.1.3. Let T be the Bruhat-Tits tree associated to P GL_{2}(E); denote by V (resp., by E) the set of its vertices (resp., oriented edges). A vertex x ∈ V is a homothety class [L] of an O-lattice L ⊂ E^{2}; an oriented edge \overrightarrow{x_{0}x_{1}} ∈ E is a homothety class of ordered pairs of O-lattices L_{1} ⊂ L_{0} ⊂ E^{2} satisfying L_{0}/L_{1} ⟶^{∼} O/ϖO. We put s(e) = x_{0}, t(e) = x_{1} and ι(e) = −e = \overrightarrow{x_{1}x_{0}} (= the homothety class of ϖL_{0} ⊂ L_{1} ⊂ E^{2}). The degree of each vertex is equal to q + 1. Let o = [O^{2}] ∈ V be the origin of T . Denote by d : V × V ⟶ N the distance function on the vertices of the tree (if L/L′ ⟶^{∼} O/ϖ^{n}O, then d([L], [L′]) = n). 3.1.4. There is a natural transitive left action (by isometries) of G on V, hence also on E. For each n ≥ 0,

put g_{n} = ( \begin{matrix}1 & 0 \\ 0 & ϖ^{n} \end{matrix} ) and o_{n} = g_{n}o ∈ V; let e_{0} ∈ E be the edge satisfying s(e_{0}) = o, t(e_{0}) = o_{1}. As

Stab_{G}(o) = KZ, Stab_{G}(e_{0}) = K_{0}(1)Z,

there are natural G-equivariant bijections

(3.1.4.1)  \begin{matrix}G/KZ & ⟶^{∼} V \\ gKZ & ↦ gO, \end{matrix}  \begin{matrix}G/K_{0}(1)Z & ⟶^{∼} E \\ gK_{0}(1)Z & ↦ ge_{0}, \end{matrix}

which can be interpreted in a coordinate-free form as G-equivariant bijections

(3.1.4.2) {maximal orders in M_{2}(E)} ⟶^{∼} V R  ↦ V^{R^{∗}},  {Eichler orders of level one in M_{2}(E)} ⟶^{∼} E/{id, ι} R_{1}  ↦ E^{R_{1}^{∗}}

(with G acting on the respective sets of orders by conjugation). 3.1.5. For n ∈ N (resp., n ∈ N_{>0}), put

V_{≤n} = {x ∈ V | d(o, x) ≤ n}, V_{n} = {x ∈ V | d(o, x) = n},

resp.,

E_{n} = {e ∈ E | s(e) ∈ V_{n−1}, t(e) ∈ V_{n}}.

For n ≥ m ≥ 0, denote by r_{n,m} : V_{n} ⟶ V_{m} the geodesic projection: r_{n,m}(x) is the unique point x′ ∈ V_{m} satisfying d(x, x′) = n − m. Similarly, let r_{n,m} : E_{n} ⟶ E_{m} be the map satisfying t ∘ r_{n,m} = r_{n,m} ∘ t (for n ≥ m ≥ 1).

If x ∈ V_{≤n}, then x = [L] for some lattice O^{2} ⊃ L ⊃ ϖ^{n}O^{2}, which implies that K(n)x = x. As the fibres of r_{n,m} coincide with the orbits of K(m) on V_{n}, we obtain

(3.1.5.1)  V^{K(n)} = V_{≤n} (n ≥ 0),  E^{K(n)} = ⋃_{k=1}^{n}(E_{k} ∪ ιE_{k}) (n ≥ 1).

Moreover,

Stab_{K}(o_{n}) = K ∩ g_{n}Kg_{n}^{−1} = K_{0}(n) (n ≥ 0),

which yields K-equivariant bijections

(3.1.5.2)  K/K_{0}(n) ⟶^{∼} V_{n},  kK_{0}(n) ↦ k o_{n} = kg_{n}o (n ≥ 0).

3.1.6. More generally, for each n ≥ 0 the group G acts transitively on the set P_{n}(T ) of oriented paths of length n (without backtracking) in T (above, V = P_{0}(T ), E = P_{1}(T )). As

Stab_{G}(o → o_{1} → ⋯ → o_{n}) = K_{0}(n)Z,

there are natural G-equivariant bijections

(3.1.6.1)  \begin{matrix}G/K_{0}(n)Z & ⟶^{∼} P_{n}(T ) \\ gK_{0}(n)Z & ↦ g(o → o_{1} → ⋯ → o_{n}), \end{matrix}

(3.1.6.2)

{Eichler orders of level n in M_{2}(E)} ⟶^{∼} P_{n}(T )/{id, ι}
R_{n}  ↦ P_{n}(T )^{R_{n}^{∗}},

where ι inverses the orientation of every path. 3.1.7. Degeneracy maps. For 0 ≤ m ≤ n and 0 ≤ j ≤ n−m there are natural G-equivariant “degeneracy maps”

(3.1.7.1)  \begin{matrix}d_{j} : P_{n}(T ) & ⟶ P_{m}(T ) \\ (x_{0} → ⋯ → x_{n}) & ↦ (x_{j} → ⋯ → x_{j+m}), \end{matrix}

which can also be written using bijections (3.1.6.1) as

[·g_{j}] : G/K_{0}(n)Z ⟶ G/K_{0}(m)Z, gK_{0}(n)Z ↦ gg_{j}K_{0}(m)Z,

since

d_{j}(o → o_{1} → ⋯ → o_{n}) = g_{j}(o → o_{1} → ⋯ → o_{m}).

If π is an infinite-dimensional irreducible admissible representation of G/Z and m = o(π), then dim π^{K_{0}(m)} = 1. Moreover, if π^{K_{0}(m)} = Cf and n ≥ m, then the elements g_{j} · f (0 ≤ j ≤ n − m) form a basis of π^{K_{0}(n)} ([Ca, §1]). 3.1.8. Unramified induced representations. Given α_{1}, α_{2} ∈ C^{∗}, let μ_{1}, μ_{2} : E^{∗} ⟶ C^{∗} be the unramified characters given by μ_{j}(a) = α_{j}^{v(a)}; define

μ : B ⟶ C^{∗}, μ( ( \begin{matrix}a_{1} & t \\ 0 & a_{2} \end{matrix} ) ) ↦ μ_{1}(a_{1})μ_{2}(a_{2}),

ρ = ρ(μ_{1}, μ_{2}) = Ind_{B}^{G}(μ) = {f : G ⟶ C locally constant | f(bg) = μ(b)f(g) (b ∈ B, g ∈ G)}

with the G-action (g · f)(g′) = f(g′g). This is a smooth representation of G with central character μ_{1}μ_{2}, which is irreducible ⟺ α_{1}/α_{2} ≠ 1, q^{−2}. Moreover, the Iwasawa decomposition G = BK together with B ∩ K ⊆ Ker(μ) imply that

(3.1.8.1)  ρ^{K} = Cf_{o},  f_{o}(bk) = μ(b)  (b ∈ B, k ∈ K).

More generally, for n ≥ 1, the Iwasawa decomposition implies that

(3.1.8.2)  B\G/K(n) = B\BK/K(n) = (B ∩ K)\K/K(n) = \underline{B}(O/ϖ^{n}O)\\underline{G}(O/ϖ^{n}O) = K_{0}(n)\K,

hence

(3.1.8.3)  dim ρ^{K(n)} = |K_{0}(n)\K| = |V_{n}| = q^{n} + q^{n−1}.

More precisely, putting together the identifications (3.1.5.2) and (3.1.8.2), we obtain an isomorphism of C-vector spaces

(3.1.8.4)  \begin{matrix}Iw_{n} : ρ^{K(n)} & ⟶^{∼} C[V_{n}] \\ f & ↦ ∑_{k∈K_{0}(n)\K} f(k) (k^{−1}g_{n}o) \end{matrix}

(note that, for any k ∈ K, both f(k) ∈ C and k^{−1}g_{n}o ∈ V_{n} depend only on the class K_{0}(n)k ∈ K_{0}(n)\K).

3.2. Unramified principal series with trivial central character

3.2.1. Assume that, in the notation of 3.1.8, α_{1}α_{2} = 1 and α_{1}^{2} ≠ 1, q^{−2}. The representation ρ is then irreducible and Z acts trivially. The orbit map

\begin{matrix}V & ⟶^{∼} & G/KZ & ⟶ & ρ \\ go & ↦ & gKZ & ↦ & g · f_{o} \end{matrix}

gives rise to a G-equivariant map

\begin{aligned}F : & C[V] & ⟶ & ρ \\ & ∑ n_{g} go & ↦ & ∑ n_{g}(g · f_{o}) \end{aligned}

between two smooth representations of G. The Hecke operator

(3.2.1.1)  \begin{matrix}T : C[V] & ⟶ C[V] \\ x & ↦ ∑_{\overrightarrow{xy}∈E} y \end{matrix}

commutes with the G-action and

(3.2.1.2)  F ∘ T = (qα_{1} + α_{2})F.

3.2.2. Proposition (cf. [Gr-Pr, Lemma 3.1]). (1) F : C[V_{0}] ⟶^{∼} ρ^{K}.

(2) ∀m ≥ n ≥ 0 F ∘ r_{m,n}^{∗}(C[V_{≤n}]) ⊆ F (C[V_{≤n}]).

(3) ∀n ≥ 0 the map C[V_{≤n}] = C[V^{K(n)}] ⟶^{F} ρ^{K(n)} is surjective.

(4) ∀n ≥ 1 the map C[V_{n}] = C[V_{≤n}]/C[V_{≤n−1}] ⟶^{F} ρ^{K(n)}/ρ^{K(n−1)} induces an isomorphism

C[V_{n}]/r_{n,n−1}^{∗}(C[V_{n−1}]) ⟶^{∼} ρ^{K(n)}/ρ^{K(n−1)}.

(5) ∀n ≥ 1 F induces an isomorphism

Ker ((r_{n,n−1})_{∗} : C[V_{n}] ⟶ C[V_{n−1}]) ⟶^{∼} Ker (Tr_{K(n−1)/K(n)} : ρ^{K(n)} ⟶ ρ^{K(n−1)}) ,

where Tr_{H/H′} := ∑_{h∈H/H′} h.

Proof. The statement (1) is just (3.1.8.1), while (2) follows by induction on m − n from (3.2.1.2) and the geometry of T . In order to show (3), assume that f ∈ ρ^{K(n)}; then f = F (φ) for some φ ∈ V_{≤m}, m ≥ n (as ρ is irreducible, F is surjective). As f is K(n)-invariant, we have

(K(n) : K(m)) F (φ) = Tr_{K(n)/K(m)}F (φ) = F (Tr_{K(n)/K(m)}φ).

Writing φ = ∑_{i=0}^{m} φ_{i}, φ_{i} ∈ C[V_{i}], the claim follows from the fact that, for each i > n, F (Tr_{K(n)/K(m)}φ_{i}) is a rational multiple of F (r_{i,n}^{∗} ∘ (r_{i,n})_{∗}φ) (as the orbits of K(n)/K(m) on V_{m} coincide with the fibres of r_{m,n}), and the last term is contained in F (C[V_{≤n}]), by (2).

(4) The map in question is surjective (by (3)) and its kernel contains r_{n,n−1}^{∗}(C[V_{n−1}]) (by (2)); we conclude by dimension count, using the equality (3.1.8.3) for n and n − 1.

(5) This follows from (4), the decomposition (3.1.1.1) for f = r_{n,n−1} and the analogous decomposition

ρ^{K(n)} = ρ^{K(n−1)} ⊕ Ker (Tr_{K(n−1)/K(n)})

(since the fibres of r_{n,n−1} coincide with the orbits of K(n − 1)/K(n) on V_{n}).

3.2.3. Proposition. For each m ≥ 0, C[V_{m}] = r_{m,0}^{∗}(C[V_{0}]) ⊕ ⨁_{k=1}^{m} r_{m,k}^{∗} (Ker(r_{k,k−1})_{∗}).

Proof. Apply successively (3.1.1.1) to f = r_{1,0}, …, r_{m,m−1}. 3.2.4. Iwasawa decomposition revisited. Our next goal is to describe, for 0 ≤ m ≤ n, the image of each term in the decomposition 3.2.3 under the map

C[V_{m}] ⟶^{F} ρ^{K(m)} ↪ ρ^{K(n)}.

Composing it with the isomorphism Iw_{n} from (3.1.8.4), we obtain a linear map

(3.2.4.1)  \begin{matrix}C[V_{m}] & ⟶^{F} & ρ^{K(m)} ↪ ρ^{K(n)} & ⟶^{Iw_{n}} & C[V_{n}] \\ [k_{1}^{−1}g_{m}o] & ↦ & k_{1}^{−1}g_{m} · f_{o} & ↦ & ∑_{k_{2}∈K_{0}(n)\K}(k_{1}^{−1}g_{m} · f_{o})(k_{2}) [k_{2}^{−1}g_{n}o] \end{matrix}

(where k_{1} ∈ K_{0}(m)\K) with matrix elements

(3.2.4.2)  M_{x_{2},x_{1}} = (k_{1}^{−1}g_{m} · f_{o})(k_{2}) = f_{o}(k_{2}k_{1}^{−1}g_{m}),  x_{1} = k_{1}^{−1}g_{m}o ∈ V_{m}, x_{2} = k_{2}^{−1}g_{n}o ∈ V_{n}.

Note that the composite map (3.2.4.1) makes sense for principal series representations with arbitrary central characters. We compute its matrix coefficients (3.2.4.2) in the general context of 3.1.8.

3.2.5. Proposition. For all 0 ≤ m ≤ n,

M_{x,y} = α_{1}^{d(x′,y)/2}α_{2}^{m−d(x′,y)/2} (x ∈ V_{n}, y ∈ V_{m}, x′ = r_{n,m}(x) ∈ V_{m})

[which implies that the eigenvalue of T acting on ρ^{K} is, indeed, equal to ∑_{x∈V_{1}} M_{x,o_{1}} = qα_{1} + α_{2}].

Proof. As f_{o}(k_{2}k_{1}^{−1}g_{m}) does not change if we replace k_{i} by k_{i}k (k ∈ K), we have M_{kx,ky} = M_{x,y} (k ∈ K). In particular, we can suppose that x corresponds to K_{0}(n)k_{2} = K_{0}(n). Identifying K_{0}(m)\K with P^{1}(O/ϖ^{m}O)

via K_{0}(m) ( \begin{matrix}a & b \\ c & d \end{matrix} ) ↦ [c : d], we obtain the following set of representatives k_{1} of K_{0}(m)\K:

( \begin{matrix}1 & 0 \\ a & 1 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ 1 & b \end{matrix} ) ,

where a runs through a set of representatives of O/ϖ^{m}O in O (say, with ϖ^{m+1} ∤ a) and b runs through a set of representatives of ϖO/ϖ^{m}O in O (if m ≥ 1). Expressing k_{2}k_{1}^{−1}g_{m} = k_{1}^{−1}g_{m} in terms of the Iwasawa decomposition

\begin{aligned}( \begin{matrix}1 & 0 \\ a & 1 \end{matrix} )^{−1} ( \begin{matrix}1 & 0 \\ 0 & ϖ^{m} \end{matrix} ) & = ( \begin{matrix}ϖ^{m}/a & −1 \\ 0 & a \end{matrix} ) ( \begin{matrix}0 & 1 \\ −1 & ϖ^{m}/a \end{matrix} ) \\ ( \begin{matrix}0 & 1 \\ 1 & b \end{matrix} )^{−1} ( \begin{matrix}1 & 0 \\ 0 & ϖ^{m} \end{matrix} ) & = ( \begin{matrix}ϖ^{m} & −b \\ 0 & 1 \end{matrix} ) ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , \end{aligned}

we obtain, respectively,

M_{x,y} = \{ \begin{matrix}α_{1}^{m−v(a)}α_{2}^{v(a)}, \\ α_{1}^{m}, \end{matrix} x′ = [0 : 1], y = \{ \begin{matrix}[a : 1] \\ [1 : b] \end{matrix} ∈ P^{1}(O/ϖ^{m}O).

As

r_{m,k}(x′) = r_{m,k}(y) ⟺ \{ \begin{matrix}v(a) ≥ k, \\ k = 0, \end{matrix}

it follows that

d(x′, y)/2 = \{ \begin{matrix}m − v(a) \\ m, \end{matrix}

proving the claim.

3.2.6. Proposition. For all 1 ≤ k ≤ m ≤ n, the restriction of the map

F_{m,n} : C[V_{m}] ⟶^{F} ρ^{K(m)} ↪ ρ^{K(n)} ⟶^{Iw_{n}} C[V_{n}]

to the subspace r_{m,k}^{∗} (Ker(r_{k,k−1})_{∗}) is equal to C_{k,m}r_{n,m}^{∗}, where

C_{k,m} = α_{2}^{k−1}(α_{2} − α_{1}) ∏_{{ζ^{m−k+1}=1 \atop ζ≠1}}(α_{2} − qζα_{1}).

Proof. The space r_{m,k}^{∗} (Ker(r_{k,k−1})_{∗}) is generated by elements of the form u(y, y′) = r_{m,k}^{∗}([y] − [y′]), where y, y′ ∈ V_{k} and r_{k,k−1}(y) = r_{k,k−1}(y′). It follows from Proposition 3.2.5 that the coefficient of x ∈ V_{n} in F_{m,n}(u(y, y′)) ∈ C[V_{n}] depends only on z = r_{n,k}(x) ∈ V_{k}, and is equal to

\{ \begin{matrix}0, & z ≠ y, y′ \\ C, & z = y \\ −C, & z = y′, \end{matrix}

where

C = α_{2}^{m} + α_{1}α_{2}^{m−1}(q − 1) + ⋯ + α_{1}^{m−k}α_{2}^{k}(q^{m−k} − q^{m−k−1}) − α_{1}^{m−k+1}α_{2}^{k−1}q^{m−k} =

= α_{2}^{k−1}(α_{2} − α_{1}) ∑_{j=0}^{m−k} q^{j}α_{1}^{j}α_{2}^{m−k−j} = α_{2}^{k−1}(α_{2} − α_{1}) ∏_{{ζ^{m−k+1}=1 \atop ζ≠1}}(α_{2} − qζα_{1}),

hence F_{m,n}(u(y, y′)) = C r_{n,m}^{∗}(u(y, y′)), proving the claim.

3.2.7. Corollary. For all 1 ≤ k ≤ m ≤ n, the image of the map

r_{m,k}^{∗} (Ker(r_{k,k−1})_{∗}) ↪ C[V_{m}] ⟶^{F} ρ^{K(m)} ↪ ρ^{K(n)} ⟶^{Iw_{n}} C[V_{n}]

is equal to r_{n,k}^{∗} (Ker(r_{k,k−1})_{∗}) (resp., to zero) if [(qα_{1}/α_{2})^{m−k+1} ≠ 1 or qα_{1} = α_{2}] (resp., if [qα_{1} ≠ α_{2} and (qα_{1}/α_{2})^{m−k+1} = 1].

Proof. According to Proposition 3.2.6, the two cases are distinguished by the (non-)vanishing of C_{k,m}; we recall that α_{1} ≠ α_{2}, as ρ is irreducible.

3.2.8. Corollary (the case of the trivial central character). Assume that α_{1}α_{2} = 1 and put λ = q^{1/2}α_{1}. The eigenvalue of the Hecke operator T acting on ρ^{K} is equal to q^{1/2}(λ+λ^{−1}) and, for 1 ≤ k ≤ m ≤ n, the image of the map

r_{m,k}^{∗} (Ker(r_{k,k−1})_{∗}) ↪ C[V_{m}] ⟶^{F} ρ^{K(m)} ↪ ρ^{K(n)} ⟶^{Iw_{n}} C[V_{n}]

is equal to r_{n,k}^{∗} (Ker(r_{k,k−1})_{∗}) (resp., to zero) if [λ^{2(m−k+1)} ≠ 1 or λ^{2} = 1] (resp., if [λ^{2} ≠ 1 and λ^{2(m−k+1)} = 1].

3.3. Unramified special representations with trivial central character

3.3.1. Unramified special representations. Assume that, in the notation of 3.1.8, α_{1} = α_{2} = α. The function f_{o} is then given by f_{o}(g) = α^{v(det(g))} and the subspace Cf_{o} ⊂ ρ is G-stable. Put \overline{ρ} = ρ/Cf_{o} and denote by \overline{f} the image in \overline{ρ} of any f ∈ ρ. The (irreducible) representation \overline{ρ} = St ⊗ μ of G is a twist of the Steinberg representation by the unramified character μ = μ_{1} = μ_{2} satisfying μ(ϖ) = α.

The double coset space

B\G/K_{0}(1) = B\BK/K_{0}(1) = (B ∩ K)\K/K_{0}(1) = \underline{B}(O/ϖO)\\underline{G}(O/ϖO)/\underline{B}(O/ϖO),

contains two elements, represented by the matrices

I = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) , J = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) .

It follows that the space

ρ^{K_{0}(1)} = Cf_{I} ⊕ Cf_{J},

is two-dimensional, with basis given by the functions

f_{X}(bY k_{0}) = \{ \begin{matrix}μ(b), & X = Y \\ 0, & X ≠ Y \end{matrix} (X, Y ∈ \{I, J\}, b ∈ B, k_{0} ∈ K_{0}(1)).

As f_{o} = f_{I} + f_{J}, we have \overline{f}_{I} = −\overline{f}_{J} in \overline{ρ} and

\overline{ρ}^{K_{0}(1)} = C\overline{f}_{I} = C\overline{f}_{J}.

An easy calculation shows that the matrix w = ( \begin{matrix}0 & 1 \\ ϖ & 0 \end{matrix} ) acts on ρ^{K_{0}(1)} by

w · f_{I} = αf_{J}, w · f_{J} = αf_{I},

hence

(3.3.1.1)  w acts on \overline{ρ}^{K_{0}(1)} by multiplication by − α = −μ(ϖ).

3.3.2. The case of the trivial central character. Assume that α = ±1 ( ⟺ μ^{2} = 1 ⟺ the central character of \overline{ρ} = St ⊗ μ is trivial). As in 3.2.1, the orbit map

\begin{matrix}E & ⟶^{∼} & G/K_{0}(1)Z & ⟶ & \overline{ρ} \\ ge_{0} & ↦ & gK_{0}(1)Z & ↦ & g · f_{I} \end{matrix}

gives rise to a G-equivariant map

\begin{aligned}\overline{F} : & C[E] & ⟶ & \overline{ρ} \\ & ∑ n_{g} ge_{0} & ↦ & ∑ n_{g}(g · f_{I}). \end{aligned}

It follows from (3.3.1.1) that

(3.3.2.1)  \overline{F} ∘ ι = −α\overline{F} = −μ(ϖ)\overline{F}  ( ⟺ ∀e ∈ E  \overline{F} (−e) = −α\overline{F} (e)).

Moreover, \overline{F} satisfies the “harmonicity condition”

(3.3.2.2)  ∀x ∈ V  \overline{F} ( ∑_{s(e)=x} e) = \overline{F} ( ∑_{t(e)=x} e) = 0

(for x = o, the two terms on the L.H.S. are contained in \overline{ρ}^{K} = 0; the case of general x follows from the G-equivariance of \overline{F}).

3.3.3. Proposition. (1) ∀n ≥ 1 \overline{F} (C[E_{n}]) ⊆ \overline{F} (C[E_{n+1}]).

(2) ∀n ≥ 1 the map \overline{F} : C[E_{n}] ⟶ \overline{ρ}^{K(n)} is surjective.

(3) ∀n ≥ 1 the sequence

0 ⟶ C ∑_{e∈E_{n}} e ⟶ C[E_{n}] ⟶^{\overline{F}} \overline{ρ}^{K(n)} ⟶ 0

is exact.

Proof. (1) This follows from the harmonicity condition (3.3.2.2).

(2) The argument used in the proof of Proposition 3.2.2(3) shows that the map

\overline{F} : C[ ⋃_{k=1}^{n} E_{k}] ⟶ \overline{ρ}^{K(n)}

is surjective; we conclude by (1).

(3) The element ∑_{e∈E_{n}} e lies in the kernel of \overline{F}, by a repeated application of (3.3.2.2). On the other hand, dim C[E_{n}] = |E_{n}| = |V_{n}| and dim \overline{ρ}^{K(n)} = dim ρ^{K(n)} − 1 = |V_{n}| − 1 (by (3.1.8.3)), hence dim(Ker(\overline{F} : C[E_{n}] ⟶ \overline{ρ}^{K(n)})) = 1, by (2).

3.4. New vectors as test vectors

In this section we apply the results of 3.1-3.3 to the question formulated in 2.4.13.

3.4.1. Orders and conductors. Return to the abstract situation of 2.4.13 in the case B_{v} = M_{2}(F_{v}). We are going to use the results of 3.1-3.3 over E = F_{v} (hence q = Nv). The given embedding j_{v} : K_{v} ↪ M_{2}(F_{v}) induces a map

ℒ : V ⟶^{∼} {maximal orders in M_{2}(E)} ⟶^{j_{v}^{∗}} {O_{F,v} − orders in K_{v}} ⟶^{ord_{v}(c(−))} N

(where j^{∗}(R) = j_{v}^{−1}(R)); for x ∈ V, we say that “x has level ℒ (x)”. The j_{v}(K_{v}^{∗}/F_{v}^{∗})-orbits on V are precisely the fibres of the level map ℒ. They can be easily visualised on the tree T , if we take into account the following rules ([Co 1, Cor. 2.2.2], [Co-Va 1, Lemma 6.2,6.5]):

If x ∈ V has level n > 0, then x has 1 neighbour of level n − 1 and q neighbours of level n + 1. If x ∈ V has level 0, then x has 1 + a neighbours of level 0 and q − a neighbours of level 1, where

a = \{ \begin{matrix}−1, & if v is inert in K/F \\ 0, & if v is ramified in K/F \\ 1, & if v splits in K/F. \end{matrix}

In particular, ℒ^{−1}(0) consists of a single point if v is inert, resp., of the two endpoints of an edge if v is ramified, resp., of the vertices of an infinite path (infinite in both directions) if v splits.

3.4.2. Unramified principal series. Assume that we are in the situation of 2.4.14.3, and that c(χ_{v}) = v^{m}, c(j_{v}^{−1}(R_{v})) = v^{n}, m ≤ n (hence (2.4.13.4) holds). We are going to investigate the validity of (2.4.13.6) using the results and the notation of 3.2. Firstly, π_{v} = ρ and R_{v} corresponds to the vertex x ∈ V fixed by R_{v}^{∗} (by (3.1.4.2)), hence n = ℒ (x) and ρ^{R_{v}^{∗}} = F (Cx). We need to decide whether the subspace (of dimension at most one)

e_{\overline{χ}_{v}}(ρ^{R_{v}^{∗}}) = e_{\overline{χ}_{v}}(F (Cx)) = CF (e_{\overline{χ}_{v}}x) = CF ( ∑_{a∈U′_{v}} χ_{v}(a)(j_{v}(a)x)) ⊂ ρ

is non-zero.

In order to conform to the notation of 3.2 we must change the origin of T (by conjugating everything in sight by suitable g ∈ G) so that ℒ (o) = 0 and x = o_{n} ∈ V_{n} (equivalently, we could have stated everything in 3.2 in a coordinate-free form).

Denote by X ⊆ ℒ^{−1}(n) ⊆ V_{n} the j_{v}^{∗}((K_{v}^{∗})^{∘}/F_{v}^{∗})-orbit of x. The description of the function ℒ recalled in 3.4.1 implies the following:

(3.4.2.1)  X = \{ \begin{matrix}V_{n}, & if v is inert in K/F \\ (V_{n} − r_{n,1}^{−1}(o′)) ∪ r_{n+1,1}^{−1}(o′), & if v is ramified in K/F \\ V_{n} − r_{n,1}^{−1}(o′, o′′), & if v splits in K/F, \end{matrix}

where o′ ≠ r_{n,1}(x) (resp., o′, o′′ ≠ r_{n,1}(x)) is (resp., are) the unique neighbour(s) of o of level 0 if v is ramified (resp., if v splits).

3.4.3. Proposition. In the situation of 3.4.2, write the eigenvalue of the Hecke operator T = T_{v} acting on ρ^{K} as (Nv)^{1/2}(λ + λ^{−1}) (λ ∈ C). Assume that 1 ≤ m = ord_{v}(c(χ_{v})) ≤ n = ord_{v}(c(j_{v}^{−1}(R_{v}))) and [λ^{2(n−m+1)} ≠ 1 or λ^{2} = 1]. Then

e_{\overline{χ}_{v}}(ρ^{R_{v}^{∗}}) = CF (e_{\overline{χ}_{v}}x) ≠ 0.

Proof. If v is unramified in K/F, it follows from (3.4.2.1) that 0 ≠ e_{\overline{χ}_{v}}x ∈ r_{n,m}^{∗}(Ker(r_{m,m−1})_{∗}), hence

Iw_{n} ∘ F (e_{\overline{χ}_{v}}x) = C_{m,n}e_{\overline{χ}_{v}}x ≠ 0,

by Proposition 3.2.6 (as C_{m,n} ≠ 0 by the assumptions on λ).

If v is ramified in K/F, then (3.4.2.1) implies that e_{\overline{χ}_{v}}x = φ_{0} + φ_{1}, where

0 ≠ φ_{0} ∈ r_{n,m}^{∗}(Ker(r_{m,m−1})_{∗}), 0 ≠ φ_{1} ∈ r_{n+1,m+1}^{∗}(Ker(r_{m+1,m})_{∗})

and r_{n+1,n}^{∗}(φ_{0}), φ_{1} ∈ C[V_{n+1}] have disjoint supports. Proposition 3.2.6 then implies that

Iw_{n+1} ∘ F (e_{\overline{χ}_{v}}x) = C_{m,n}r_{n+1,n}^{∗}(φ_{0}) + C_{m+1,n+1}φ_{1} ≠ 0,

(as C_{m,n}C_{m+1,n+1} ≠ 0 by the assumptions on λ). 3.4.4. Unramified special representations. We now turn to the situation considered in 2.4.14.5. In this case π_{v} = \overline{ρ} and R_{v} is an Eichler order of level one. Let e ∈ E be the unique (up to ι) edge stabilised by R_{v}^{∗}. As in 3.4.2, write c(χ_{v}) = v^{m}, c(j_{v}^{−1}(R_{v})) = v^{n} (hence n = max(ℒ (s(e)), ℒ (t(e)))) and assume that m ≤ n. As \overline{ρ}^{R_{v}^{∗}} = \overline{F} (Ce), our goal is to decide whether the subspace

e_{\overline{χ}_{v}}(\overline{ρ}^{R_{v}^{∗}}) = e_{\overline{χ}_{v}}(\overline{F} (Ce)) = C\overline{F} (e_{\overline{χ}_{v}}e) = C\overline{F} ( ∑_{a∈U′_{v}} χ_{v}(a)(j_{v}(a)e)) ⊂ \overline{ρ}

is non-zero. As in 3.4.2, we change the origin of T so that ℒ (o) = 0 and (possibly after replacing e by ι(e)) e ∈ E_{n} in the case n > 0. Let Y ⊂ E be the j_{v}^{∗}((K_{v}^{∗})^{∘}/F_{v}^{∗})-orbit of e. As in 3.4.2, let {o′} = V_{1} ∩ ℒ^{−1}(0) (resp., {o′, o′′} = V_{1} ∩ ℒ^{−1}(0)) if v is ramified (resp., if v splits) in K/F.

If n = 0, then v is not inert and

(3.4.4.1)  Y = \{ \begin{matrix}\{e\}, & if v splits in K/F \\ \{e, ιe\} = \{\overrightarrow{oo′}, \overrightarrow{o′o}\}, & if v is ramified in K/F. \end{matrix}

If n > 0, then

(3.4.4.2)  Y = \{ \begin{matrix}E_{n}, & if v is inert in K/F \\ (E_{n} − r_{n,1}^{−1}(\overrightarrow{oo′})) ∪ r_{n+1,1}^{−1}(\overrightarrow{oo′}) = Y_{0} ∪ Y_{1}, & if v is ramified in K/F \\ E_{n} − r_{n,1}^{−1}(\overrightarrow{oo′}, \overrightarrow{oo′′}), & if v splits in K/F, \end{matrix}

3.4.5. Proposition. If, in the situation of 3.4.4, either (2.4.14.5.1) or (2.4.14.5.2) holds, then

e_{\overline{χ}_{v}}(\overline{ρ}^{R_{v}^{∗}}) = C\overline{F} (e_{\overline{χ}_{v}}e) ≠ 0.

Proof. Assume first n = 0. We have to consider only the case of v ramified in K/F, when vO_{K} = w^{2}, m = 0 and χ_{v}^{2} = 1. By (3.4.4.1) and (3.3.1.1),

\overline{F} (e_{\overline{χ}_{v}}e) = \overline{F} (e + χ_{v}(w)ι(e)) = (1 − μ(v)χ_{v}(w))\overline{F} (e),

which is non-zero precisely when χ_{v}(w) = −μ(v) (as χ_{v}(w), μ(v) ∈ {±1}), which is guaranteed by (2.4.14.5.2). We now turn to the case n > 0. If v is inert in K/F, then χ_{v} ≠ 1 by the assumption (2.4.14.5.2), which implies that m ≥ 1, hence e_{\overline{χ}_{v}}e ∉ C ∑_{e′∈E_{n}} e′ by (3.4.4.2); thus \overline{F} (e_{\overline{χ}_{v}}e) ≠ 0, by Proposition 3.3.3(3). If v splits in K/F, then Y is a proper subset of E_{n} by (3.4.4.2), hence e_{\overline{χ}_{v}}e ∉ C ∑_{e′∈E_{n}} e′ for any χ_{v}; we conclude again by Proposition 3.3.3(3). Assume, finally, that v is ramified in K/F, vO_{K} = w^{2}. It follows from (3.4.4.2) that e_{\overline{χ}_{v}}e = φ_{0} + φ_{1}, with φ_{0} ∈ C[E_{n}], φ_{1} ∈ C[E_{n+1}] such that the supports of r_{n+1,n}^{∗}(φ_{0}), φ_{1} ∈ C[E_{n+1}] are disjoint. If χ_{v} is ramified, then the coefficients of φ_{1} are not all equal to each other, which implies that e_{\overline{χ}_{v}}e ∉ C ∑_{e′∈E_{n}} e′ in this case, too. If χ_{v} is unramified, then

φ_{0} = ∑_{e′∈Y_{0}} e′, φ_{1} = χ_{v}(w) ∑_{e′∈Y_{1}} e′,

hence

\overline{F} (φ_{0} + φ_{1}) = (−1)^{n}\overline{F} (\overrightarrow{oo′}) + (−1)^{n}χ_{v}(w)F (\overrightarrow{o′o}) = (−1)^{n}(1 − χ_{v}(w)μ(v))F (\overrightarrow{oo′}),

which is again non-zero, thanks to the assumption (2.4.14.5.2).

4. Strong non-vanishing results

In §4 we generalise the main result of [Co-Va 1] in the “indefinite case” ([Co-Va 1, Thm. 4.2]) in our setting. Assume that we are given the data (D1) - (D5) from 2.4.1.

4.1. Primitive characters

In this section we generalise [Co-Va 1, Lemma 2.8]. 4.1.1. For m ≤ n ∈ N^{s}, put

G_{0}^{(c)}(n) = Im(G_{0}^{(c)} ⟶ G^{(c)}(n)), Z(n, m) = Ker(G^{(c)}(n) ⟶ G^{(c)}(n − m)), H(n) = G^{(c)}(n)/G_{0}^{(c)}(n)

(by 1.2.4-5, H(n) does not depend on c, as the notation suggests, for all but finitely many n ∈ N^{s}). For any subset I ⊆ {1, …, s} denote by δ_{I} ∈ N^{s} the characteristic function of I: δ_{I} = (m_{1}, …, m_{s}) with m_{i} = 1 (resp., m_{i} = 0) if i ∈ I (resp., if i ∉ I). If n ∈ N^{s} is large enough, then the following commutative diagram has exact rows and columns (for any I ⊆ {1, …, s}):

(4.1.1.1)  \begin{aligned}& & & & 0 & & 0 \\ & & & & ↓ & & ↓ \\ & & & & G_{0}^{(c)} & = & G_{0}^{(c)} \\ & & & & ↓ & & ↓ \\ 0 & ⟶ & Z(n, δ_{I}) & ⟶ & G^{(c)}(n) & ⟶ & G^{(c)}(n − δ_{I}) & ⟶ & 0 \\ & & ∥ & & ↓ & & ↓ \\ 0 & ⟶ & Z(n, δ_{I}) & ⟶ & H(n) & ⟶ & H(n − δ_{I}) & ⟶ & 0 \\ & & & & ↓ & & ↓ \\ & & & & 0 & & 0 \end{aligned}

In particular, if n is large enough, then there is an exact sequence

0 ⟶ Z(n, δ_{I}) ⊕ G_{0}^{(c)} ⟶ G^{(c)}(n) ⟶ H(n − δ_{I}) ⟶ 0,

in which

Z(n, δ_{I}) ⟶^{∼} ⨁_{i∈I} Z(n, δ_{{i}}), Z(n, δ_{{i}}) ⟶^{∼} O_{F} /P_{i}.

4.1.2. Fix an embedding L ↪ C and consider χ_{0} as a character χ_{0} : G_{0}^{(c)} ⟶ C^{∗}. For any n ∈ N^{s}, put (following [Co-Va 1, 2.3])

P (χ_{0}, n) = {χ : G^{(c)}(n) ⟶ C^{∗} | χ induces χ_{0} on G_{0}^{(c)} and P^{n} | c(χ)}
= {χ : G^{(c)}(n) ⟶ C^{∗} | χ induces χ_{0} on G_{0}^{(c)} and c(χ) = c(χ_{0})P^{n}}.
e(χ_{0}, n) = ∑_{χ∈P (χ_{0},n)} e\overline{χ},  e\overline{χ} = ∑_{g∈G^{(c)}(n)} χ(g) g ∈ C[G^{(c)}(n)]

(where c(χ_{0}) = c(χ_{0}|_{G_{1}^{(c)}}), in the notation of 1.2.8).

4.1.3. Lemma ([Co-Va 1, Lemma 2.8] if s = 1). If n ∈ N^{s} is sufficiently large, then

e(χ_{0}, n) = |H(n)| e_{\overline{χ}_{0}} ∏_{i=1}^{s} (1 − (NP_{i})^{−1}Tr_{Z(n,δ_{\{i\}})}) ,

where

Tr_{Z(n,δ_{I})} = ∑_{h∈Z(n,δ_{I})} h ∈ C[G^{(c)}(n)], e_{\overline{χ}_{0}} = ∑_{g∈G_{0}^{(c)}} χ(g) g ∈ C[G^{(c)}(n)].

Proof (following [Co-Va 1]). For any finite abelian group Δ denote by Δ^{∨} = Hom(Δ, C^{∗}) its group of characters. Let n ∈ N^{s} be sufficiently large. It follows from (4.1.1.1) that there exists χ′_{0} ∈ Im(G^{(c)}(n − δ_{{1,...,s}})^{∨} ⟶ G^{(c)}(n)^{∨}) which induces χ_{0} on G_{0}^{(c)}; then χ′_{0} = 1 on each Z(n, δ_{I}) and

{χ : G^{(c)}(n) ⟶ C^{∗} | χ induces χ_{0} on G_{0}^{(c)}} = H(n)^{∨} χ′_{0}.

The inclusion-exclusion principle implies that

P (χ_{0}, n) = ∑_{I⊆{1,...,s}}(−1)^{|I|} H(n − δ_{I})^{∨} χ′_{0}

(as sets with multiplicities), hence

e(χ_{0}, n) = ∑_{I⊆{1,...,s}}(−1)^{|I|} ∑_{χ∈H(n−δ_{I})^{∨}} ∑_{g∈G^{(c)}(n)}(χχ′_{0})(g) g =

= ∑_{I⊆{1,...,s}}(−1)^{|I|} Tr_{Z(n,δ_{I})} ∑_{g∈G^{(c)}(n−δ_{I})} ∑_{χ∈H(n−δ_{I})^{∨}}(χχ′_{0})(g) g =

= ∑_{I⊆\{1,...,s\}}(−1)^{|I|} Tr_{Z(n,δ_{I})}|H(n − δ_{I})| ∑_{g∈G_{0}^{(c)}} χ_{0}(g) g = |H(n)| ∏_{i=1}^{s} (1 − (NP_{i})^{−1}Tr_{Z(n,δ_{\{i\}})}) e_{\overline{χ}_{0}}.

4.1.4. Corollary. If n ∈ N^{s} is sufficiently large, then each CM point x ∈ CM(N_{H}^{∗}, K) with c(x) | cP^{n} satisfies

∑_{χ∈P (χ_{0},n)} e\overline{χ}(α(x)) = \frac{|H(n)|}{N(P_{1} ⋯ P_{s})} e_{\overline{χ}_{0}}(y) ∈ C ⊗_{O_{L}} A(K[cP^{n}]),

where

y = ∏_{i=1}^{s} (NP_{i} − Tr_{Z(n,δ_{{i}})}) α(x) ∈ A_{0}(K[cP^{n}]).

4.1.5. The next step in the argument from [Co-Va 1, 4.4] is to rewrite the point y ∈ A_{0}(K[cP^{n}]) in terms of another Shimura curve N_{H^{+}}, a new CM point x^{+} ∈ CM(N_{H^{+}}^{∗}, K) and a new map α_{+} : N_{H^{+}}^{∗} ⟶ J(N_{H^{+}}^{∗}) ↠ A_{0} as y = α_{+}(x^{+}) (provided that c(x) = c_{0}P^{n}, c_{0} | c). This will be done in §4.2.

4.2. Reparameterisation

In this section we generalise [Co-Va 1, 4.4]. 4.2.1. As in [Co-Va 1], we impose the following additional assumptions.

(A1) For each i = 1, …, s, R_{P_{i}} is an Eichler order in B_{P_{i}} (B_{P_{i}} ⟶^{∼} M_{2}(F_{P_{i}}), by (D1)); let a_{i} ≥ 0 be its level. (A2) For each i = 1, …, s, the quotient J(N_{H}^{∗}) ↠ A_{0} is P_{i}-new in the sense of [Co-Va 1, Def. 3.13], which amounts to saying that a_{i} = o(^{σ}π_{P_{i}}) (for one, hence for all σ : L ↪ C). 4.2.2. We are going to describe the elements of the (P_{1} ⋯ P_{s})-isogeny class C together with a large part of the Galois action on them in terms of the Bruhat-Tits trees T_{i} associated to P GL_{2}(F_{P_{i}}) (i = 1, …, s).

Fix a CM point x_{0} = [z, b]_{H} ∈ C. The map

(4.2.2.1)  ∏_{i=1}^{s} B_{P_{i}}^{∗}/F_{P_{i}}^{∗}(b_{P_{i}}R_{P_{i}}b_{P_{i}}^{−1})^{∗} ↠ C,  (g_{1}, …, g_{s}) ↦ [z, g_{1} ⋯ g_{s}b]_{H}

is surjective and ∏_{i=1}^{s} K_{P_{i}}^{∗}-equivariant, with K_{P_{i}}^{∗} acting on the L.H.S. (resp., on the R.H.S.) via t_{P_{i}} : K_{P_{i}} ↪ B_{P_{i}} and left multiplication (resp., via the reciprocity map).

For each i = 1, …, s, fix an isomorphism φ_{i} : B_{P_{i}} ⟶^{∼} M_{2}(F_{v}) which maps b_{P_{i}}B_{P_{i}}b_{P_{i}}^{−1} to the standard Eichler order

(4.2.2.2)  R_{0}(P_{i}^{a_{i}}) = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ M_{2}(O_{F,P_{i}}) | c ∈ P_{i}^{a_{i}}\};

together with (3.1.6.1), this yields identifications

∏_{i=1}^{s} B_{P_{i}}^{∗}/F_{P_{i}}^{∗}(b_{P_{i}}R_{P_{i}}b_{P_{i}}^{−1})^{∗} ⟶^{∼} ∏_{i=1}^{s} GL_{2}(F_{P_{i}})/F_{P_{i}}^{∗}R_{0}(P_{i}^{a_{i}})^{∗} ⟶^{∼} ∏_{i=1}^{s} P_{a_{i}}(T_{i}),

which allow us to rewrite (4.2.2.1) as a ∏_{i=1}^{s} K_{P_{i}}^{∗}-equivariant surjection

(4.2.2.3)  X : ∏_{i=1}^{s} P_{a_{i}}(T_{i}) ↠ C,

with K_{P_{i}}^{∗} acting on T_{i} and P_{a_{i}}(T_{i}) via φ_{i} ∘ t_{P_{i}} : K_{P_{i}} ↪ M_{2}(F_{v}).

As in 3.4.1, the embedding φ_{i} ∘ t_{P_{i}} gives rise to a “level function”

ℒ_{i} : V(T_{i}) ⟶ N;

define

\begin{matrix}\overline{ℒ} _{i} : P_{a_{i}}(T_{i}) & ⟶ N \\ (x_{0} → ⋯ → x_{a_{i}}) & ↦ max_{0≤j≤a_{i}} ℒ_{i}(x_{j}); \end{matrix}

then

(4.2.2.4)  ∀i = 1, …, s ∀γ = (γ_{1}, …, γ_{s}) ∈ ∏_{i=1}^{s} P_{a_{i}}(T_{i})  ord_{P_{i}} (c (X (γ))) = \overline{ℒ} _{i}(γ).

For each i = 1, …, s there is a natural K_{P_{i}}^{∗}-equivariant map

ℒ_{i}^{−1} (N_{>0}) ⟶ V(T_{i}), x ↦ x_{l},

which associates to a vertex x ∈ V(T_{i}) with ℒ_{i}(x) > 0 its unique “lower neighbour” x_{l} satisfying

d(x, x_{l}) = 1, ℒ_{i}(x_{l}) = ℒ_{i}(x) − 1

(this makes sense, thanks to the discussion in 3.4.1).

4.2.3. Definition ([Co-Va 1, Def. 1.6]). A CM point x ∈ C is P_{i}-good if either a_{i} = 0, or if x = X(γ), γ = (γ_{1}, …, γ_{s}), γ_{i} = (x_{0} → ⋯ → x_{a_{i}}), ℒ_{i}(x_{0}) ≠ ℒ_{i}(x_{a_{i}}) (for one, hence for all γ such that x = X(γ)). [This is automatic if a_{i} = 1 and P_{i} | c(x).] The point x is good if it is P_{i}-good for all i = 1, …, s.

4.2.4. Assume that x ∈ C, c(x) = c_{0}P^{n}, c_{0} | c and n ∈ N^{s} is sufficiently large. Following [Co-Va 1, §6], we are going to describe the Galois orbits Z(n, δ_{{i}}) x ⊂ C. Writing x = X(γ), γ = (γ_{1}, …, γ_{s}), γ_{i} ∈ P_{a_{i}}(T_{i}), it is enough to describe the orbits

Z_{i}(n_{i}) γ_{i} ⊂ P_{a_{i}}(T_{i}),

where

Z_{i}(n_{i}) = Ker (U(P_{i}^{a_{i}}) ⟶ U(P_{i}^{a_{i}−1})) ⊆ K_{P_{i}}^{∗}/F_{P_{i}}^{∗}(O_{F,P_{i}} + P_{i}^{a_{i}}O_{K,P_{i}})^{∗}

maps isomorphically onto Z(n, δ_{{i}}) by the reciprocity map. 4.2.4.1. [Co-Va 1, Cor. 6.6]. If a_{i} = 0, then γ_{i} = y ∈ V(T_{i}); we can assume that n_{i} = ℒ_{i}(y) ≥ 2. In this case

Z_{i}(n_{i}) y = {x ∈ V(T_{i}) | d(x, y_{l}) = 1, x ≠ (y_{l})_{l}}.

4.2.4.2. [Co-Va 1, Lemma 6.11]. If a_{i} = 1, then γ_{i} = (y_{0} → y_{1}) ∈ E(T_{i}); we can again assume that n_{i} = max(ℒ_{i}(y_{0}), ℒ_{i}(y_{1})) ≥ 2. There are two possible cases: If γ_{i} has positive orientation (notation: γ_{i} > 0), i.e., n_{i} = ℒ_{i}(y_{1}) = ℒ_{i}(y_{0}) + 1, then

Z_{i}(n_{i}) (y_{0} → y_{1}) = {(y_{0} → y) | y ≠ (y_{0})_{l}}.

If γ_{i} has negative orientation (notation: γ_{i} < 0), i.e., n_{i} = ℒ_{i}(y_{0}) = ℒ_{i}(y_{1}) + 1, then

Z_{i}(n_{i}) (y_{0} → y_{1}) = {(y → y_{1}) | y ≠ (y_{1})_{l}}.

4.2.4.3. [Co-Va 1, §6.4]. If a_{i} ≥ 2, then γ_{i} = (y_{0} → ⋯ → y_{a_{i}}). If x is P_{i}-good, then there are two possibilities, as in 4.2.4.2: If n_{i} = ℒ_{i}(y_{a_{i}}) > ℒ_{i}(y_{0}), then

Z_{i}(n_{i}) γ_{i} = {(y_{0} → ⋯ → y_{a_{i}−1} → y)} = d_{0}^{−1}(y_{0} → ⋯ → y_{a_{i}−1}).

If n_{i} = ℒ_{i}(y_{0}) > ℒ_{i}(y_{a_{i}}), then

Z_{i}(n_{i}) γ_{i} = {(y → y_{1} → ⋯ → y_{a_{i}})} = d_{1}^{−1}(y_{1} → ⋯ → y_{a_{i}}),

where

d_{0}, d_{1} : P_{a_{i}}(T_{i}) ⟶ P_{a_{i}−1}(T_{i})

are the degeneracy maps from (3.1.7.1). 4.2.5. Change of orientation. In the situation of 4.2.4.2, we have (for any σ_{0} : L_{0} ↪ C) π(σ_{0})_{P_{i}} = St⊗μ_{i}, where μ_{i} : F_{P_{i}}^{∗} ⟶ {±1} is unramified. Put

ι_{i}(γ_{i}) = (y_{1} → y_{0}), ι_{i}(γ) = (γ_{1}, …, ι_{i}(γ_{i}), …, γ_{s});

then

X(ι_{i}(γ)) = [·g] (X(γ)) , g = φ_{i}^{−1}( ( \begin{matrix}0 & 1 \\ ϖ_{P_{i}} & 0 \end{matrix} ) ) ∈ B_{P_{i}}^{∗},

hence

(4.2.5.1)  α (X(ι_{i}(γ))) = −μ_{i}(P_{i}) α(X(γ)),

by (3.3.2.1). 4.2.6. Change of the Shimura curve. For i = 1, …, s, put

a_{i}^{+} = max(2, a_{i}),  J_{i} = {0, …, a_{i}^{+} − a_{i}},  J = ∏_{i=1}^{s} J_{i}.

Let R^{+} ⊂ B be the order whose localisations are equal to

R_{v}^{+} = \{ \begin{matrix}R_{v}, & v ∤ P_{1} ⋯ P_{s} \\ b_{P_{i}}^{−1}(φ_{i}^{−1}(R_{0}(P_{i}^{a_{i}^{+}})))b_{P_{i}}, & v = P_{i}, \end{matrix}

in the notation of (4.2.2.2); put H^{+} = (R̂^{+})^{∗}. For j = (j_{1}, …, j_{s}) ∈ J, put

g(j) = (g_{1}(j_{1}), …, g_{s}(j_{s})) ∈ ∏_{i=1}^{s} B_{P_{i}}^{∗}, g_{i}(j_{i}) = b_{P_{i}}^{−1}φ_{i}^{−1}( ( \begin{matrix}1 & 0 \\ 0 & ϖ_{P_{i}}^{j_{i}} \end{matrix} ) )b_{P_{i}} ∈ B_{P_{i}}^{∗}.

As g(j)Rg(j)^{−1} ⊇ R^{+}, we obtain, for each j ∈ J, a degeneracy map

d(j) : N_{H^{+}}^{∗} ⟶^{[·g(j)]} N_{g(j)^{−1}H^{+}g(j)}^{∗} ⟶^{pr} N_{H}^{∗}.

Let C^{+} ⊂ CM(N_{H^{+}}, K) be the (P_{1} ⋯ P_{s})-isogeny class of the CM point [z, b]_{H^{+}}. The construction from 4.2.2 (with the same isomorphisms φ_{i}) yields a ∏_{i=1}^{s} K_{P_{i}}^{∗}-equivariant surjection

X^{+} : ∏_{i=1}^{s} P_{a_{i}^{+}}(T_{i}) ↠ C^{+}.

It follows from 3.1.7 that the following diagram is commutative, for any j ∈ J:

∏_{i=1}^{s} P_{a_{i}^{+}}(T_{i}) ⟶^{X^{+}} C^{+}
↓^{(d_{j_{1}} ,...,d_{js})}  ↓^{d(j)}
∏_{i=1}^{s} P_{a_{i}}(T_{i}) ⟶^{X} C.

4.2.7. Lift of CM points. For each i = 1, …, s there is a natural K_{P_{i}}^{∗}-equivariant map (commuting with \overline{ℒ} _{i})

(4.2.7.1)  \overline{ℒ} _{i}^{−1}(N_{>1}) ⟶ P_{a_{i}^{+}}(T_{i}),  γ_{i} ↦ γ_{i}^{+},

defined as follows: If a_{1} ≥ 2, then a_{i}^{+} = a_{i} and γ_{i}^{+} = γ_{i}. If a_{i} = 1 and γ_{i} = (y_{0} → y_{1}), then

γ_{i}^{+} = \{ \begin{matrix}((y_{0})_{l} → y_{0} → y_{1}), & if γ_{i} > 0 \\ (y_{0} → y_{1} → (y_{1})_{l}), & if γ_{i} < 0. \end{matrix}

If a_{i} = 0 and γ_{i} = y, then γ_{i}^{+} = (y → y_{l} → (y_{l})_{l}). By construction,

(4.2.7.2)  d_{0}(γ_{i}^{+}) = γ_{i}, unless a_{i} = 1 and γ_{i} > 0 (in which case d_{1}(γ_{i}^{+}) = γ_{i}).

Putting the maps (4.2.7.1) together, we obtain ∏_{i=1}^{s} K_{P_{i}}^{∗}-equivariant maps

\overline{ℒ} ^{−1}(N_{>1}^{s}) ⟶ ∏_{i=1}^{s} P_{a_{i}^{+}}(T_{i}), γ = (γ_{1}, …, γ_{s}) ↦ γ^{+} = (γ_{1}^{+}, …, γ_{s}^{+})

and

{x ∈ C; (P_{1} ⋯ P_{s})^{2} | c(x)} ⟶ C^{+}, x = X(γ) ↦ x^{+} = X^{+}(γ^{+}).

4.2.8. P_{i}-new quotients. The assumption (A2) of 4.2.1 implies that the map

Z[∏_{i=1}^{s} P_{a_{i}}(T_{i})] = ⨂_{i=1}^{s} Z[P_{a_{i}}(T_{i})] ⟶^{X_{∗}} Z[C] ⟶^{α_{∗}} A_{0}(K[cP^{∞}])

factors through the “new quotient”

Z[∏_{i=1}^{s} P_{a_{i}}(T_{i})]_{new} = ⨂_{i=1}^{s} Z[P_{a_{i}}(T_{i})]_{new} = ⨂_{i=1}^{s} Z[P_{a_{i}}(T_{i})]/M_{i},

where

M_{i} = ∑_{{0≤k<a_{i} \atop 0≤j≤a_{i}−k}} Im(d_{j}^{∗} : Z[P_{k}(T_{i})] ⟶ Z[P_{a_{i}}(T_{i})]).

4.2.9. Lemma. Assume that γ_{i} = (y_{0} → ⋯ → y_{a_{i}}) ∈ P_{a_{i}}(T_{i}), \overline{ℒ} _{i}(γ_{i}) = n_{i} ≥ 2 and γ_{i} is P_{i}-good (i.e., either a_{i} = 0, or ℒ_{i}(y_{0}) ≠ ℒ_{i}(y_{a_{i}})). Then the image of Tr_{Z_{i}(n_{i})}(γ_{i}) = ∑_{a∈Z_{i}(n_{i})} a · γ_{i} in Z[P_{a_{i}}(T_{i})]_{new} is equal to

T_{P_{i}}(d_{1}(γ_{i}^{+})) − d_{2}(γ_{i}^{+}),  if a_{i} = 0
−ι_{i}(d_{0}(γ_{i}^{+})),  if a_{i} = 1 and γ_{i} > 0
−ι_{i}(d_{1}(γ_{i}^{+})),  if a_{i} = 1 and γ_{i} < 0
0,  if a_{i} ≥ 2.

Above, T_{P_{i}} is the Hecke operator (3.2.1.1) on T_{i}, which associates to each vertex the sum of its neighbours.

Proof. This follows from 4.2.4.1-3 and the definition of the new quotient in 4.2.8. 4.2.10. Change of the parameterisation α. For j = (j_{1}, …, j_{s}) ∈ J, define

u(j) = ∏_{i=1}^{s} u_{i}(j_{i}) ∈ End_{F} (J(N_{H}^{∗})), u_{i}(j_{i}) = \{ \begin{matrix}NP_{i}, & if j_{i} = 0 \\ −T_{P_{i}}, & if j_{i} = 1, a_{i} = 0 \\ −μ_{i}(P_{i}), & if j_{i} = 1, a_{i} = 1 \\ 1, & if j_{i} = 2, a_{i} = 0, \end{matrix}

where this time T_{P_{i}} denotes the image of the eponymous Hecke operator in End_{F} (J(N_{H}^{∗})), and μ_{i} is as in 4.2.5. The map

α_{+} : N_{H^{+}}^{∗} ⟶^{(d(j))_{j∈J}} (N_{H}^{∗})^{J} ⟶^{ι_{H}^{J}} J(N_{H}^{∗})^{J} \underrightarrow{∑ u(j)(−)} J(N_{H}^{∗}) ↠ A_{0}

is an algebraic morphism defined over F. Functoriality of the Hodge class [Co-Va 1, §3.5] implies that, if we multiply the Hodge classes of H and H^{+} by the same integer in order to define ι_{H} and ι_{H^{+}}, then α_{+} is also given by

(4.2.10.1)  α_{+} : N_{H^{+}}^{∗} ⟶^{ι_{H^{+}}} J(N_{H^{+}}^{∗})⟶^{(d(j))_{∗}} J(N_{H}^{∗})^{J} \underrightarrow{∑ u(j)(−)} J(N_{H}^{∗}) ↠ A_{0}.

4.2.11. Proposition. Assume that x ∈ C is a good CM point which is negatively oriented at each P_{i} with a_{i} = 1, c(x) = c_{0}P^{n} = c_{0}P_{1}^{n_{1}} ⋯ P_{s}^{n_{s}}, c_{0} | c, and n ∈ N^{s} is large enough. Then

∏_{i=1}^{s} (NP_{i} − Tr_{Z(n,δ_{{i}})}) α(x) = α_{+}(x^{+}) ∈ A_{0}(K[cP^{∞}]).

Proof. This follows from (4.2.7.2), Lemma 4.2.9 and (3.3.2.1).

4.2.12. Proposition. The morphism α_{+} is not constant.

Proof. As the map ∑ u(j)(−) in (4.2.10.1) is surjective, it is enough to show that the map

(d(j))^{∗} : Γ(N_{H}^{∗an}, Ω^{an})^{J} ⟶ Γ(N_{H^{+}}^{∗an}, Ω^{an})

is injective. However, this map is given by

⨁_{σ_{0}:L_{0}↪C} (⨂_{i=1}^{s} ((π(σ_{0})_{P_{i}}^{R_{i}^{∗}})^{⊕|J_{i}|} ⟶ π(σ_{0})_{P_{i}}^{(R_{i}^{+})^{∗}}) ⊗ id) ,

where |J_{i}| = a_{i}^{+} − a_{i} + 1, and each local map

(d_{0}, …, d_{a_{i}^{+}−a_{i}})^{∗} : (π(σ_{0})_{P_{i}}^{K_{0}(P_{i}^{a_{i}})})^{⊕|J_{i}|} ⟶ π(σ_{0})_{P_{i}}^{K_{0}(P_{i}^{a{+ \atop i}})}

is injective, by Casselman’s result recalled at the end of 3.1.7.

4.3. The main result

4.3.1. Theorem. Assume that we are given the data (D1) - (D5) from 2.4.1 satisfying the assumptions (A1)

- (A2) from 4.2.1. Assume also that there exist σ : L ↪ C, ω ∈ Γ(A^{an}, Ω^{an}) ∩^{σ}π^{H} and a j(∏_{v∈S}(K_{v}^{∗})^{∘}/F_{v}^{∗})- invariant linear map ^{σ}ℓ : ^{σ}π ⟶ C(^{σ}χ_{1}^{−1}) (i.e., such that ^{σ}ℓ(j(a)u) = ^{σ}χ_{1}(a)^{−1} ^{σ}ℓ(u)) satisfying ^{σ}ℓ(ω) ≠ 0. If {x_{m}}_{m∈N} ⊆ C is a sequence of good CM points (in the sense of 4.2.3) contained in the (P_{1} ⋯ P_{s})-isogeny class C such that (P_{1} ⋯ P_{s})^{m} | c(x_{m}) for all m ∈ N, then, for each sufficiently large m ∈ N, there exists a character χ_{m} : G^{(c)} ⟶ O_{L′_{m}}^{∗} of finite order (where L′_{m} is a number field containing L) such that:

(1) the restriction of χ_{m} to G_{0}^{(c)} is equal to χ_{0};

(2) ∀i = 1, …, s ord_{P_{i}}(c(χ_{m})) = ord_{P_{i}}(c(x_{m}));

(3) the element

e_{\overline{χ}_{m}}(1 ⊗ α(x_{m})) = ∑_{g∈G^{(c)}/Ker(χ_{m})} χ_{m}(g) ⊗ α(g(x_{m})) ∈ O_{L′_{m}} ⊗_{O_{L_{0}}} A_{0}(K[cP^{∞}])

is not torsion.

Proof. Applying, if necesary, the construction from 4.2.5 (which multiplies α(x_{m}) by ±1), we can assume that each point x_{m} is negatively oriented at each P_{i} with a_{i} = 1. Combining Proposition 4.2.11 with Lemma 4.1.3, we obtain that, for m ∈ N large enough,

e(χ_{0}, n(m)) α(x_{m}) = \frac{|H(n(m))|}{N(P_{1} ⋯ P_{s})} e_{\overline{χ}_{0}} α_{+}(x_{m}^{+}),

where c(x_{m}) = c_{0}P^{n(m)}, c_{0} | c, n(m) ∈ N^{s}. Applying Theorem 2.5.1 to the modified data {x_{m}^{+}} ⊆ C^{+} and α_{+} : N_{H^{+}}^{∗} ⟶ A_{0}, we deduce that

e(χ_{0}, n(m)) α(x_{m}) ≠ 0 ∈ C ⊗_{O_{L_{0}}} A_{0}(K[cP^{∞}])

for all sufficiently large m ∈ N, which proves the claim.

4.3.2. Corollary. If A_{0} does not acquire complex multiplication over any totally imaginary quadratic extension of F contained in K[c(χ_{0})P^{∞}], then, for all sufficiently large m ∈ N, the following O_{L′_{m}}-modules are finite:

e_{\overline{χ}_{m}} (O_{L′_{m}} ⊗_{O_{L_{0}}} A_{0}(K[cP^{n(m)}])/O_{L′_{m}}(1 ⊗ α(x_{m}))) , e_{\overline{χ}_{m}} (O_{L′_{m}} ⊗_{O_{L_{0}}} � (A_{0}/K[cP^{n(m)}])) ,
e_{χ_{m}} (O_{L′_{m}} ⊗_{O_{L_{0}}} A_{0}(K[cP^{n(m)}])/O_{L′_{m}}(1 ⊗ α(ρ(x_{m})))) , e_{χ_{m}} (O_{L′_{m}} ⊗_{O_{L_{0}}} � (A_{0}/K[cP^{n(m)}])) ,

where ρ ∈ Gal(\overline{F} /F) is the complex conjugation w.r.t. some embedding \overline{F} ↪ C extending τ_{1}. Proof. This follows from Theorem 4.3.1 and [Ne 2, Thm. 3.2].

4.3.3. Theorem (an explicit special case of 4.3.1). Assume that we are given the data (D1) - (D5) from 2.4.1 satisfying the assumptions (A1) - (A2) from 4.2.1. Assume also that, for each v ∈ S, the following conditions (i)–(iv) hold:

(i) o(π_{v}) ≤ 1. (ii) If o(π_{v}) = 0, then R_{v} is a maximal order in B_{v} = M_{2}(F_{v}); if o(π_{v}) = 1 and B_{v} = M_{2}(F_{v}), then R_{v} is an Eichler order of level one in B_{v}. (iii) (⋆_{v}) holds. (iv) If o(π_{v}) = 0, write the eigenvalue of T_{v} on A_{0} as (Nv)^{1/2}(λ_{v} + λ_{v}^{−1}). Put n_{v} = ord_{v}(c(x)) (for any x ∈ C) and m_{v} = ord_{v}(c(χ_{0})). Assume that 1 ≤ m_{v} ≤ n_{v} and [λ_{v}^{2} = 1 or λ_{v}^{2(n_{v}−m_{v}+1)} ≠ 1] (which is automatic if m_{v} = n_{v} ≥ 1). Then: if {x_{m}}_{m∈N} ⊆ C is a sequence of good CM points (in the sense of 4.2.3) contained in the (P_{1} ⋯ P_{s})- isogeny class C such that (P_{1} ⋯ P_{s})^{m} | c(x_{m}) for all m ∈ N, then, for each sufficiently large m ∈ N, there exists a character χ_{m} : G^{(c)} ⟶ O_{L′_{m}}^{∗} of finite order (where L′_{m} is a number field containing L) such that:

(1) the restriction of χ_{m} to G_{0}^{(c)} is equal to χ_{0};

(2) ∀i = 1, …, s ord_{P_{i}}(c(χ_{m})) = ord_{P_{i}}(c(x_{m}));

(3) the element

∑_{g∈G^{(c)}/Ker(χ_{m})} χ_{m}(g) ⊗ α(g(x_{m})) ∈ O_{L′_{m}} ⊗_{O_{L_{0}}} A_{0}(K[cP^{∞}])

is not torsion.

Proof. In the proof of Theorem 4.3.1, refer to Theorem 2.5.2 instead of Theorem 2.5.1.

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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