Hidden symmetries in the theory of complex multiplication

Jan Nekovář

To Yuri Manin on the occasion of his 70th birthday, with admiration

0. Introduction

(0.1) Let F be a totally real number field of degree d. It is well known that one can associate to any cuspidal Hilbert eigenform f over F of parallel weight 2 a compatible system of two-dimensional l-adic Galois representations V_{l}(f) of Γ_{F} = Gal(\overline{Q}/F) over \overline{Q}_{l} (having fixed embeddings \overline{Q} ↪ C and \overline{Q} ↪ \overline{Q}_{l}). (0.2) On the other hand, the Shimura variety X associated to R_{F/Q}GL(2)_{F} has reflex field Q, which means that its étale cohomology groups give rise to l-adic representations of Γ_{Q} = Gal(\overline{Q}/Q). The action of Γ_{Q} on the intersection cohomology of the Baily-Borel compactification X^{∗} of X was determined, up to semisimplification, by Brylinski and Labesse [Br-La]: non-primitive cohomology (into which we include IH^{0}) occurs in even degrees and decomposes as

IH_{et}^{2j}(X^{∗} ⊗_{Q} \overline{Q}, \overline{Q}_{l})_{non−prim} ⟶^{∼} ⨁_{χ} χ(−j),

where each χ is a finite-order character of Γ_{Q}. Primitive cohomology occurs only in degree d; it decomposes as

IH_{et}^{d}(X^{∗} ⊗_{Q} \overline{Q}, \overline{Q}_{l})_{prim} ⟶^{∼} ⨁_{f} π(f) ⊗ W_{l}(f),

where f is as above, π(f) is the (non-archimedean part of the) automorphic representation of GL(2, A_{F}) associated to f, and W_{l}(f) is a 2^{d}-dimensional l-adic representation of Γ_{Q} whose semi-simplification W_{l}(f)^{ss} is isomorphic to the tensor induction of V_{l}(f),

⨂ Ind_{F/Q}V_{l}(f),

which is defined as follows. A choice of coset representatives

Γ_{Q} = ∐_{i=1}^{d} g_{i}Γ_{F}  (0.2.1)

defines an injective group homomorphism (see §1.1 below)

Γ_{Q} ↪ S_{d} ⋉ Γ_{F}^{d} ,  g ↦ (σ, (h_{1}, …, h_{d})),  gg_{i} = g_{σ(i)}h_{i},  (0.2.2)

and ⨂ Ind_{F/Q}V_{l}(f) is obtained from the (S_{d} ⋉ Γ_{F}^{d})-module V_{l}(f)^{⊗d} by pull-back via the map (0.2.2).

(0.3) In particular, the action of Γ_{Q} on IH_{et}^{d}(X^{∗} ⊗_{Q} \overline{Q}, \overline{Q}_{l})_{prim}^{ss} extends to an action of S_{d} ⋉ Γ_{F}^{d}. The same should be true for the action on IH_{et}^{d}(X^{∗} ⊗_{Q} \overline{Q}, \overline{Q}_{l})_{prim}, since general conjectures predict that Γ_{Q} should act semi-simply on IH_{et}^{∗}(Y ⊗_{Q} \overline{Q}, \overline{Q}_{l}), for any proper scheme Y over Spec(Q). The representations χ(−j) of Γ_{Q} occurring in the non-primitive cohomology of X^{∗} do not extend to representations of S_{d} ⋉ Γ_{F}^{d}, but they extend to representations of the group (S_{d} ⋉ Γ_{F}^{d})_{0}, which is defined as the fibre product

\begin{matrix}(S_{d} ⋉ Γ_{F}^{d})_{0} & ⟶ & S_{d} ⋉ Γ_{F}^{d} \\ ↓ & & ↓ \\ Γ_{Q}^{ab} & ⟶^{V_{F/Q}} & Γ_{F}^{ab}, \end{matrix}  (0.3.1)

where the right vertical arrow is trivial on S_{d} and is given by the product map on Γ_{F}^{d}. As the field F is totally real, the transfer map V_{F/Q} is injective (see 1.2.5 below), which means that we can (and will) consider (S_{d} ⋉ Γ_{F}^{d})_{0} as a subgroup of S_{d} ⋉ Γ_{F}^{d}. The inclusion (0.2.2) factors through an inclusion Γ_{Q} ↪ (S_{d} ⋉ Γ_{F}^{d})_{0}.

(0.4) Question. To sum up: the results of [Br-La] combined with the semi-simplicity conjecture imply that the action of Γ_{Q} on IH_{et}^{∗}(X^{∗} ⊗_{Q} \overline{Q}, \overline{Q}_{l}) should extend to an action of (S_{d} ⋉ Γ_{F}^{d})_{0}. Is there a geometric explanation of this hidden symmetry of IH_{et}^{∗}(X^{∗} ⊗_{Q} \overline{Q}, \overline{Q}_{l})?

(0.5) This question admits a more invariant formulation. Recall that the inclusion (0.2.2) depends on the choice of coset representatives (0.2.1). The same choice defines an isomorphism of F -algebras

F ⊗_{Q} \overline{Q} ⟶^{∼} \overline{F}^{d}, a ⊗ b ↦ (a ⊗ g_{i}^{−1}(b))_{i},

hence a group isomorphism

S_{d} ⋉ Γ_{F}^{d} ⟶^{∼} Aut_{F −alg}(F ⊗_{Q} \overline{Q}),  (0.5.1)

the composition of which with (0.2.2) coincides with the canonical injective map

Γ_{Q} = Aut_{Q−alg}(\overline{Q}) ↪ Aut_{F −alg}(F ⊗_{Q} \overline{Q}),  g ↦ id_{F} ⊗ g.  (0.5.2)

The subgroup Aut_{F −alg}(F ⊗_{Q} \overline{Q})_{0} of Aut_{F −alg}(F ⊗_{Q} \overline{Q}) corresponding to (S_{d} ⋉Γ_{F}^{d})_{0} under the isomorphism (0.5.1) is independent of any choices, which means that we should restate Question 0.4 as follows.

(0.6) Question. Is there a geometric explanation of the fact that the action of Γ_{Q} on IH_{et}^{∗}(X^{∗} ⊗_{Q} \overline{Q}, \overline{Q}_{l}) extends to an action of Aut_{F −alg}(F ⊗_{Q} \overline{Q})_{0}? For example, does X^{∗} ⊗_{Q} \overline{Q} (or a related space) admit an action of Aut_{F −alg}(F ⊗_{Q} \overline{Q})_{0}?

(0.7) Idle speculation. The recipe (0.2.2) defines an inclusion

G ↪ S_{d} ⋉ H^{d}  (0.7.1)

(depending on chosen coset representatives of H in G) whenever H is a subgroup of index d of a group G. If p : Y ⟶ Z is an unramified covering of degree d between “nice” connected topological spaces and H = π_{1}(Y, y), G = π_{1}(Z, p(y)), then there are at least two geometric incarnations of (0.7.1).

Firstly, if Z̃ is the universal covering of Z, then

G ⟶^{∼} Aut(Z̃/Z), S_{d} ⋉ H^{d} ⟶^{∼} Aut(Y ×_{Z} Z̃/Y)

and (0.7.1) comes from the canonical map

Aut(Z̃/Z) ⟶ Aut(Y ×_{Z} Z̃/Y),  g ↦ id_{Y} × g.  (0.7.2)

In our situation, the rôle of p (resp., by Z̃) is played by the structure map Spec(F) ⟶ Spec(Q) (resp., by Spec(\overline{Q})), and (0.7.2) is nothing but (0.5.2). Secondly, S_{d} ⋉ H^{d} is closely related to π_{1}(Y^{d}/S_{d}, p^{−1}(p(y))), and there is a canonical map

Z ⟶ Y^{d}/S_{d},  z ↦ p^{−1}(z).  (0.7.3)

In other words, the map induced by (0.7.3),

π_{1}(Z, z) ⟶ π_{1}(Y^{d}/S_{d}, p^{−1}(z)),

is an approximate version of (0.7.1). In our situation, in which the rôle of Y (resp., of Z) is played by Spec(F) (resp., by Spec(Q)), we are confronted with the fact that the analogue of Y^{d}/S_{d} should be the d-th symmetric power of Spec(F) over the elusive absolute point Spec(F_{1}). A Grothendieckean approach to Question 0.6 would then involve

• making sense of the d-th symmetric power Sym^{d}(F/F_{1}) of Spec(F) over Spec(F_{1});

• extending X^{∗} to an object X̃^{∗} defined over (a desingularisation of) Sym^{d}(F/F_{1});

• relating l-adic intersection cohomology groups^{1} of X^{∗} and X̃^{∗}. At present, this seems beyond reach, but as A. Genestier pointed out to us, everything makes sense for Drinfeld modular varieties over global fields of positive characteristic. (0.8) Leaving speculations aside, in the present article we test Question 0.6 by studying the action of Γ_{Q} on the set of CM points. It is convenient to replace R_{F/Q}GL(2)_{F} by the group G defined as the fibre product

G ⟶ R_{F/Q}(GL(2)_{F})
↓  ↓^{det}
G_{m,Q} ⟶ R_{F/Q}(G_{m,F}),

since the corresponding Shimura variety is a moduli space for polarised Hilbert-Blumenthal abelian varieties (HBAV) equipped with adelic level structures. The first main result of the present article (see 2.2.5 below) is the following.

(0.9) Theorem. The group Aut_{F −alg}(F ⊗_{Q} \overline{Q})_{0} acts naturally on the set of CM points of the Shimura variety Sh(G, 𝒳 ) associated to G. This action extends the natural action of Γ_{Q} and commutes with the action of G(A_{f}) = G(Q̂) on Sh(G, 𝒳 ).

The key point in the proof is to show that the (reverse) 1-cocycle f_{Φ} : Γ_{Q} ⟶ K̂^{∗}/K^{∗} (“the Taniyama element”), which describes the Galois action on the set of CM points by K, naturally extends to a 1-cocycle f̃_{Φ} : Aut_{F −alg}(F ⊗_{Q} \overline{Q})_{0} ⟶ K̂^{∗}/K^{∗} (above, K is a totally imaginary quadratic extension of F, K̂ is the ring of finite adèles of K and Φ is a CM type of K). In fact, f_{Φ} extends to a 1-cocycle f̃_{Φ} defined on a slightly bigger subgroup Aut_{F −alg}(F ⊗_{Q} \overline{Q})_{1} of Aut_{F −alg}(F ⊗_{Q} \overline{Q}), which corresponds to the fibre product

(S_{d} ⋉ Γ_{F}^{d})_{1} ⟶  S_{d} ⋉ Γ_{F}^{d}
↓  ↓^{(1,prod)}
Γ_{Q}^{ab}/⟨c⟩ ↪^{\overline{V}_{F/Q}} Γ_{F}^{ab}/⟨c_{1}, …, c_{d}⟩,

where c ∈ Γ_{Q}^{ab} (resp., c_{1}, …, c_{d} ∈ Γ_{F}^{ab}) is the complex conjugation (resp., are the complex conjugations at the infinite primes of F). We have

Aut_{F −alg}(F ⊗_{Q} \overline{Q})_{1}/Aut_{F −alg}(F ⊗_{Q} \overline{Q})_{0} ⟶^{∼} (Z/2Z)^{d−1},

but only the elements of Aut_{F −alg}(F ⊗_{Q} \overline{Q})_{0} preserve the positivity of the polarisations. (0.10) A more abstract formulation of this result (§2.4) involves a generalisation of the Taniyama group 𝒯 and its finite-level quotients _{K}𝒯 . More precisely, in the special case when K is a Galois extension of Q, the maps f̃_{Φ} factor through Aut_{F −alg}(F ⊗_{Q} K^{ab})_{1} = Im (Aut_{F −alg}(F ⊗_{Q} \overline{Q})_{1} ⟶ Aut_{F −alg}(F ⊗_{Q} K^{ab})) and can be put together, yielding a 1-cocycle

f̃: Aut_{F −alg}(F ⊗_{Q} K^{ab})_{1} ⟶ _{K}𝒮 (K̂)/_{K}𝒮 (K),  (0.10.1)

where _{K}𝒮 is the Serre torus associated to K (see §1.5).

Our second main result (see 2.4.2-3 below) states that the coboundary of f̃ gives rise to an exact sequence of affine group schemes over Q

1 ⟶ _{K}𝒮 \underrightarrow{_{ı}} _{K} 𝒯̃ \underrightarrow{_{π}} Aut_{F −alg}(F ⊗_{Q} K^{ab})′_{1} ⟶ 1,  (0.10.2)

where Aut_{F −alg}(F ⊗_{Q} K^{ab})′_{1} is a certain F/Q-form of the constant group scheme Aut_{F −alg}(F ⊗_{Q} K^{ab})_{1}. Moreover, there is a group homomorphism s̃p : Aut_{F −alg}(F ⊗_{Q} K^{ab})_{1} ⟶ _{K} 𝒯̃(F̂) satisfying π̃ ∘ s̃p = id. The pull-back of (0.10.2) to Aut_{Q−alg}(K^{ab}) = Gal(K^{ab}/Q) is the level-K Taniyama extension

1 Establishing a relation between de Rham cohomology of X^{∗} and X̃^{∗} would also be of interest, in view of potential applications to period relations for Hilbert modular forms.

1 ⟶ _{K}𝒮 ⟶^{i} _{K}𝒯 ⟶^{π} Gal(K^{ab}/Q) ⟶ 1.

For varying K, the 1-cocycles f̃ are compatible. When put together, they give rise to an exact sequence of affine group schemes over Q

1 ⟶ 𝒮 ⟶ 𝒯̃ ⟶ \underrightarrow{lim}_{F} Aut_{F −alg}(F ⊗_{Q} \overline{Q})′_{1} ⟶ 1  (0.10.3)

(where 𝒮 is the inverse limit of the tori _{K}𝒮 with respect to the norm maps, and the direct limit is taken with respect to the transition maps id_{F′} ⊗_{F} −, for F ⊆ F′), whose pull-back to Γ_{Q} coincides with the Taniyama extension

1 ⟶ 𝒮 ⟶ 𝒯 ⟶ Γ_{Q} ⟶ 1.

(0.11) Question. As shown in [De], the Taniyama group 𝒯 has a natural Tannakian interpretation. Does 𝒯̃, or its subgroup scheme 𝒯̃_{0} ⊂ 𝒯̃ sitting in the exact sequence

1 ⟶ 𝒮 ⟶ 𝒯̃_{0} ⟶ \underrightarrow{lim}_{F} Aut_{F −alg}(F ⊗ \overline{Q})′_{0} ⟶ 1,

have a similar interpretation?

(0.12) If A is a polarised HBAV over \overline{Q}, then H_{dR}^{1}(A/\overline{Q}) is a free F ⊗_{Q} \overline{Q}-module of rank 2, and for each prime p the F ⊗_{Q} \overline{Q} ⊗_{Q} Q_{p}-module H_{dR}^{1}(A/\overline{Q}) ⊗_{Q} Q_{p} has an additional crystalline structure. The comparison theorems between étale and crystalline cohomology together with Faltings’s isogeny theorem imply that the F -linear isogeny class of A is determined by H_{dR}^{1}(A/\overline{Q}) with this additional structure. It is very likely (even though we have not checked this) that the action (0.9) of Aut_{F −alg}(F ⊗_{Q} \overline{Q})_{0} on the set of CM points of Sh(G, 𝒳 ) is compatible, via the functor A ↦ H_{dR}^{1}(A/\overline{Q}), with the natural action of Aut_{F −alg}(F ⊗_{Q} \overline{Q}) on the category of F ⊗_{Q} \overline{Q}-modules.

(0.13) Question. What happens for non-CM points? In other words, for what g ∈ Aut_{F −alg}(F ⊗_{Q} \overline{Q})_{0} is there a polarised HBAV A′ over \overline{Q} such that

H_{dR}^{1}(A′/\overline{Q}) = g^{∗} H_{dR}^{1}(A/\overline{Q}),

with all the additional structure?

1. Background material

In §1.4-1.7 of this chapter we recall the main results of the theory of complex multiplication. In §1.1-1.3 we collect some elementary background material.

Notation and conventions: An action of a group on a set always means a left action. We write A ⊗ B instead of A⊗_{Z} B and denote by \overline{Q} the algebraic closure of Q in C. By a number field we always understand a subfield of \overline{Q} of finite degree over Q. The ring of integers of a number field k will be denoted by O_{k}. For each subfield L of \overline{Q} we put Γ_{L} = Gal(\overline{Q}/L) and X(L) = Hom_{Q−alg}(L, \overline{Q}). The restriction map g ↦ g|_{L} defines an isomorphism of left Γ_{Q}-sets Γ_{Q}/Γ_{L} ⟶^{∼} X(L). Denote by c ∈ Γ_{Q} the complex conjugation. For any abelian group A, put Â = A ⊗ Ẑ. If A is a ring, so is Â (if k is a number field, then k̂ is the ring of finite adèles of k).

1.1 Wreath products and Galois theory

(1.1.1) Notation. If X and Y are sets, we denote by Y^{X} = {f : X ⟶ Y } the set of maps from X to Y. If Y is a group, so is Y^{X}. The group of permutations of the set X, denoted by S_{X} = {bijective maps σ :

X ⟶ X}, acts on Y^{X} by ^{σ}f = f ∘ σ^{−1}. For any group H, the semi-direct product of H^{X} and S_{X} (with respect to this action of S_{X} on H^{X}) is equal to

S_{X} ⋉ H^{X} = {(σ, h) | σ ∈ S_{X}, h : X ⟶ H}, (σ, h)(σ′, h′) = (σσ′, (h ∘ σ′)h′).

If Y is a left H-set, then Y^{X} is a left (S_{X} ⋉ H^{X})-set via the action

(σ, h)(y) = (hy) ∘ σ^{−1},  h ∈ H^{X}, y ∈ Y^{X}, (hy)(x) = (h(x))(y(x)).  (1.1.1.1)

(1.1.2) Basic construction. Let H be a subgroup of a group G. Fix a section s : X = G/H ⟶ G of the natural projection G ⟶ G/H. Left multiplication by g ∈ G defines a permutation σ = (x ↦ gx) ∈ S_{X}. For each x ∈ X,

gs(x) = s(gx)h(x), h(x) ∈ H,

and the map

g ↦ (σ, h) = ((x ↦ gx), (x ↦ s(gx)^{−1}gs(x))) ∈ S_{X} ⋉ H^{X}

is an injective group homomorphism

ρ_{s} : G ↪ S_{X} ⋉ H^{X}  (X = G/H).  (1.1.2.1)

If s′ : X = G/H ⟶ G is another section, then s′ = st, t ∈ H^{X}, and

∀g ∈ G  ρ_{s′}(g) = (1, t)^{−1}ρ_{s}(g)(1, t).  (1.1.2.2)

If (G : H) < ∞, then the diagram

\begin{matrix}G & ⟶^{ρ_{s}} & S_{X} ⋉ H^{X} \\ ↓ & & ↓^{(1,prod)} \\ G^{ab} & ⟶^{V} & H^{ab} \end{matrix}  (1.1.2.3)

is commutative, where prod is the product map h ↦ ∏_{x∈X} h(x) (mod [H, H]) and V is the transfer. The map ρ_{s} factors through an injective group homomorphism

G ↪ (S_{X} ⋉ H^{X})_{0},

where (S_{X} ⋉ H^{X})_{0} is the group defined as the fibre product

\begin{matrix}(S_{X} ⋉ H^{X})_{0} & ⟶ & S_{X} ⋉ H^{X} \\ ↓ & & ↓^{(1,prod)} \\ G^{ab} & ⟶^{V} & H^{ab}. \end{matrix}  (1.1.2.4)

If V is injective, we can (and will) identify (S_{X} ⋉ H^{X})_{0} with its image in S_{X} ⋉ H^{X}.

(1.1.3) Proposition. Let k′/k be a Galois extension (not necessarily finite) and X a finite set. The action of Γ_{k′/k} = Gal(k′/k) = Aut_{k−alg}(k′) on k′ gives rise, as in (1.1.1.1), to an action of S_{X} ⋉ Γ_{k′/k}^{X} on (k′)^{X} by k-algebra automorphisms, and each k-algebra automorphism of (k′)^{X} arises in this way:

S_{X} ⋉ Γ_{k′/k}^{X} = Aut_{k−alg}((k′)^{X}), (σ, h) ↦ (a ↦ (ha) ∘ σ^{−1}).

Proof. Any k-algebra automorphism f of (k′)^{X} must permute the set of irreducible idempotents {1_{x} | x ∈ X} of (k′)^{X}: f(1_{x}) = 1_{σ(x)}, σ ∈ S_{X}. This implies that (σ, 1)∘f preserves the decomposition (k′)^{X} = ∏_{x∈X} k′·1_{x}, hence (σ, 1) ∘ f ∈ Aut_{k−alg}(k′)^{X} = Γ_{k′/k}^{X}, which implies that f ∈ S_{X} ⋉ Γ_{k′/k}^{X}.

(1.1.4) Proposition. Let k′/k be as in Proposition 1.1.3. Let F/k be a finite subextension of k′/k; put X = Hom_{k−alg}(F, k′). Fix a section s : X ⟶ Γ_{k′/k} of the restriction map Γ_{k′/k} ⟶ Γ_{k′/k}/Γ_{k′/F} = X, g ↦ g|_{F}. The chosen section induces an isomorphism of k-algebras

s : (k′)^{X} ⟶ (k′)^{X}, u ↦ (x ↦ s(x)(u(x))).

(i) The map

α : F ⊗_{k} k′ ⟶ (k′)^{X}, a ⊗ b ↦ (x ↦ x(a)b)

is an isomorphism of k-algebras. (ii) The map

β_{s} : F ⊗_{k} k′ ⟶^{α} (k′)^{X} ⟵^{s} (k′)^{X}, a ⊗ b ↦ (x ↦ as(x)^{−1}(b))

is an isomorphism of F -algebras. (iii) The map β_{s} satisfies

∀g ∈ Aut_{k−alg}(k′) = Γ_{k′/k} β_{s} ∘ (id_{F} ⊗ g) = ρ_{s}(g)β_{s},

hence induces a group isomorphism

β_{s∗} : Aut_{F −alg}(F ⊗_{k} k′) ⟶^{∼} Aut_{F −alg}((k′)^{X}) = S_{X} ⋉ Γ_{k′/F}^{X} , f ↦ β_{s} ∘ f ∘ β_{s}^{−1}

satisfying β_{s∗}(id_{F} ⊗ g) = ρ_{s}(g), for all g ∈ Γ_{k′/k}. (iv) If s′ = st : X ⟶ Γ_{k′/k} is another section of the restriction map g ↦ g|_{F} (t : X ⟶ Γ_{k′/F}), then

∀g ∈ Aut_{F −alg}(F ⊗_{k} k′) β_{s′∗}(g) = (1, t)^{−1} β_{s∗}(g) (1, t),

i.e., β_{s∗} = Ad(1, t) ∘ β_{s′∗}.

Proof. (i) This is a well-known fact from Galois theory. (ii) The map β_{s} is an isomorphism of k-algebras, by (i). For each a ∈ F, we have β_{s}(a) : x ↦ a, which means that β_{s} is a morphism of F -algebras. (iii) Let a ∈ F, b ∈ k′, g ∈ Γ_{k′/k} = G, H = Γ_{k′/F}; put ρ_{s}(g) = (σ, h). For each x ∈ X we have

σ(x) = gx, h(x) = s(gx)^{−1}gs(x) = s(σ(x))^{−1}gs(x) ∈ H, β_{s}(a ⊗ b)(x) = as(x)^{−1}(b),

hence

β_{s} ∘ (id_{F} ⊗ g)(a ⊗ b) = β_{s}(a ⊗ g(b)) : x ↦ as(x)^{−1}(g(b)).

On the other hand,

(σ, h) ∘ β_{s}(a ⊗ b) : x ↦ h(σ^{−1}(x)) (a s(σ^{−1}(x))^{−1}(b)) = a (s(x)^{−1}g) (b),

which proves that β_{s} ∘ (id_{F} ⊗ g) = ρ_{s}(g) ∘ β_{s}, as claimed. (iv) We have β_{s′} = t^{−1}β_{s}, since

∀x ∈ X β_{s′}(a ⊗ b)(x) = at(x)^{−1} ∘ s(x)^{−1}(x) = t(x)^{−1} (as(x)^{−1}(b)) = t(x)^{−1} (β_{s}(a ⊗ b)(x)) ,

in the notation of the proof of (iii). It follows that

β_{s′∗}(g) = β_{s′} ∘ g ∘ β_{s′}^{−1} = t^{−1}β_{s} ∘ g ∘ β_{s}^{−1}t = t^{−1}β_{s∗}(g)t,

as claimed. (1.1.5) To sum up the discussion from 1.1.3-4, the natural map

(id_{F} ⊗ −) : Γ_{k′/k} = Aut_{k−alg}(k′) ⟶ Aut_{F −alg}(F ⊗_{k} k′), g ↦ id_{F} ⊗ g

is a canonical incarnation of the morphism ρ_{s} : Γ_{k′/k} ↪ S_{X} ⋉ Γ_{k′/F}^{X}, since β_{s∗} ∘ (id_{F} ⊗ −) = ρ_{s}.

(1.1.6) Proposition. Let k ⊂ F ⊂ k′ and s : X ⟶ Γ_{k′/k} be as in Proposition 1.1.4. Given ũ ∈ Γ_{k′/k}, put u = ũ|_{F}, F′ = u(F) and X′ = Hom_{k−alg}(F′, k′). The bijection X ⟶^{∼} X′ (x ↦ x′ = xu^{−1}) gives rise to a section s′ : X′ ⟶ Γ_{k′/k} of the restriction map g ↦ g|_{F′}, given by s′(x′) = s(x)u^{−1}.

(i) The map

ũ_{∗} : S_{X} ⋉ Γ_{k′/F}^{X} ⟶ S_{X′} ⋉ Γ_{k′/F′}^{X′}, (σ, h) ↦ (σ′, h′)

σ′(x′) = σ(x)′ ( ⟺ σ′(xu^{−1}) = σ(x)u^{−1}), h′(x′) = ũh(x)u^{−1} ( ⟺ h′(xu^{−1}) = ũh(x)u^{−1})

is a group isomorphism satisfying ũ_{∗} ∘ ρ_{s} = ρ_{s′}. (ii) ∀ũ, u′ ∈ Γ_{k′/k} u′_{∗}ũ_{∗} = (u′ũ)_{∗}.

Proof. Easy calculation.

(1.1.7) Proposition. In the situation of Proposition 1.1.6,

(i) the map

[u] : Aut_{F −alg}(F ⊗_{k} k′) ⟶  Aut_{F′−alg}(F′ ⊗_{k} k′)
g  ↦ (u ⊗ id_{k′}) ∘ g ∘ (u^{−1} ⊗ id_{k′})

is a group isomorphism satisfying [u′u] = [u′] ∘ [u] and

∀g ∈ Γ_{k′/k} [u](id_{F} ⊗ g) = id_{F′} ⊗ g.

(ii) The following diagram is commutative.

Aut_{F −alg}(F ⊗_{k} k′) ⟶^{β_{s∗}} S_{X} ⋉ Γ_{k′/F}^{X}
↓^{[u]}  ↓_{u_{∗}}
Aut_{F′−alg}(F′ ⊗_{k} k′) ⟶^{β_{s′∗}} S_{X′} ⋉ Γ_{k′/F′}^{X′}

(iii) If F′ = F, then the group automorphism

β_{s∗} ∘ [u] ∘ β_{s∗}^{−1} : S_{X} ⋉ Γ_{k′/F}^{X} ⟶ S_{X} ⋉ Γ_{k′/F}^{X}

is given by the formula (σ, h) ↦ (σ_{u}, h_{u}), where

∀x ∈ X  σ_{u}(x) = σ(xu)u^{−1},  h_{u}(x) = s (σ_{u}(x))^{−1} s (σ_{u}(x)u) h(xu)s(xu)^{−1}s(x).

(iv) If F is a Galois extension of k, then the maps [u] define an action of Γ_{F/k} on Aut_{F −alg}(F ⊗_{k} k′), whose set of fixed points is equal to id_{F} ⊗ Γ_{k′/k}.

Proof. (i) Straightforward. (ii) Let g ∈ Aut_{F −alg}(F ⊗_{k} k′); put (σ, h) = β_{s∗}(g) and (σ′, h′) = ũ_{∗}(σ, h). For a ⊗ b ∈ F ⊗_{k} k′, write g(1 ⊗ b) = ∑ a_{i} ⊗ b_{i}; then g(a ⊗ b) = ∑ aa_{i} ⊗ b_{i}. As β_{s}(a ⊗ b)(x) = as(x)^{−1}(b), the equalities

β_{s}(g(a ⊗ b))(x) = ((σ, h)β_{s}(a ⊗ b))(x) (x ∈ X)

read as

∑ aa_{i}s(x)^{−1}(b_{i}) = ah(σ^{−1}(x))s(σ^{−1}(x))^{−1}(b)  (x ∈ X).  (1.1.7.1)

As ([u](g))(1 ⊗ b) = ∑ u(a_{i}) ⊗ b_{i}, the statement to be proved, namely

∀x′ ∈ X′ ∀a′ ∈ F′ ∀b ∈ k′ β_{s′}(([u](g))(a′ ⊗ b))(x′) =^{?} ((σ′, h′)β_{s′}(a′ ⊗ b))(x′),

reads as

∑ a′u(a_{i})s′(x′)^{−1}(b_{i}) =^{?} a′h′(σ^{′−1}(x′))s′(σ^{′−1}(x′))^{−1}(b),

which is obtained from (1.1.7.1) (with x = x′u) by applying u, since

s′(x′)^{−1} = ũs(x)^{−1},  s′(σ^{′−1}(x′))^{−1} = ũs(σ^{−1}(x))^{−1},  h′(σ^{′−1}(x′)) = ũh(σ^{−1}(x))u^{−1}.

(iii) The assumption F′ = F implies that s′ = st, where t : X ⟶ Γ_{k′/F} is given by t(x) = s(x)^{−1}s(xu)u^{−1}. It follows from (ii) and Proposition 1.1.4(iv) that

β_{s∗} ∘ [u] ∘ β_{s∗}^{−1} = β_{s∗} ∘ β_{s′∗}^{−1} ∘ ũ_{∗} = Ad(1, t) ∘ ũ_{∗},

hence

(σ_{u}, h_{u}) = (1, t)(σ′, h′)(1, t)^{−1} = (σ′, (t ∘ σ′)h′t^{−1}), σ_{u}(x) = σ′(x) = σ(xu)u^{−1},
h_{u}(x) = t(σ_{u}(x))h′(x)t(x)^{−1} = s (σ_{u}(x))^{−1} s (σ_{u}(x)u) h(xu)s(xu)^{−1}s(x).

(1.1.8) Proposition. In the situation of Proposition 1.1.4, let F′/F be a subextension of k′/F; put X′ = Hom_{k−alg}(F′, k′) and fix a section s′ : X′ ⟶ Γ_{k′/k} of the restriction map g ↦ g|_{F′}. For each x′ ∈ X′, define t(x′) ∈ Γ_{k′/F} by the relation s′(x′) = s(x′|_{F})t(x′).

(i) The map

ρ_{s,s′} : S_{X} ⋉ Γ_{k′/F}^{X} ⟶ S_{X′} ⋉ Γ_{k′/F′}^{X′}, (σ, h) ↦ (σ′, h′),
σ′(x′) = s(σ(x))h(x)s(x)^{−1}x′,  h′(x′) = t(σ′(x′))^{−1}h(x)t(x′),  x = x′|_{F}

is a group homomorphism satisfying

σ′(x′)|_{F} = σ(x), s′(σ′(x′))h′(x′)s′(x′)^{−1} = s(σ(x))h(x)s(x)^{−1}.

(ii) The following diagram is commutative:

Aut_{F −alg}(F ⊗_{k} k′) ⟶^{β_{s∗}} S_{X} ⋉ Γ_{k′/F}^{X}
↓^{(id_{F′} ⊗_{F} −)}  ↓^{ρ_{s,s′}}
Aut_{F′−alg}(F′ ⊗_{k} k′) ⟶^{β_{s′∗}} S_{X′} ⋉ Γ_{k′/F′}^{X′}

Proof. (i) Easy calculation. (ii) As in the proof of Proposition 1.1.7, fix a⊗b ∈ F ⊗_{k} k′, g ∈ Aut_{F −alg}(F ⊗_{k} k′) and put (σ, h) = β_{s∗}(g). Writing g(1 ⊗ b) = ∑ a_{i} ⊗ b_{i}, then (1.1.7.1) (for σ(x) instead of x) reads as

∑ a_{i}s(σ(x))^{−1}(b_{i}) = h(x)s(x)^{−1}(b)  (x ∈ X).  (1.1.8.1)

Define (σ′, h′) := ρ_{s,s′}(σ, h); we must show that

∀x′ ∈ X′ ∀a′ ∈ F′ ∀b ∈ k′ β_{s′} ((id_{F′} ⊗_{F} g)(a′ ⊗ b)) (x′) =^{?} ((σ′, h′)β_{s′}(a′ ⊗ b))(x′),

which can be rewritten (again using (1.1.7.1) and replacing x′ by σ′(x′)) as follows:

∑ a′a_{i}s′(σ′(x′))^{−1}(b_{i}) =^{?} a′h′(x′)s′(x′)^{−1}(b)  (x′ ∈ X′).  (1.1.8.2)

As σ′(x′)|_{F} = σ(x′|_{F}), the equality (1.1.8.2) is obtained by multiplying (1.1.8.1) (for x = x′|_{F}) by t(σ′(x′))^{−1} on the left.

1.2 Class Field Theory

(1.2.1) Let k be a number field. Denote by

k_{+}^{∗} = Ker (k^{∗} ⟶ π_{0}((k ⊗ R)^{∗})) , O_{k,+}^{∗} = O_{k}^{∗} ∩ k_{+}^{∗}

the set of totally positive elements and the set of totally positive units of k, respectively. Let A_{k} be the adèle ring of k and C_{k} = A_{k}^{∗}/k^{∗} the idèle class group of k. The reciprocity map

rec_{k} : C_{k} ⟶ Γ_{k}^{ab}

will be normalised by letting local uniformisers correspond to geometric Frobenius elements. As rec_{k} induces an isomorphism π_{0}(C_{k}) ⟶^{∼} Γ_{k}^{ab}, its restriction to the group of finite idèles gives rise to a surjective continuous morphism

r_{k} : k̂^{∗}/k_{+}^{∗} ⟶ Γ_{k}^{ab}.

(1.2.2) It follows from the structure of the connected component of C_{k} ([Ar-Ta], ch. 9, Thm. 3) that the kernel of r_{k} is isomorphic, as an Aut(k/Q)-module, to O_{k,+}^{∗} ⊗ (Ẑ/Z) = O_{k,+}^{∗} ⊗ (Q̂/Q). (1.2.3) For k = Q, the map r_{Q} is an isomorphism, and its composition with the canonical isomorphism Ẑ^{∗} ⟶^{∼} Q̂^{∗}/Q_{+}^{∗} (induced by the inclusion of Ẑ into Q̂) is inverse to the cyclotomic character

χ : Γ_{Q}^{ab} ⟶^{∼} Ẑ^{∗},  g(ζ) = ζ^{χ(g)}  (∀ζ a root of unity in \overline{Q}).

(1.2.4) If k′/k is a finite extension of number fields, then the inclusion k ↪ k′ and the norm N_{k′/k} : k^{′∗} ⟶ k^{∗} induce commutative diagrams

\begin{matrix}k̂^{∗}/k_{+}^{∗} & ⟶^{i_{k′/k}} & k̂^{′∗}/k_{+}^{′∗} \\ ↓ r_{k} & & ↓ r′_{k} \\ Γ_{k}^{ab} & ⟶^{V_{k′/k}} & Γ_{k′}^{ab} \end{matrix}  \begin{matrix}k̂^{′∗}/k_{+}^{′∗} & ⟶^{N_{k′/k}} & k̂^{∗}/k_{+}^{∗} \\ ↓ r′_{k} & & ↓ r_{k} \\ Γ_{k′}^{ab} & ⟶^{j_{k′/k}} & Γ_{k}^{ab}, \end{matrix}  (1.2.4.1)

where V_{k′/k} is the transfer map and j_{k′/k} is given by the restriction map g ↦ g|_{k^{ab}}.

(1.2.5) Proposition. For any number field L,

Ker (V_{L/Q} : Γ_{Q}^{ab} ⟶ Γ_{L}^{ab}) = \{ \begin{matrix}\{1, c\}, & if L is totally complex, \\ \{1\}, & otherwise. \end{matrix}

Proof. Let L′ be the Galois closure of L over Q. As

Im (i_{L/Q}) ∩ Ker(r_{L}) ⊆ (O_{L′,+}^{∗} ⊗ Q̂/Q)^{Gal(L′/Q)} = Z_{+}^{∗} ⊗ Q̂/Q = {1},

the first commutative diagram (1.2.4.1) for L/Q implies that i_{L/Q}^{−1} (Ker(r_{L})) = Ker(r_{L} ∘ i_{L/Q}) is equal to

Ker (i_{L/Q}) = (Q^{∗} ∩ L_{+}^{∗}) /Q_{+}^{∗} = \{ \begin{matrix}Q^{∗}/Q_{+}^{∗} = \{±1\}, & if L is totally complex, \\ \{1\}, & otherwise. \end{matrix}

As r_{Q} is an isomorphism and r_{Q}(−1) = c, the statement follows.

1.3 CM fields

Let K be a CM number field; let F be its maximal totally real subfield (in other words, c(K) = K, τc = cτ ≠ τ for all τ ∈ X(K), and F = K^{c=1}). Put X = X(F). (1.3.1) Complex conjugations. Fix a section s : X ⟶ Γ_{Q} of the restriction map g ↦ g|_{F}. For each x ∈ X, the image of the element s(x)^{−1}cs(x) ∈ Γ_{F} in Γ_{F}^{ab} is independent of the chosen section; denote it by c_{x} ∈ Γ_{F}^{ab} (this is the complex conjugation defined by the real place x of F). Denote by ⟨c_{X}⟩ the subgroup of Γ_{F}^{ab} generated by all c_{x} (x ∈ X). The signs at the real places induce an isomorphism

(sgn ∘ x)_{x∈X} : F^{∗}/F_{+}^{∗} ⟶^{∼} {±1}^{X}.

Compatibility of the local and global reciprocity maps implies that

∀a ∈ F^{∗}  r_{F} (aF_{+}^{∗}) = ∏_{x∈X} c_{x}^{a_{x}},  (−1)^{a_{x}} = sgn(x(a)).

As Ker(r_{F}) is a Q-vector space, we have Ker(r_{F}) ∩ F^{∗}/F_{+}^{∗} = {1}, which means that r_{F} induces an isomor-

phism F^{∗}/F_{+}^{∗} ⟶^{∼} ⟨c_{X}⟩.

(1.3.2) Transfer maps. If we denote by

R : Γ_{F} ⟶ Γ_{K},  g, cg ↦ g  (g ∈ Γ_{K})

the “retraction map” from Γ_{F} to Γ_{K}, then

∀h ∈ Γ_{F}  V_{K/F} (h|_{F^{ab}}) = V_{K/F} (ch|_{F^{ab}}) = hchc|_{K^{ab}} = ^{1+c} (R(h)|_{K^{ab}}) .  (1.3.2.1)

As noted in 1.2.5,

Ker (V_{K/Q} : Γ_{Q}^{ab} ⟶ Γ_{K}^{ab}) = r_{Q} (Ker(i_{K/Q})) = r_{Q} (Q^{∗}/Q_{+}^{∗}) = {1, c} = ⟨c⟩.  (1.3.2.2)

The equality Ker(r_{F}) = O_{F,+}^{∗} ⊗ Q̂/Q = O_{K}^{∗} ⊗ Q̂/Q = Ker(r_{K}) implies, thanks to (1.2.4.1), that

Ker (V_{K/F} : Γ_{F}^{ab} ⟶ Γ_{K}^{ab}) = r_{F} (Ker(i_{K/F})) = r_{F} (F^{∗}/F_{+}^{∗}) = ⟨c_{X}⟩.  (1.3.2.3)

As a result, the map

\overline{V}_{F/Q} : Γ_{Q}^{ab}/⟨c⟩ ↪ Γ_{F}^{ab}/⟨c_{X}⟩  (1.3.2.4)

induced by V_{F/Q} is injective and

{h ∈ Γ_{F}^{ab} | V_{K/F} (h) ∈ V_{K/Q}(Γ_{Q}^{ab})} = ⟨c_{X}⟩ V_{F/Q}(Γ_{Q}^{ab}).  (1.3.2.5)

It also follows that

V_{F/Q}(Γ_{Q}^{ab}) ∩ ⟨c_{X}⟩ = ⟨V_{F/Q}(c)⟩  (1.3.2.6)

is the cyclic group of order 2 generated by V_{F/Q}(c) = ∏_{x∈X} c_{x}. (1.3.3) As observed in [Ta, Lemma 1], the finiteness of O_{K}^{∗}/O_{F,+}^{∗} implies that c (resp., 1 + c) acts trivially (resp., invertibly) on the Q-vector space Ker(r_{K}).

(1.3.4) Proposition. (i) The continuous homomorphism (induced by r_{K})

{a ∈ K̂^{∗} | ^{1+c}a ∈ Ẑ^{∗}K^{∗}}/K^{∗} ⟶ {g ∈ Γ_{K}^{ab} | g|_{F^{ab}} ∈ ⟨c_{X}⟩ V_{F/Q}(Γ_{Q}^{ab})}

is bijective. Denote by ℓ_{K} its inverse; then ^{1+c}ℓ_{K}(g) = χ(u(g))K^{∗}, where u(g) ∈ Γ_{Q}^{ab}/⟨c⟩ is the (unique) element satisfying \overline{V}_{F/Q} (u(g)) = ⟨c_{X}⟩g|_{F^{ab}} (equivalently, V_{K/Q} (u(g)) = ^{1+c}g). (ii) More precisely, if g ∈ Γ_{K}^{ab} satisfies

g|_{F^{ab}} = V_{F/Q}(u(g)) ∏_{x∈X} c_{x}^{a_{x}} (u(g) ∈ Γ_{Q}^{ab}, a_{x} ∈ Z/2Z),

then N_{K/F} (ℓ_{K}(g)) = χ(u(g))αF_{+}^{∗} ∈ F̂^{∗}/F_{+}^{∗}, where α ∈ F^{∗} and

∀x ∈ X sgn(x(α)) = (−1)^{a_{x}}.

(iii) The canonical morphism (induced by the inclusion Ô_{K} ↪ K̂)

{x ∈ Ô_{K}^{∗} | ^{1+c}x ∈ Ẑ^{∗}} ⟶ {a ∈ K̂^{∗} | ^{1+c}a ∈ Ẑ^{∗}K^{∗}}/K^{∗}

has finite kernel and cokernel. (iv) The morphism ℓ_{K} defined in (i) admits a lift

ℓ̃_{K} : {g ∈ Γ_{K}^{ab} | g|_{F^{ab}} ∈ ⟨c_{X}⟩ V_{F/Q}(Γ_{Q}^{ab})} ⟶ {a ∈ K̂^{∗} | ^{1+c}a ∈ Ẑ^{∗}K^{∗}},

which is a homomorphism when restricted to a suitable open subgroup.

Proof. (i) In the following commutative diagram the right column is exact and r_{Q} is an isomorphism:

0 ↓ Ker(r_{K}) ↓

Ẑ^{∗} ⟶ K̂^{∗}/K^{∗}
↓ ^{r_{Q}}  ↓ ^{r_{K}}
Γ_{Q}^{ab} ⟶^{V_{K/Q}}  Γ_{K}^{ab}

↓ 0

As 1 + c acts invertibly on Ker(r_{K}), the snake lemma implies that r_{K} induces an isomorphism

Ker (K̂^{∗}/K^{∗} ⟶^{1+c} K̂^{∗}/K^{∗}Ẑ^{∗}) ⟶^{∼} Ker (1 + c = V_{K/F} ∘ j_{K/F} : Γ_{K}^{ab} ⟶ Γ_{K}^{ab}/V_{K/Q}(Γ_{Q}^{ab}));

by (1.3.2.5), the second group is equal to {g ∈ Γ_{K}^{ab} | g|_{F^{ab}} ∈ ⟨c_{X}⟩ V_{F/Q}(Γ_{Q}^{ab})}. The remaining statement follows from the fact that

r_{K} (^{1+c}ℓ_{K}(g)) = ^{1+c}g = V_{K/F} ∘ j_{K/F} (g) = V_{K/F} (g|_{F^{ab}}) = V_{K/F} ∘ V_{F/Q} (u(g)) =
= V_{K/Q} (u(g)) = r_{K} ∘ i_{K/Q} ∘ r_{Q}^{−1} (u(g)) = r_{K} (χ(u(g))) .

(ii) Let a ∈ K̂^{∗} be a lift of ℓ_{K}(g) such that ^{1+c}a = bα′, where b ∈ Ẑ^{∗}, α′ ∈ K^{∗}; then α′ ∈ (K^{∗})^{c=1} = F^{∗}. As

g|_{F^{ab}} = r_{F} (N_{K/F} (a)) = r_{F} (b)r_{F} (α′) = V_{F/Q}(r_{Q}(b)) ∏_{x∈X} c_{x}^{a′_{x}}, (−1)^{a′_{x}} = sgn(x(α′)),

it follows from (1.3.2.6) that there is t ∈ Z/2Z such that

u(g) = r_{Q}(b)c^{t}, ∀x ∈ X a′_{x} = a_{x} + t.

This implies that χ(u(g)) = b(−1)^{t} and

N_{K/F} (ℓ_{K}(g)) = ^{1+c}aF_{+}^{∗} = χ(u(g))αF_{+}^{∗}

with α = α′(−1)^{t}, hence

∀x ∈ X sgn(x(α)) = sgn(x(α′))(−1)^{t} = (−1)^{a′_{x}+t} = (−1)^{a_{x}}.

(iii) This follows from the finiteness of the groups Ker, Coker(1 + c : O_{K}^{∗} ⟶ O_{K}^{∗}) and Cl_{K} = K̂^{∗}/Ô_{K}^{∗}K^{∗}, combined with the snake lemma applied to the diagrams

0 ⟶ O_{K}^{∗} ⟶ Ô_{K}^{∗} ⟶ Ô_{K}^{∗}/O_{K}^{∗} ⟶ 0
↓^{1+c}  ↓^{1+c}  ↓^{1+c}
0 ⟶ O_{K}^{∗}/Z^{∗} ⟶ Ô_{K}^{∗}/Ẑ^{∗} ⟶ Ô_{K}^{∗}/Ẑ^{∗}O_{K}^{∗} ⟶ 0

and

0 ⟶ Ô_{K}^{∗}/O_{K}^{∗} ⟶ K̂^{∗}/K^{∗} ⟶ Cl_{K} ⟶ 0
↓^{1+c}  ↓^{1+c}  ↓^{1+c}
0 ⟶ Ô_{K}^{∗}/Ẑ^{∗}O_{K}^{∗} ⟶ K̂^{∗}/Ẑ^{∗}K^{∗} ⟶ Cl_{K} ⟶ 0.

Above, Ô_{K}^{∗} is a shorthand for (Ô_{K})^{∗}. Note also that Ẑ^{∗} ∩ O_{K}^{∗} = Z^{∗} inside Ô_{K}^{∗}. (iv) By (i) and (iii), r_{K} induces a continuous homomorphism of pro-finite abelian groups

f : A = {x ∈ Ô_{K}^{∗} | ^{1+c}x ∈ Ẑ^{∗}} ⟶ B = {g ∈ Γ_{K}^{ab} | g|_{F^{ab}} ∈ ⟨c_{X}⟩ V_{F/Q}(Γ_{Q}^{ab})}

with finite kernel and cokernel. This implies that there exists an open subgroup (= a compact subgroup of finite index) A′ ⊂ A such that A′ ∩ Ker(f) = {1}. Then B′ = f(A′) is a compact subgroup of finite index (= an open subgroup) of B, and f induces a topological isomorphism f′ : A′ ⟶^{∼} B′. Fix coset representatives B = ⋃_{i} b_{i}B′ (disjoint union) and lifts ã_{i} ∈ K̂^{∗} of ℓ_{K}(b_{i}) ∈ K̂^{∗}/K^{∗} such that b_{i_{0}} = 1 and ã_{i_{0}} = 1; the map

ℓ̃_{K} : B ⟶ K̂^{∗},  b_{i}f′(a′) ↦ ã_{i}a′  (a′ ∈ A′)

has the required properties.

1.4 Tate’s construction

Let Φ be a CM type of K, i.e., a subset Φ ⊂ X(K) such that X(K) = Φ ∪ cΦ (disjoint union). (1.4.1) Tate’s half-transfer [Ta] is the continuous map F_{Φ} : Γ_{Q} ⟶ Γ_{K}^{ab} defined by the formula

F_{Φ}(g) = ∏_{φ∈Φ} w(gφ)^{−1}gw(φ) (mod Γ_{K^{ab}}),  (1.4.1.1)

where w : X(K) ⟶ X(\overline{Q}) = Γ_{Q} is any section of the restriction map g ↦ g|_{K} satisfying w(cy) = cw(y), for all y ∈ X(K). The restriction map g ↦ g|_{F} defines a bijection Φ ⟶^{∼} X(F). Composing its inverse with w, we obtain a section t : X(F) ⟶ X(\overline{Q}) = Γ_{Q} of the restriction map to F, which implies that

F_{Φ}(g)|_{F^{ab}} = ∏_{x∈X(F)} t(gx)^{−1}c^{a(g,x)}gt(x) (mod Γ_{F^{ab}}) ∈ ⟨c_{X}⟩V_{F/Q}(g)  (a(g, x) ∈ Z/2Z). (1.4.1.2)

The maps F_{Φ} satisfy

F_{Φ}(gg′) = F_{g′Φ}(g)F_{Φ}(g′)  (g, g′ ∈ Γ_{Q})  (1.4.1.3)

and

u ∘ F_{Φ}(g) ∘ u^{−1} = F_{Φu^{−1}}(g)  (g ∈ Γ_{Q}),  (1.4.1.4)

for any isomorphism of CM number fields u : K ⟶^{∼} K′. In addition, if K′ is a CM number field containing K and Φ′ = {y ∈ X(K′) | y|_{K} ∈ Φ} is the CM type of K′ induced from Φ, then

F_{Φ′}(g) = V_{K′/K} (F_{Φ}(g))  (g ∈ Γ_{Q}).  (1.4.1.5)

(1.4.2) The Taniyama element is the map f_{Φ} : Γ_{Q} ⟶ K̂^{∗}/K^{∗} defined as

f_{Φ}(g) = ℓ_{K} (F_{Φ}(g)) ,  (1.4.2.1)

where

ℓ_{K} : {g ∈ Γ_{K}^{ab} | g|_{F^{ab}} ∈ ⟨c_{X}⟩ V_{F/Q}(Γ_{Q}^{ab})} ⟶^{∼} {a ∈ K̂^{∗} | ^{1+c}a ∈ Ẑ^{∗}K^{∗}}/K^{∗} ⊂ K̂^{∗}/K^{∗}

is the morphism from 1.3.4(i). It follows that

1+cF_{Φ}(g) = V_{K/F} (F_{Φ}(g)|_{F^{ab}}) = V_{K/F} ∘ V_{F/Q}(g) = V_{K/Q}(g) =
= r_{K} ∘ i_{K/Q} ∘ r_{Q}^{−1} (g|_{Q^{ab}}) = r_{K}(χ(g)).

As in the proof of 1.3.4(i), this implies that

1+cf_{Φ}(g) = χ(g)K^{∗},  r_{K} (f_{Φ}(g)) = F_{Φ}(g).  (1.4.2.2)

In Tate’s original definition, the properties (1.4.2.2) were used to characterise f_{Φ}(g). The identities (1.4.1.3-5) imply that

f_{Φ}(gg′) = f_{g′Φ}(g)f_{Φ}(g′)  (g, g′ ∈ Γ_{Q}),  (1.4.2.3)

uf_{Φ}(g) = f_{Φu^{−1}}(g)  (g ∈ Γ_{Q}, u : K ⟶^{∼} K′)  (1.4.2.4)

and

f_{Φ′}(g) = i_{K′/K} (f_{Φ}(g))  (K ⊂ K′, Φ′ induced from Φ).  (1.4.2.5)

(1.4.3) Tate [Ta] conjectured that the idèle class f_{Φ}(g) determines the action of g ∈ Γ_{Q} on abelian varieties with complex multiplication and on their torsion points (this was, essentially, the zero-dimensional case of an earlier conjecture of Langlands [L] about conjugation of Shimura varieties). Building upon earlier results of Shimura and Taniyama, Tate proved his conjecture up to an element of F̂^{∗} of square 1. The full conjecture was subsequently proved by Deligne [La, ch. 7, §4]. More precisely, if A is a CM abelian variety of type (K, Φ, a, t) in the sense of [La, ch. 7, §3] (see 2.2.5 below), then ^{g}A is of type (K, gΦ, af, tχ(g)/^{1+c}f), where f ∈ K̂^{∗} is any lift of f_{Φ}(g). Furthermore, for each complex uniformisation

θ : C^{Φ}/a ⟶^{∼} A(C),

there is a unique uniformisation

θ′ : C^{gΦ}/af ⟶^{∼} ^{g}A(C)

such that the action of g on A(\overline{Q})_{tors} = A(C)_{tors} is given by

g : A(\overline{Q})_{tors} ⟶^{θ^{−1}} K/a ⟶^{[×f]} K/af ⟶^{θ′} ^{g}A(\overline{Q})_{tors}.

This implies that, for each full level structure η : (F/O_{F})^{2} ⟶^{∼} A(\overline{Q})_{tors}, the level structure ^{g}η is equal to

gη : (F/O_{F})^{2} ⟶^{η} A(\overline{Q})_{tors} ⟶^{θ^{−1}} K/a ⟶^{[×f]} K/af ⟶^{θ′} ^{g}A(\overline{Q})_{tors}.  (1.4.3.1)

1.5 The Serre torus

Let K be as in 1.3.

(1.5.1) The torus _{K}T = R_{K/Q} (G_{m}) represents the functor A ↦ _{K}T (A) = (K ⊗_{Q} A)^{∗} on Q-algebras A. The Γ_{Q}-equivariant bijections

KT (\overline{Q}) = (K ⊗_{Q} \overline{Q})^{∗} ⟶^{∼} (\overline{Q}^{∗})^{X(K)} = Hom_{Sets}(X(K), \overline{Q}^{∗}) ⟶^{∼} Hom_{Z}(Z[X(K)], \overline{Q}^{∗})
a ⊗ b  ↦  (y ↦ y(a)b)  (y ∈ X(K))

imply that the character group of _{K}T is equal to

X^{∗}(_{K}T) = Z[X(K)] = { ∑_{y∈X(K)} n_{y}[y] | n_{y} ∈ Z},

with g ∈ Γ_{Q} acting on X^{∗}(_{K}T) by

λ = ∑ n_{y}[y] ↦ ^{g}λ = ∑ n_{y}[gy] = ∑ n_{g^{−1}y}[y].  (1.5.1.1)

(1.5.2) The Serre torus of K is the quotient _{K}𝒮 of _{K}T (defined over Q) whose character group is equal to

X^{∗}(_{K}𝒮) = {λ ∈ X^{∗}(_{K}T) | ^{1+c}λ ∈ Z · N_{K/Q}} (N_{K/Q} = ∑_{y∈X(K)}[y]).

Each CM type Φ of K defines a character λ_{Φ} ∈ X^{∗}(_{K}𝒮): λ_{Φ}(y) = 1 (resp., = 0) if y ∈ Φ (resp., if y ∈ cΦ). Moreover, the abelian group X^{∗}(_{K}𝒮) is generated by the characters λ_{Φ} ([Sch, 1.3.2]), and

∀g ∈ Γ_{Q} ^{g}λ_{Φ} = λ_{gΦ}.

(1.5.3) Tate’s half-transfer satisfies the following identity: if n is a function

n : {CM types of K} ⟶ Z, Φ ↦ n_{Φ},

such that ∑_{Φ} n_{Φ}λ_{Φ} = 0, then

∀g ∈ Γ_{Q}  ∏_{Φ} F_{Φ}(g)^{n_{Φ}} = 1 ∈ Γ_{K}^{ab}.  (1.5.3.1)

Applying ℓ_{K}, we deduce from (1.5.3.1) that

∀g ∈ Γ_{Q}  ∏_{Φ} f_{Φ}(g)^{n_{Φ}} = 1 ∈ K̂^{∗}/K^{∗}.  (1.5.3.2)

(1.5.4) In the special case when K is a Galois extension of Q, the action (1.5.1.1) of Γ_{Q} factors through Gal(K/Q), which implies that the tori _{K}T and _{K}𝒮 are split over K. In addition, the action of Gal(K/Q) on K induces an action of Gal(K/Q) on the Q-group scheme _{K}T, which will be denoted by t ↦ g ∗ t (g ∈ Gal(K/Q)). The corresponding action on the character group

(h ∗ λ)(t) = λ(h^{−1} ∗ t)  (λ ∈ X^{∗}(_{K}T))  (1.5.4.1)

is given by

λ = ∑ n_{y}[y] ↦ h ∗ λ = ∑ n_{y}[yh^{−1}] = ∑ n_{yh}[y].

The two actions are related by

ι(^{h}λ) = h ∗ ι(λ)  (h ∈ Gal(K/Q), λ ∈ X^{∗}(_{K}T)),  (1.5.4.2)

where

ι : X^{∗}(_{K}T) ⟶ X^{∗}(_{K}T),  ∑ n_{y}[y] ↦ ∑ n_{y}[y^{−1}] = ∑ n_{y^{−1}}[y]  (1.5.4.3)

is the involution induced by the inverse map g ↦ g^{−1} on Gal(K/Q) = X(K). As ι(λ_{Φ}) = λ_{Φ^{−1}}, the involution ι and the action (1.5.4.1) preserve X^{∗}(_{K}𝒮), and we have

h ∗ λ_{Φ} = λ_{Φh^{−1}}.  (1.5.4.4)

We denote by

ι : _{K}𝒮_{K} = _{K}𝒮 ⊗_{Q} K ⟶ _{K}𝒮_{K}

the morphism corresponding to ι.

1.6 Universal Taniyama elements [Mi], [Sch]

In this section, we assume that K is a CM number field which is a Galois extension of Q. (1.6.1) The two actions of Gal(K/Q) on X^{∗}(_{K}𝒮) correspond to two actions of Gal(K/Q) on _{K}𝒮 (K̂): the Galois action t ↦ ^{g}t and the algebraic action t ↦ h ∗ t, which commute with each other and satisfy

(^{g}λ)(^{g}t) = ^{g}(λ(t)),  (h ∗ λ)(h ∗ t) = λ(t)  (λ ∈ X^{∗}(_{K}𝒮), t ∈ _{K}𝒮 (K̂)),

respectively.

(1.6.2) Proposition. (i) There exists a unique map f′ : Γ_{Q} ⟶ _{K}𝒮 (K̂)/_{K}𝒮 (K) such that λ_{Φ} ∘ f′ = f_{Φ}, for all CM types Φ of K. The map f′ factors through Gal(K^{ab}/Q). (ii) For each λ ∈ X^{∗}(_{K}𝒮), put f′_{λ} = λ ∘ f′ : Γ_{Q} ⟶ K̂^{∗}/K^{∗}; then f′_{λ+μ}(g) = f′_{λ}(g)f′_{μ}(g). (iii) ∀λ ∈ X^{∗}(_{K}𝒮) ∀g, g′ ∈ Γ_{Q} f′_{λ}(gg′) = f′_{^{g′}λ}(g)f′_{λ}(g′). (iv) ∀h ∈ Gal(K/Q) ^{h}(f′_{λ}(g)) = f′_{h∗λ}(g).

Proof. (i) As the torus _{K}𝒮 is split over K and X^{∗}(_{K}𝒮) is a free abelian group generated by the CM characters λ_{Φ}, we have

K𝒮 (K̂)/_{K}𝒮 (K) = Hom_{Z}(X^{∗}(_{K}𝒮), K̂^{∗})/Hom_{Z}(X^{∗}(_{K}𝒮), K^{∗}) = Hom_{Z}(X^{∗}(_{K}𝒮), K̂^{∗}/K^{∗}) =
= {α : {CM types of K} ⟶ K̂^{∗}/K^{∗} | ∏ α(Φ)^{n_{Φ}} = 1 whenever ∑ n_{Φ}λ_{Φ} = 0}.

The existence and uniqueness of f′ then follow from (1.5.3.2). As K is a Galois extension of Q, the maps F_{Φ} (hence f_{Φ}, too) factor through Gal(K^{ab}/Q). (ii) This is a consequence of (the proof of) (i). (iii), (iv) If λ = λ_{Φ}, the statement of (iii) (resp., of (iv)) is just (1.4.2.3) (resp., (1.4.2.4)). The general case then follows from (ii).

(1.6.3) Proposition. (i) Define the map f : Γ_{Q} ⟶ _{K}𝒮 (K̂)/_{K}𝒮 (K) by the formula f(g) = (ι (f′(g)))^{−1}. The map f factors through Gal(K^{ab}/Q) and has the following properties. (ii) The maps f_{λ} = λ ∘ f : Γ_{Q} ⟶ K̂^{∗}/K^{∗} (λ ∈ X^{∗}(_{K}𝒮)) satisfy

f_{λ+μ}(g) = f_{λ}(g)f_{μ}(g),  f_{λ}(g) = f′_{ι(λ)}(g)^{−1},  f_{λ}(gg′) = f_{g′∗λ}(g)f_{λ}(g′).
(iii) ∀h ∈ Gal(K/Q) ∀g ∈ Γ_{Q}  ^{h}(f_{λ}(g)) = f_{^{h}λ}(g),  ^{h}(f(g)) = f(g).

(iv) ∀g, g′ ∈ Γ_{Q} f(gg′) = (g^{′−1} ∗ f(g)) f(g′).

Proof. The statements of (i), (ii) and the first part of (iii) are immediate consequences of 1.6.2, thanks to (1.5.4.2). The second part of (iii) follows from

(^{h}λ) (^{h}(f(g))) = ^{h}(λ (f(g))) =^{(iii)} (^{h}λ) (f(g)) (λ ∈ X^{∗}(_{K}𝒮)),

while (iv) is a consequence of the last formula from (ii) and

λ (g^{′−1} ∗ f(g)) = (g′ ∗ λ) (f(g)) .

(1.6.4) For each CM type Φ of K, the map f_{λ_{Φ}} is given by

f_{λ_{Φ}}(g) = f_{Φ^{−1}}(g)^{−1},

which implies that

r_{K} ∘ f_{λ_{Φ}}(g) = F_{Φ^{−1}}(g)^{−1}.

In the notation of ([Sch], 4.2), we have f_{λ}(g) = f_{K}(g, λ). The map f is the “universal Taniyama element” of ([Mi], I.5.7).

(1.6.5) Proposition. If K′ is a CM number field, which is a Galois extension of Q and contains K, then the universal Taniyama elements f_{K} : Γ_{Q} ⟶ _{K}𝒮 (K̂)/_{K}𝒮 (K) and f_{K′} : Γ_{Q} ⟶ _{K′}𝒮 (K̂′)/_{K′}𝒮 (K′) over K and K′, respectively, satisfy f_{K} = N_{K′/K} ∘ f_{K′}.

Proof. As the map i_{K′/K} : K̂^{∗}/K^{∗} ⟶ K̂^{′∗}/K^{′∗} is injective, it is enough to check that, for any CM type Φ of K and g ∈ Γ_{Q},

i_{K′/K} ∘ λ_{Φ} ∘ f_{K}(g) =^{?} i_{K′/K} ∘ λ_{Φ} ∘ N_{K′/K} ∘ f_{K′}(g) ∈ K̂^{′∗}/K^{′∗},

which follows from (1.4.2.5), since

i_{K′/K} ∘ λ_{Φ} ∘ f_{K}(g) = i_{K′/K} (f_{Φ^{−1}}(g)^{−1}) =^{(1.4.2.5)} f_{Φ^{′−1}}(g)^{−1} = λ_{Φ′} ∘ f_{K′}(g) = i_{K′/K} ∘ λ_{Φ} ∘ N_{K′/K} ∘ f_{K′}(g),

where Φ′ is the CM type of K′ induced from Φ.

1.7 The Taniyama group ([Mi], [Mi-Sh], [Sch])

Let K be as in §1.6. (1.7.1) The Taniyama group of level K sits in an exact sequence of affine group schemes over Q,

1 ⟶ _{K}𝒮 ⟶^{i} _{K}𝒯 ⟶^{π} Gal(K^{ab}/Q) ⟶ 1,

such that the action of (the constant group scheme) Gal(K^{ab}/Q) on _{K}𝒮 defined by this exact sequence is given by the algebraic action (g, t) ↦ g ∗ t. In addition, there exists a continuous group homomorphism

sp : Gal(K^{ab}/Q) ⟶ _{K}𝒯 (Q̂)

satisfying π ∘ sp = id. (1.7.2) Choose a section

α : Gal(K^{ab}/Q) ⟶ _{K}𝒯 (K)

of the map_{K}𝒯 (K) ⟶ Gal(K^{ab}/Q) (which is surjective, since the torus_{K}𝒮 is split over K and H^{1}(K, G_{m}) = 0); the map

b : Gal(K^{ab}/Q) ⟶ _{K}𝒮 (K̂), b(g) = sp(g)α(g)^{−1},

has the following properties:

(1.7.2.1) The induced map \overline{b} : Gal(K^{ab}/Q) ⟶ _{K}𝒮 (K̂)/_{K}𝒮 (K) does not depend on the choice of α. (1.7.2.2) ∀g, g′ ∈ Gal(K^{ab}/Q) \overline{b}(gg′) = (g^{′−1} ∗ \overline{b}(g)) \overline{b}(g′). (1.7.2.3) ∀h ∈ Gal(K/Q) ∀g ∈ Gal(K^{ab}/Q) ^{h}(\overline{b}(g)) = \overline{b}(g).

(1.7.2.4) The “coboundary” d_{g,g′} = (g^{′−1} ∗ b(g)) b(g′) b(gg′)^{−1} is a locally constant function on Gal(K^{ab}/Q)^{2}. (1.7.3) Conversely, any map b satisfying (1.7.2.1-4) gives rise to an object from 1.7.1 ([Mi-Sh], Prop. 2.7): firstly, the reverse 2-cocycle d_{g,g′} with values in _{K}𝒮 (K) defines an exact sequence of affine group schemes over K,

1 ⟶ _{K}𝒮_{K} ⟶^{i} G′ ⟶^{π} Gal(K^{ab}/Q) ⟶ 1,  (1.7.3.1)

equipped with a section α : Gal(K^{ab}/Q) ⟶ G′(K) such that

∀g, g′ ∈ Gal(K^{ab}/Q) α(gg′) = α(g)α(g′)d_{g,g′}.

Secondly, the map

sp : Gal(K^{ab}/Q) ⟶ G′(K̂), sp(g) = b(g)α(g),

is a group homomorphism satisfying π ∘ sp = id. Thirdly, each element h ∈ Γ_{K} acts on G′(\overline{Q}) by

h(s α(g)) = ^{h}s α(g)  (s ∈ _{K}𝒮 (\overline{Q})).  (1.7.3.2)

In order to descend the sequence (1.7.3.1) to an exact sequence of group schemes over Q

1 ⟶ _{K}𝒮 ⟶^{i} G ⟶^{π} Gal(K^{ab}/Q) ⟶ 1,

it is enough to extend the action of Γ_{K} from (1.7.3.2) to an action of Γ_{Q} compatible with i and π. This is done by putting

h(s α(g)) = c_{h}(g)^{h}s α(g),  c_{h}(g) = b(g)^{h}(b(g))^{−1} ∈ _{K}𝒮 (K)  (h ∈ Γ_{Q}, g ∈ Gal(K^{ab}/Q)).

As ^{h}(sp(g)) = sp(g) for all h ∈ Γ_{Q} and g ∈ Gal(K^{ab}/Q), the map sp has values in G(Q̂). Up to isomorphism, the quadruple (G, i, π, sp) obtained by this method depends only on \overline{b}, not on its lift b. (1.7.4) The Taniyama group _{K}𝒯 of level K is defined by applying the construction from 1.7.3 to the universal Taniyama element f, which satisfies (1.7.2.2-3), by Proposition 1.6.3. The existence of a lift b of f satisfying (1.7.2.4) is established in the following proposition.

(1.7.5) Proposition. There exists a lift b : Gal(K^{ab}/Q) ⟶ _{K}𝒮 (K̂) of f whose “coboundary” d_{g,g′} = (g^{′−1} ∗ b(g)) b(g′) b(gg′)^{−1} is a locally constant function on Gal(K^{ab}/Q)^{2}.

Proof. Let ℓ̃_{K} be as in 1.3.4(iv). As the maps F_{Φ} (which factor through Gal(K^{ab}/Q)) are continuous, there exists an open subgroup U ⊂ Γ_{K}^{ab} such that ℓ̃_{K}, when restricted to ⋃_{Φ} F_{Φ}(U), is a homomorphism. If n_{Φ} ∈ Z satisfy ∑_{Φ} n_{Φ}λ_{Φ} = 0, then the relation (1.5.3.1) implies that

∀u ∈ U  ∏_{Φ} ℓ̃_{K} (F_{Φ}(u))^{n_{Φ}} = 1 ∈ K̂^{∗}.

As in the proof of 1.6.2(i), we conclude that, for each u ∈ U, there exists a unique element b′(u) ∈ _{K}𝒮 (K̂) satisfying λ_{Φ}(b′(u)) = ℓ̃_{K} (F_{Φ}(u)). Fix coset representatives Gal(K^{ab}/Q) = ⋃_{j} g_{i}U (disjoint union) and lifts s̃_{j} ∈ _{K}𝒮 (K̂) of f′(g_{j}) ∈ _{K}𝒮 (K̂)/_{K}𝒮 (K) such that g_{j_{0}} = 1 and s̃_{j_{0}} = 1; define a map b′ : Gal(K^{ab}/Q) ⟶ K𝒮 (K̂) by

b′(g_{j}u) = s̃_{j}b′(u) (u ∈ U).

The map b(g) := (ι(b′(g)))^{−1} then has the required property. (1.7.6) Proposition 1.6.5 implies that the pull-backs of the group schemes _{K}𝒯 via Γ_{Q} ⟶ Gal(K^{ab}/Q) form, for varying K, a projective system compatible with the norm maps N_{K′/K} : _{K′}𝒮 ⟶ _{K}𝒮. In the limit, they give rise to an exact sequence

1 ⟶ 𝒮 ⟶^{i} 𝒯 ⟶^{π} Γ_{Q} ⟶ 1  (1.7.6.1)

equipped with a splitting sp : Γ_{Q} ⟶ 𝒯 (Q̂). The main result of [De] states that the affine group scheme 𝒯 (= the Taniyama group) is the Tannaka dual of the category CM_{Q} of CM motives (for absolute Hodge cycles) defined over Q. The group scheme _{K}𝒯 corresponds to the full Tannakian subcategory of CM_{Q} consisting of objects with reflex field in K.

2. Hidden symmetries in the CM theory

Throughout this chapter, K and F are as in 1.3. Put X = X(F). In §2.1 (resp., §2.2) we extend Tate’s half-transfer F_{Φ} (resp., the Taniyama element f_{Φ}) from Γ_{Q} to Aut_{F −alg}(F ⊗\overline{Q}) (resp., to Aut_{F −alg}(F ⊗\overline{Q})_{1}). In §2.3-2.4 we use our generalisation of the Taniyama element to construct a generalised Taniyama group.

2.1 Generalised half-transfer

(2.1.1) Fix a section s : X ⟶ Γ_{Q} of the restriction map g ↦ g|_{F}. As in 1.1.2-4, the choice of s determines the following objects:

(2.1.1.1) An injection ρ_{s} : Γ_{Q} ↪ S_{X} ⋉ Γ_{F}^{X}. (2.1.1.2) An isomorphism β_{s∗} : Aut_{F −alg}(F ⊗\overline{Q}) ⟶^{∼} Aut_{F −alg}(\overline{Q}^{X}) = S_{X} ⋉Γ_{F}^{X} satisfying β_{s∗}(id_{F} ⊗g) = ρ_{s}(g).

In addition, we obtain

(2.1.1.3) A bijection between (Z/2Z)^{X} and the set of CM types of K: a function α : X ⟶ Z/2Z corresponds

to the CM type {c^{α(x)}s(x)|_{K} = s(x)c^{α(x)}|_{K}}_{x∈X}.

(2.1.1.4) A section w_{s} : X(K) ⟶ Γ_{Q} of the restriction map g ↦ g|_{K} satisfying w_{s}(cy) = cw_{s}(y), namely w_{s}(c^{a}s(x)|_{K}) = c^{a}s(x) (x ∈ X, a ∈ Z/2Z).

For h ∈ Γ_{F}^{X}, we denote by \overline{h} : X ⟶ Z/2Z the image of h in Gal(K/F)^{X} ⟶^{∼} (Z/2Z)^{X}. In other words,

∀x ∈ X  h(x)|_{K} = c^{\overline{h}(x)},  R(h(x)) = c^{\overline{h}(x)}h(x),

where R : Γ_{F} ⟶ Γ_{K} is the retraction map from 1.3.2. We let S_{X} ⋉ Γ_{F}^{X} act on (Z/2Z)^{X} via (1.1.1.1) and the natural projection (σ, h) ↦ (σ, \overline{h}):

(σ, h)α = (α + \overline{h}) ∘ σ^{−1}.  (2.1.1.5)

(2.1.2) Rewriting Tate’s half-transfer in terms of ρ_{s}. Let Φ be a CM type of K. If g ∈ Γ_{Q}, then

ρ_{s}(g) = (σ, h) ∈ S_{X} ⋉ Γ_{F}^{X},  ∀x ∈ X σ(x) = gx,  h(x) = s(gx)^{−1}gs(x) = s(σ(x))^{−1}gs(x) ∈ Γ_{F} .

Let α ∈ (Z/2Z)^{X} correspond to Φ, as in (2.1.1.3). For each x ∈ X, the element

φ_{x} = c^{α(x)}s(x)|_{K} = s(x)c^{α(x)}|_{K} ∈ Φ

satisfies w_{s}(φ_{x}) = c^{α(x)}s(x) and

gφ_{x} = gs(x)c^{α(x)}|_{K} = s(σ(x))h(x)c^{α(x)}|_{K} = c^{α(x)+\overline{h}(x)}s(σ(x))|_{K},

which implies that w_{s}(gφ_{x}) = c^{α(x)+\overline{h}(x)}s(σ(x)) and

\begin{matrix}w_{s}(gφ_{x})^{−1}gw_{s}(φ_{x}) = s(σ(x))^{−1}c^{α(x)+\overline{h}(x)}gc^{α(x)}s(x) & = s(σ(x))^{−1}c^{α(x)+\overline{h}(x)}s(σ(x))h(x)s(x)^{−1}c^{α(x)}s(x) = \\ = [s(σ(x))^{−1}c^{α(x)+\overline{h}(x)}s(σ(x))c^{α(x)+\overline{h}(x)}] & [c^{α(x)+\overline{h}(x)}h(x)c^{α(x)}] [c^{α(x)}s(x)^{−1}c^{α(x)}s(x)] . \end{matrix}

Denote by γ_{x,s} the image of s(x)^{−1}cs(x)c ∈ Γ_{K} in Γ_{K}^{ab}. As each of the three elements in square brackets lies in Γ_{K}, we have

F_{Φ}(g) = ∏_{x∈X} w_{s}(gφ_{x})^{−1}gw_{s}(φ_{x})|_{K^{ab}} = ∏_{x∈|(σ,h)α|} γ_{x,s} ∏_{x∈|α|} γ_{x,s}^{−1} ∏_{x∈X} c^{α(x)}R(h(x))c^{α(x)}|_{K^{ab}},

where we have denoted by |α| = {x ∈ X | α(x) ≠ 0} the support of α. This calculation justifies the following:

(2.1.3) Proposition-Definition. For each α ∈ (Z/2Z)^{X}, the formula

sF̃_{α}(σ, h) = ∏_{x∈X} s(σ(x))^{−1}c^{α(x)+\overline{h}(x)}s(σ(x))h(x)s(x)^{−1}c^{α(x)}s(x)|_{K^{ab}} =

= ∏_{x∈|(σ,h)α|} γ_{x,s} ∏_{x∈|α|} γ_{x,s}^{−1} ∏_{x∈X} c^{α(x)}R(h(x))c^{α(x)}|_{K^{ab}}

defines a map

sF̃_{α} : S_{X} ⋉ Γ_{F}^{X} ⟶ Γ_{K}^{ab}

(depending on s and α) satisfying _{s}F̃_{α} ∘ρ_{s} = F_{Φ}, where Φ is the CM type corresponding to α, as in (2.1.1.3).

(2.1.4) Proposition. The maps _{s}F̃_{α} have the following properties:

(i) ∀g, g′ ∈ S_{X} ⋉ Γ_{F}^{X} _{s}F̃_{α}(gg′) = _{s}F̃_{g′α}(g)_{s}F̃_{α}(g′). (ii) For each (σ, h) ∈ S_{X} ⋉ Γ_{F}^{X},

sF̃_{α}(σ, h)|_{F^{ab}} = ∏_{x∈|(σ,h)α|} c_{x} ∏_{x∈|α|} c_{x} ∏_{x∈X} h(x)|_{F^{ab}}, ^{1+c} (_{s}F̃_{α}(σ, h)) = Ṽ_{K/F} (σ, h) = ∏_{x∈X} ^{1+c}R(h(x))|_{K^{ab}},

where Ṽ_{K/F} (σ, h) = ∏_{x∈X} V_{K/F} (h(x)|_{F^{ab}}). (iii) Each map _{s}F̃_{α} factors through S_{X} ⋉ Gal(K^{ab}/F)^{X}. (iv) If g = (σ, h) ∈ S_{X} ⋉ Γ_{F}^{X} satisfies gα = α, then

sF̃_{α}(g) = ∏_{x∈X} c^{α(x)}R(h(x))c^{α(x)}|_{K^{ab}}.

(v) ∀(σ, h) ∈ S_{X} ⋉ Γ_{K}^{X} _{s}F̃_{0}(σ, h) = ∏_{x∈X} h(x)|_{K^{ab}}. (vi) ∀α ∈ (Z/2Z)^{X} _{s}F̃_{0}(1, c^{α}) = ∏_{x∈|α|} γ_{x,s}.

Proof. (i) If g = (σ, h) and g′ = (σ′, h′), then gg′ = (σσ′, (h ∘ σ′)h′) and α′ := g′α = (α + \overline{h}′) ∘ σ^{′−1}, which implies that _{s}F̃_{α}(gg′)_{s}F̃_{α}(g′)^{−1}_{s}F̃_{g′α}(g)^{−1} is equal to

∏_{x∈X} (c^{α(x)+\overline{h}(σ′(x))+\overline{h}′(x)}h(σ′(x))h′(x)c^{α(x)}) (c^{α(x)+\overline{h}′(x)}h′(x)c^{α(x)})^{−1} (c^{α′(x)+\overline{h}(x)}h(x)c^{α′(x)})^{−1} =

= ∏_{x∈X} (c^{α′(σ′(x))+\overline{h}(σ′(x))}h(σ′(x))c^{α′(σ′(x))}) (c^{α′(x)+\overline{h}(x)}h(x)c^{α′(x)})^{−1} = 1.

(ii) The first formula is a consequence of the fact that

∀x ∈ X  γ_{x,s}|_{F^{ab}} = c_{x}c,  c^{α(x)}R(h(x))c^{α(x)}|_{F^{ab}} = c^{\overline{h}(x)}h(x)|_{F^{ab}};

applying (1.3.2.1), we obtain the second formula. The statements (iii)-(vi) follow directly from the definitions.

(2.1.5) Change of s. Let s, s′ ⟶ Γ_{Q} be two sections of the restriction map g ↦ g|_{F}. We have s′ = st, where t : X ⟶ Γ_{F}. As in 2.1.1, we write, for each x ∈ X, t(x)|_{K} = c^{\overline{t}(x)} (\overline{t}(x) ∈ Z/2Z); then R(t(x)) = c^{\overline{t}(x)}t(x) ∈ Γ_{K}. The recipe (2.1.1.3), applied to s and s′, respectively, associates to each CM type Φ of K two functions α = α_{Φ,s}, α′ = α_{Φ,s′} : X ⟶ Z/2Z such that

Φ = \{c^{α(x)}s(x)|_{K}\} = \{c^{α′(x)}s′(x)|_{K}\} (⟹ α′ = α + \overline{t}).

According to Proposition 1.1.4, the following diagram is commutative:

\xymatrix{ &  & S_{X} ⋉ Γ_{F}^{X} \ar[dd]^{Ad(1,t)^{−1}} \\ Γ_{Q} \ar[urr]^{ρ_{s}} \ar[r] \ar[drr]^{ρ_{s′}} & Aut_{F −alg}(F ⊗ \overline{Q}) \ar[ur]_{β_{s∗}} \ar[dr]^{β_{s′∗}} \\  &  & S_{X} ⋉ Γ_{F}^{X}}  (2.1.5.1)

For (σ, h) ∈ S_{X} ⋉ Γ_{F}^{X}, put

(σ′, h′) := Ad(1, t)^{−1}(σ, h) = (1, t)^{−1}(σ, h)(1, t) = (σ, (t ∘ σ)^{−1}ht) ∈ S_{X} ⋉ Γ_{F}^{X}.  (2.1.5.2)

The map Ṽ_{K/F} from Proposition 2.1.4(ii) satisfies Ṽ_{K/F} (σ, h) = Ṽ_{K/F} (σ′, h′), which means that the map

Ṽ_{K/F} ∘ β_{s∗} : Aut_{F −alg}(F ⊗ \overline{Q}) ⟶ Γ_{K}^{ab}  (2.1.5.3)

does not depend on s; we denote it again by Ṽ_{K/F}. The equalities

(σ, h)α = (α + \overline{h}) ∘ σ^{−1}, (σ′, h′)α′ = (α′ + \overline{h}′) ∘ σ^{−1} = (α + \overline{h}) ∘ σ^{−1} + \overline{t} ∈ (Z/2Z)^{X}

imply that the action of S_{X} ⋉ Γ_{F}^{X} on (Z/2Z)^{X} defined in (2.1.1.5) gives rise to an action of the group Aut_{F −alg}(F ⊗ \overline{Q}) on the set of CM types of K, which is characterised by

∀g ∈ Aut_{F −alg}(F ⊗ \overline{Q})  α_{gΦ,s} = β_{s∗}(g)α_{Φ,s},  (2.1.5.4)

but which does not depend on s.

(2.1.6) Proposition. In the notation of (2.1.5.2), we have _{s}F̃_{α}(σ, h) = _{s′} F̃_{α′}(σ′, h′) ∈ Γ_{K}^{ab}.

Proof. The relations s′ = st, (σ′, h′) = (σ, (t ∘ σ)^{−1}ht), \overline{h}′ = \overline{h} + \overline{t} + \overline{t} ∘ σ, α′ = α + \overline{t}, (σ, h)α = (α + \overline{h}) ∘ σ^{−1} and (σ′, h′)α′ = (σ, h)α + \overline{t} imply that

s′ F̃_{α′}(σ′, h′) = ∏_{x∈X} t(σ(x))^{−1}s(σ(x))^{−1}c^{(α+\overline{h})(x)+\overline{t}(σ(x))}s(σ(x))h(x)s(x)^{−1}c^{(α+\overline{t})(x)}s(x)t(x)|_{K^{ab}} =

= ∏_{x∈X} A′((σ, h)α, x)^{−1}B(α, x)A′(α, x),

where

A′(α, x) = c^{α(x)}s(x)^{−1}c^{(α+\overline{t})(x)}s(x)t(x)|_{K^{ab}}, B(α, x) = c^{α(x)}R(h(x))c^{α(x)}|_{K^{ab}}.

As

sF̃_{α}(σ, h) = ∏_{x∈X} A((σ, h)α, x)^{−1}B(α, x)A(α, x),

where

A(α, x) = c^{α(x)}s(x)^{−1}c^{α(x)}s(x)|_{K^{ab}},

the equality _{s}F̃_{α}(σ, h) = _{s′} F̃_{α′}(σ′, h′) follows from the fact that

∀x ∈ X A(α, x)^{−1}A′(α, x) = s(x)^{−1}c^{\overline{t}(x)}s(x)t(x)|_{K^{ab}}

does not depend on α.

(2.1.7) Proposition-Definition. In the notation of 2.1.5, the map

F̃_{Φ} = _{s}F̃_{α}(σ, h) ∘ β_{s∗} : Aut_{F −alg}(F ⊗ \overline{Q}) ⟶ Γ_{K}^{ab}

depends on Φ, but not on s; it has the following properties:

(i) ∀g ∈ Γ_{Q}  F̃_{Φ}(id_{F} ⊗ g) = F_{Φ}(g).
(ii) ∀g, g′ ∈ Aut_{F −alg}(F ⊗ \overline{Q})  F̃_{Φ}(gg′) = F̃_{g′Φ}(g)F̃_{Φ}(g′).

(iii) ∀g ∈ Aut_{F −alg}(F ⊗ \overline{Q}) ^{1+c}F̃_{Φ}(g) = Ṽ_{K/F} (g) (in the notation of (2.1.5.3)).

Proof. The independence of F̃_{Φ} of the choice of s follows from Proposition 2.1.6 and the commutative diagram (2.1.5.1). The remaining statements are consequences of Proposition 2.1.4.

(2.1.8) Galois functoriality of F̃_{Φ}. Given an element ũ ∈ Γ_{Q}, define u := ũ|_{K}, u_{F} := u|_{F}, K′ := u(K), F′ = u_{F} (F) and X′ = X(F′). As in Proposition 1.1.6 (for k = Q and k′ = \overline{Q}), a fixed section s : X ⟶ Γ_{Q} of the restriction map g ↦ g|_{F} defines a section s′ : X′ ⟶ Γ_{Q} of the restriction map g ↦ g|_{F′}, given by

s′(x′) = s′(xu_{F}^{−1}) = s(x) ∘ u^{−1} (x ∈ X).

(2.1.9) Proposition. For each α : X ⟶ Z/2Z, the diagram

S_{X} ⋉ Γ_{F}^{X}  ⟶^{_{s}_{F_{α}}} Γ_{K}^{ab}
↓_{u_{∗}}  ↓^{u}
S_{X′} ⋉ Γ_{F′}^{X′}  ⟶^{_{s′} _{F_{α′}}} Γ_{K′}^{ab}

is commutative, where ũ_{∗} is the map defined in Proposition 1.1.6, α′ : X′ ⟶ Z/2Z is given by α′(x′) = α(x) (x = x′u_{F}) and the right vertical map (which depends only on u) is given by g ↦ ũgu^{−1}.

Proof. For (σ, h) ∈ S_{X} ⋉ Γ_{F}^{X}, we have ũ_{∗}(σ, h) = (σ′, h′), where σ′(x′) = σ(x)u_{F}^{−1}, h′(x′) = ũh(x)u^{−1} (x′ = xu_{F}^{−1}). The relations s′(σ′(x′)) = s(σ(x))u^{−1}, s′(x′) = s(x)u^{−1}, \overline{h}′(x′) = \overline{h}(x) and α′(x′) = α(x) imply

s′ F̃_{α′}(σ′, h′) = ∏_{x′∈X′} s′(σ′(x′))^{−1}c^{α′(x′)+\overline{h}′(x′)}s′(σ′(x′))h′(x′)s′(x′)^{−1}c^{α′(x′)}s′(x′)|_{K^{′ab}} =

= ũ ∏_{x∈X} s(σ(x))^{−1}c^{α(x)+\overline{h}(x)}s(σ(x))h(x)s(x)^{−1}c^{α(x)}s(x)|_{K^{ab}} u^{−1} = u (_{s}F̃_{α}(σ, h)) .

(2.1.10) Corollary. For each CM type Φ of K, the diagram

Aut_{F −alg}(F ⊗ \overline{Q}) \underrightarrow{_{F_{Φ}}} Γ_{K}^{ab}
↓^{[u_{F}]}  ↓^{u}
Aut_{F′−alg}(F′ ⊗ \overline{Q}) \underrightarrow{_{F_{Φu^{−1}}}} Γ_{K′}^{ab}

is commutative, where [u_{F}] is the map defined in Proposition 1.1.7(i).

Proof. This follows from Proposition 2.1.9 combined with Proposition 1.1.7(ii) (for k = Q and k′ = \overline{Q}), if we take into account the fact that

{c^{α′(x′)}s′(x′)|_{K′}}_{x′∈X′} = {c^{α(x)}s(x)|_{K} u^{−1}}_{x∈X}.

2.2 Generalised Taniyama elements

(2.2.1) Let (S_{X} ⋉ Γ_{F}^{X})_{1} be the group defined as the fibre product

(S_{X} ⋉ Γ_{F}^{X})_{1} ⟶ S_{X} ⋉ Γ_{F}^{X}
↓  ↓^{(1,prod)}
Γ_{Q}^{ab}/⟨c⟩  ↪^{\overline{V}_{F/Q}} Γ_{F}^{ab}/⟨c_{X}⟩.

As the morphism \overline{V}_{F/Q} is injective (1.3.2.4), we can (and will) identify (S_{X} ⋉Γ_{F}^{X})_{1} with its image in S_{X} ⋉Γ_{F}^{X}. The group (S_{X} ⋉ Γ_{F}^{X})_{0}, defined in (1.1.2.4), sits in an exact sequence

1 ⟶ (S_{X} ⋉ Γ_{F}^{X})_{0} ⟶ (S_{X} ⋉ Γ_{F}^{X})_{1} ⟶ ⟨c_{X}⟩/V_{F/Q}(⟨c⟩) ⟶ 1.

For i = 0, 1, the subgroups β_{s∗}^{−1} ((S_{X} ⋉ Γ_{F}^{X})_{i}) of Aut_{F −alg}(F ⊗ \overline{Q}) are independent of the choice of a section s : X ⟶ Γ_{Q}; we denote them by

Aut_{F −alg}(F ⊗ \overline{Q})_{0} ⊂ Aut_{F −alg}(F ⊗ \overline{Q})_{1} ⊂ Aut_{F −alg}(F ⊗ \overline{Q}).

(2.2.2) Definition. For each CM type Φ of K, define a map

f̃_{Φ} : Aut_{F −alg}(F ⊗ \overline{Q})_{1} ⟶ K̂^{∗}/K^{∗}

by

f̃_{Φ}(g) = ℓ_{K} (F̃_{Φ}(g)) ,

where ℓ_{K} is the morphism from Proposition 1.3.4(i). [This definition makes sense, by Proposition 2.1.4(ii).]

(2.2.3) Proposition. The maps f̃_{Φ} have the following properties:

(i) r_{K} ∘ f̃_{Φ} = F̃_{Φ}. (ii) ∀g ∈ Γ_{Q} f̃_{Φ}(id_{F} ⊗ g) = f_{Φ}(g). (iii) Each map f̃_{Φ} factors through

Aut_{F −alg}(F ⊗ K^{ab})_{1} := Im (Aut_{F −alg}(F ⊗ \overline{Q})_{1} ⟶ Aut_{F −alg}(F ⊗ K^{ab})) .
(iv) ∀g, g′ ∈ Aut_{F −alg}(F ⊗ \overline{Q})_{1}  f̃_{Φ}(gg′) = f̃_{g′Φ}(g)f̃_{Φ}(g′).

(v) If u : K ⟶^{∼} K′ is an isomorphism of CM number fields, then

f̃_{Φu^{−1}} ∘ [u|_{F}] = u ∘ f̃_{Φ}.

(vi) For g ∈ Aut_{F −alg}(F ⊗ \overline{Q})_{1}, denote by u(g) ∈ Γ_{Q}^{ab}/⟨c⟩ the unique element satisfying V_{K/Q}(u(g)) = 1+cF̃_{Φ}(g); then ^{1+c}f̃_{Φ}(g) = χ(u(g))K^{∗}. (vii) For g ∈ Aut_{F −alg}(F ⊗\overline{Q})_{0}, denote by u(g) ∈ Γ_{Q}^{ab} the unique element satisfying V_{F/Q}(u(g)) = F̃_{Φ}(g)|_{F^{ab}}; then N_{K/F} (f̃_{Φ}(g)) = χ(u(g))αF_{+}^{∗} ∈ F̂^{∗}/F_{+}^{∗}, where α ∈ F^{∗} satisfies

∀x ∈ X sgn(x(a)) = \{ \begin{matrix}1, & if Φ and gΦ agree at x \\ −1, & if Φ and gΦ do not agree at x \end{matrix}

(we say that two CM types Φ and Φ′ of K agree at x ∈ X if the unique element of Φ whose restriction to F is x is equal to the unique element of Φ′ whose restriction to F is x).

Proof. statement (i) holds by definition, while (ii)-(v) follow from the corresponding assertions for F̃_{Φ}, proved in Proposition 2.1.7 and Corollary 2.1.10. Property (vi) (resp., (vii)) is a consequence of Proposition 1.3.4(i) (resp., 1.3.4(ii)) combined with the second (resp., the first) formula in Proposition 2.1.4(ii).

(2.2.4) Proposition. Let K′ be a CM number field containing K; put X′ = X(F′), where F′ is the maximal totally real subfield of K′. If Φ is a CM type of K and Φ′ is the induced CM type of K′, then:

(i) ∀g ∈ Aut_{F −alg}(F ⊗ \overline{Q})  F̃_{Φ′}(id_{F′} ⊗_{F} g) = V_{K′/K} (F̃_{Φ}(g)) ∈ Γ_{K′}^{ab}.

(ii) ∀i = 0, 1 ∀g ∈ Aut_{F −alg}(F ⊗ \overline{Q})_{i} id_{F′} ⊗_{F} g ∈ Aut_{F′−alg}(F′ ⊗ \overline{Q})_{i}.

(iii) ∀g ∈ Aut_{F −alg}(F ⊗ \overline{Q})_{1}  f̃_{Φ′}(id_{F′} ⊗_{F} g) = i_{K′/K} (f̃_{Φ}(g)) ∈ K̂^{′∗}/K^{′∗}.

Proof. (i) Fix a section s : X ⟶ Γ_{Q}; let α : X ⟶ Z/2Z correspond to Φ, as in (2.1.1.3). The sets Γ_{K}/Γ_{K′} and Γ_{F} /Γ_{F′} are canonically identified. Fix a section u : Γ_{K}/Γ_{K′} = Hom_{K−alg}(K′, \overline{Q}) ⟶ Γ_{K} of the restriction map g ↦ g|_{K} and define a section s′ : X′ ⟶ Γ_{Q} by

s′(s(x)y|_{F′}) = s(x)u(y) (x ∈ X, y ∈ Γ_{F} /Γ_{F′});

then Φ′ corresponds to α′ = α ∘ p : X′ ⟶ Z/2Z, where we have denoted by p : X′ ⟶ X the restriction map g ↦ g|_{F}. Proposition 1.1.8 implies that the elements

(σ, h) = β_{s∗}(g) ∈ S_{X} ⋉ Γ_{F}^{X}, (σ′, h′) = β_{s′∗}(id_{F′} ⊗_{F} g) ∈ S_{X′} ⋉ Γ_{F′}^{X′}

are related by

σ′(s(x)y|_{F′}) = s(σ(x))h(x)y|_{F′}, s′(σ′(s(x)y|_{F′})) = s(σ(x))u(h(x)y),
h′(s(x)y|_{F′}) = u(h(x)y)^{−1}h(x)u(y) (x ∈ X, y ∈ Γ_{F} /Γ_{F′}),

hence \overline{h}′ = \overline{h} ∘ p. For x ∈ X and x′ ∈ X′, put

k(x) = s(σ(x))^{−1}c^{α(x)+\overline{h}(x)}s(σ(x))h(x)s(x)^{−1}c^{α(x)}s(x) ∈ Γ_{K}
k′(x′) = s′(σ′(x′))^{−1}c^{α′(x′)+\overline{h}′(x′)}s′(σ′(x′))h′(x′)s′(x′)^{−1}c^{α′(x′)}s′(x′) ∈ Γ_{K′}.

By definition,

F̃_{Φ}(g) = _{s}F̃_{α}(σ, h) = ∏_{x∈X} k(x)|_{K^{ab}} ∈ Γ_{K}^{ab},  F̃_{Φ′}(id_{F′} ⊗_{F} g) = _{s′} F̃_{α′}(σ′, h′) = ∏_{x′∈X′} k′(x′)|_{K^{′ab}} ∈ Γ_{K′}^{ab}.

For each x ∈ X and y ∈ Γ_{F} /Γ_{F′},

k′(s(x)y|_{F′}) = u(h(x)y)^{−1}k(x)u(y) ∈ Γ_{K′},

which implies that k(x)y = k(x)u(y)|_{K′} = u(h(x)y)|_{K′} = h(x)y, hence u(h(x)y) = u(k(x)y) and

∏_{x′∈p^{−1}(x)} k′(x′)|_{K^{′ab}} = ∏_{y∈Γ_{K}/Γ_{K′}} u(k(x)y)^{−1}k(x)u(y)|_{K^{′ab}} = V_{K′/K} (k(x)|_{K^{ab}}) .

Taking the product over all x ∈ X yields (i). The statement (ii) follows from the fact that, in the notation used in the proof of (i),

∏_{x′∈p^{−1}(x)} h′(x′)|_{F^{′ab}} = ∏_{y∈Γ_{F} /Γ_{F′}} u(h(x)y)^{−1}h(x)u(y)|_{F^{′ab}} = V_{F′/F} (h(x)|_{F^{ab}}) .

Finally, (iii) follows by applying ℓ_{K′} to the statement of (i) (which makes sense, by (ii) for i = 1). (2.2.5) Action of Aut_{F −alg}(F ⊗ \overline{Q})_{0} on CM points of Hilbert modular varieties. In this section we prove Theorem 0.9. Given a polarised HBAV (Hilbert-Blumenthal abelian variety) A relative to F with CM, then A is defined over \overline{Q}, and there exist • a CM field K of degree 2 over F;

• a CM type Φ of K (which defines an embedding K ↪ C^{Φ}, α ↦ (φ ↦ φ(α))_{φ∈Φ});

• a fractional ideal a of K;

• an element t ∈ K^{∗} such that t ∉ F^{∗}, t^{2} ∈ F^{∗} and ∀φ ∈ Φ Im(φ(t)) < 0;

• an O_{K}-linear isomorphism θ : C^{Φ}/a ⟶^{∼} A(C) such that the Riemann form of the pull-back of the polarisation of A by θ is induced by the form E_{t}(x, y) = Tr_{K/Q}(tx^{c}y) on K.

One says that A is a CM abelian variety of type (K, Φ, a, t) (via θ). The type is determined up to transformations (K, Φ, a, t) ↦ (K, Φ, aα, t/^{1+c}α) (α ∈ K^{∗}), and it determines A with its polarisation up to isomorphism. Given g ∈ Aut_{F −alg}(F ⊗ \overline{Q})_{0}, let u(g) ∈ Γ_{Q}^{ab} be as in Proposition 2.2.3(vii). Fix a lift f̃ ∈ K̂^{∗} of f̃_{Φ}(g) ∈ K̂^{∗}/K^{∗} and define A′ = C^{gΦ}/af̃, with polarisation given by E_{t′}, where

t′ = t χ(u(g))/^{1+c}f̃ ∈ K^{∗}

(t′ satisfies t′ ∉ F^{∗}, t^{′2} ∈ F^{∗} and ∀φ′ ∈ gΦ Im(φ′(t′)) < 0, the last condition by Proposition 2.2.3(vii)). Given, in addition, a full level structure η : (F/O_{F})^{2} ⟶^{∼} A(\overline{Q})_{tors} of A under which the Weil pairing associated to the given polarisation is a Q̂^{∗}-multiple of the standard form Tr_{_{F}/_{Q}} ∘ det_{F} on F̂^{2}, let η′ be the following level structure of A′:

η′ : (F/O_{F})^{2} ⟶^{η} A(C)_{tors} ⟶^{θ^{−1}} K/a ⟶^{[×_{f}]} K/af̃ = A′(C)_{tors}.

The isomorphism class of the triple (A′, E_{t′}, η′) depends only on g and on the isomorphism class [(A, E_{t}, η)] of (A, E_{t}, η). Proposition 2.2.3 implies that the assignment

g[(A, E_{t}, η)] = [(A′, E_{t′}, η′)]

defines an action of Aut_{F −alg}(F ⊗ \overline{Q})_{0} on the isomorphism classes of polarised HBAV (relative to F) with CM, equipped with a full level structure. Moreover, this action commutes with the action of G(F̂) on η (by γ : η ↦ η ∘ γ), where G is the fibre product

G ⟶ R_{F/Q}(GL(2)_{F})
↓  ↓^{det}
G_{m,Q} ⟶ R_{F/Q}(G_{m,F}).

In view of the results of Tate and Deligne that were recalled in 1.4.3, it follows from Proposition 2.2.3 that the action of Aut_{F −alg}(F ⊗ \overline{Q})_{0} we have just defined extends the usual Galois action of Γ_{Q}.

Recall that f̃_{Φ}(g) is defined even for g ∈ Aut_{F −alg}(F ⊗ \overline{Q})_{1}. However, the positivity of polarisations implies that the above recipe makes sense only if the conlusion of Proposition 2.2.3(vii) is satisfied, namely if g ∈ Aut_{F −alg}(F ⊗ \overline{Q})_{0}.

(2.2.6) Proposition-Definition. Fix s : X ⟶ Γ_{Q} as in 2.1.1; then X(K) = {s(x)c^{a}|_{K} | x ∈ X, a ∈ Z/2Z}.

(i) Let g ∈ Aut_{F −alg}(F ⊗ \overline{Q}); put (σ, h) = β_{s∗}(g) ∈ S_{X} ⋉ Γ_{F}^{X}. The formula

g (s(x)c^{a}|_{K}) := s(σ(x))c^{\overline{h}(x)+a}|_{K} = s(σ(x))h(x)c^{a}|_{K}

defines an action of Aut_{F −alg}(F ⊗ \overline{Q}) on X(K). The action of g on X(K) depends only on the image of (σ, h) in S_{X} ⋉ Gal(K/F)^{X}. (ii) This action does not depend on the choice of s. (iii) For each CM type Φ ⊂ X(K) of K, the set ^{g}Φ = {^{g}y | y ∈ Φ} coincides with gΦ, defined in (2.1.5.4). (iv) If g = id_{F} ⊗ u, u ∈ Γ_{Q}, then ^{g}y = u ∘ y = uy, for each y ∈ X(K).

Proof. Easy calculation.

(2.2.7) Corollary-Definition. (i) The induced action of Aut_{F −alg}(F ⊗ \overline{Q}) on X^{∗}(_{K}T) = Z[X(K)]

λ = ∑ n_{y}[y] ↦ ^{g}λ = ∑ n_{y}[^{g}y] (g ∈ Aut_{F −alg}(F ⊗ \overline{Q}))

extends the action (1.5.1.1) of Γ_{Q} and leaves stable the subgroup X^{∗}(_{K}𝒮) of X^{∗}(_{K}T) spanned by the CM characters λ_{Φ}. (ii) In the special case when K is a Galois extension of Q, the involution ι from (1.5.4.3) gives rise to another action of Aut_{F −alg}(F ⊗ \overline{Q}) on X^{∗}(_{K}T), namely

g ∗ ι(λ) = ι(^{g}λ) (λ ∈ X^{∗}(_{K}T)).

This action extends the action (1.5.4.1) of Γ_{Q} and leaves stable X^{∗}(_{K}𝒮).

(2.2.8) Proposition. Let n : {CM types of K} ⟶ Z be a function satisfying

∑_{Φ} n_{Φ}λ_{Φ} = w · N_{K/Q} = w ∑_{y∈X(K)}[y] ∈ X^{∗}(_{K}𝒮) (w ∈ Z). Then :

(i) ∀g ∈ Aut_{F −alg}(F ⊗ \overline{Q}) ∏_{Φ} F̃_{Φ}(g)^{n_{Φ}} = Ṽ_{K/F} (g)^{w}. (ii) If w = 0, then ∀g ∈ Aut_{F −alg}(F ⊗ \overline{Q})_{1} ∏_{Φ} f̃_{Φ}(g)^{n_{Φ}} = 1 ∈ K̂^{∗}/K^{∗}.

Proof. (i) Fix s : X ⟶ Γ_{Q} as in 2.1.1, and parameterise the CM types by functions α : X ⟶ Z/2Z, as in (2.1.1.3): we write Φ_{α} = {s(x)c^{α(x)}|_{K}}_{x∈X}, n_{α} = n_{Φ_{α}} and λ_{α} = λ_{Φ_{α}}. The condition ∑_{Φ} n_{Φ}λ_{Φ} = w · N_{K/Q} is equivalent to

∀x ∈ X  ∑_{α} n_{α}λ_{α}(x) = ∑_{α} n_{α}(1 − λ_{α}(x)) = w.

Statement (i) follows from the fact that, for each g = (σ, h) ∈ S_{X} ⋉ Γ_{F}^{X},

∏_{α}_{s}F̃_{α}(g)^{n_{α}} = ∏_{x∈X} γ_{x,s}^{∑_{α}(n_{α}λ_{gα}(x)−n_{α}λ_{α}(x))}R(h(x))^{∑_{α} n_{α}(1−λ_{α}(x))} (^{c}R(h(x)))^{∑_{α} n_{α}λ_{α}(x)} =

= ∏_{x∈X} ^{1+c}R(h(c))^{w} = Ṽ_{K/F} (g)^{w}.

If w = 0, statement (ii) follows by applying ℓ_{K} to (i).

2.3 Generalised universal Taniyama elements

As in §1.6, we assume that K is a CM number field which is a Galois extension of Q.

(2.3.1) Proposition. (i) There exists a unique map f̃′ : Aut_{F −alg}(F ⊗ \overline{Q})_{1} ⟶ _{K}𝒮 (K̂)/_{K}𝒮 (K) such that λ_{Φ} ∘ f̃′ = f̃_{Φ}, for all CM types Φ of K. The map f̃′ factors through Aut_{F −alg}(F ⊗ K^{ab})_{1}. (ii) For each λ ∈ X^{∗}(_{K}𝒮), put f̃′_{λ} = λ ∘ f̃′ : Aut_{F −alg}(F ⊗ \overline{Q})_{1} ⟶ K̂^{∗}/K^{∗}; then f̃′_{λ+μ}(g) = f̃′_{λ}(g)f̃′_{μ}(g). (iii) ∀λ ∈ X^{∗}(_{K}𝒮) ∀g, g′ ∈ Aut_{F −alg}(F ⊗ \overline{Q})_{1} f̃′_{λ}(gg′) = f̃′_{^{g′}λ}(g)f̃′_{λ}(g′). (iv) ∀u ∈ Gal(K/Q) ^{u}(f̃′_{λ}(g)) = f̃′_{u∗λ}([u|_{F}]g).

(v) ∀g ∈ Γ_{Q}  f̃′(id_{F} ⊗ g) = f′(g).

Proof. Statements (i) and (ii) follow from Proposition 2.2.8(ii) by the same argument as in the proof of Proposition 1.6.2. If λ = λ_{Φ}, then (iii) (resp., (iv)) is just the statement of Proposition 2.2.3 (iv) (resp., (v)); the general case follows from (ii). Finally, (v) is a consequence of the uniqueness of f′, since

∀Φ λ_{Φ}(f̃′(id_{F} ⊗ g)) = f̃_{Φ}(id_{F} ⊗ g) = f_{Φ}(g) = λ_{Φ}(f′(g)),

by Proposition 2.2.3(ii).

(2.3.2) Proposition. (i) Define the map f̃ : Aut_{F −alg}(F ⊗ \overline{Q})_{1} ⟶ _{K}𝒮 (K̂)/_{K}𝒮 (K) by the formula f̃(g) = (ι(f̃′(g)))^{−1}. This map factors through Aut_{F −alg}(F ⊗ K^{ab})_{1} and has the following properties: (ii) The maps f̃_{λ} = λ ∘ f̃: Aut_{F −alg}(F ⊗ \overline{Q})_{1} ⟶ K̂^{∗}/K^{∗} (λ ∈ X^{∗}(_{K}𝒮)) satisfy

f̃_{λ+μ}(g) = f̃_{λ}(g)f̃_{μ}(g),  f̃_{λ}(g) = f̃′_{ι(λ)}(g)^{−1},  f̃_{λ}(gg′) = f̃_{g′∗λ}(g)f̃_{λ}(g′).
(iii) ∀u ∈ Gal(K/Q) ∀g ∈ Aut_{F −alg}(F ⊗ \overline{Q})_{1}  ^{u}(f̃_{λ}(g)) = f̃_{^{u}λ}([u|_{F}]g),  ^{u}(f̃(g)) = f̃([u|_{F}]g).
(iv) ∀g, g′ ∈ Aut_{F −alg}(F ⊗ \overline{Q})_{1}  f̃(gg′) = (g^{′−1} ∗ f̃(g)) f̃(g′).
(v) ∀g ∈ Γ_{Q}  f̃(id_{F} ⊗ g) = f(g).

Proof. As in the proof of Proposition 1.6.3, everything follows from Proposition 2.3.1.

(2.3.3) Proposition. There exists a lift b̃ : Aut_{F −alg}(F ⊗ K^{ab})_{1} ⟶ _{K}𝒮 (K̂) of f̃ whose “coboundary” d̃_{g,g′} = (g^{′−1} ∗ b̃(g)) b̃(g′) b̃(gg′)^{−1} is a locally constant function on (Aut_{F −alg}(F ⊗ K^{ab})_{1})^{2}.

Proof. The argument from the proof of Proposition 1.7.5 applies.

(2.3.4) Proposition. If K′ is a CM number field, which is a Galois extension of Q and contains K, then the generalised universal Taniyama elements f̃_{K} : Aut_{F −alg}(F ⊗ \overline{Q})_{1} ⟶ _{K}𝒮 (K̂)/_{K}𝒮 (K) and f̃_{K′} : Aut_{F′−alg}(F′ ⊗ \overline{Q})_{1} ⟶ _{K′}𝒮 (K̂′)/_{K′}𝒮 (K′) over K and K′, respectively, satisfy

∀g ∈ Aut_{F −alg}(F ⊗ \overline{Q})_{1}  f̃_{K}(g) = N_{K′/K} (f̃_{K′}(id_{F′} ⊗_{F} g)) .

Proof. This follows from Proposition 2.2.4(iii), as in the proof of Proposition 1.6.5.

2.4 Generalised Taniyama group

Let K be as in §2.3. (2.4.1) Let us try to apply the method of [Mi-Sh, Prop. 2.7] (see 1.7.3 above) to the generalised universal Taniyama element f̃ and its lift b̃. The reverse 2-cocycle d̃_{g,g′} with values in _{K}𝒮 (K) gives rise to an exact sequence of affine group schemes over K

1 ⟶ _{K}𝒮_{K} \underrightarrow{_{ı}} G̃′ \underrightarrow{_{π}} Aut_{F −alg}(F ⊗ K^{ab})_{1} ⟶ 1  (2.4.1.1)

(where the term on the right is considered as a constant group scheme), equipped with a section α̃ : Aut_{F −alg}(F ⊗ K^{ab})_{1} ⟶ G̃′(K) such that

∀g, g′ ∈ Aut_{F −alg}(F ⊗ K^{ab})_{1}  α̃(gg′) = α̃(g)α̃(g′)d̃_{g,g′}.

The map

s̃p : Aut_{F −alg}(F ⊗ K^{ab})_{1} ⟶ G̃′(K̂),  s̃p(g) = b̃(g)α̃(g),

is a group homomorphism satisfying π̃ ∘ s̃p = id. (2.4.2) Each element u ∈ Γ_{K} acts on G̃′(\overline{Q}) by

u(s α̃(g)) = ^{u}s α̃(g)  (s ∈ _{K}𝒮 (\overline{Q})).  (2.4.2.2)

We extend this action to an action of Γ_{Q}: for u ∈ Γ_{Q} and g ∈ Aut_{F −alg}(F ⊗ K^{ab})_{1}, put

c̃_{u}(g) = b̃([u|_{F}]g)^{u}(b̃(g))^{−1} ∈ _{K}𝒮 (K).

As

∀u, u′ ∈ Γ_{Q} ∀g ∈ Aut_{F −alg}(F ⊗ K^{ab})_{1} c_{uu′}(g) = c̃_{u}([u′|_{F}]g)^{u}(c_{u′}(g)),

the formula

u(s α̃(g)) = c̃_{u}(g)^{u}s α̃(g)  (s ∈ _{K}𝒮 (\overline{Q}), g ∈ Aut_{F −alg}(F ⊗ K^{ab})_{1})  (2.4.2.3)

defines an action of Γ_{Q} on G̃′(\overline{Q}) which extends the action (2.4.2.2) of Γ_{K}.

We define _{K} 𝒯̃ to be the affine group scheme over Q such that _{K} 𝒯̃(\overline{Q}) = G̃′(\overline{Q}), with the Γ_{Q}-action given by (2.4.2.3). The exact sequence (2.4.1.1) descends to an exact sequence

1 ⟶ _{K}𝒮 \underrightarrow{_{ı}} _{K} 𝒯̃ \underrightarrow{_{π}} Aut_{F −alg}(F ⊗ K^{ab})′_{1} ⟶ 1,  (2.4.2.4)

where we have denoted by Aut_{F −alg}(F ⊗ K^{ab})′_{1} a twisted form of the constant group scheme Aut_{F −alg}(F ⊗ K^{ab})_{1}, for which u ∈ Γ_{Q} acts on

Aut_{F −alg}(F ⊗ K^{ab})′_{1}(\overline{Q}) = Aut_{F −alg}(F ⊗ K^{ab})_{1}

by [u|_{F}]. Note that

Aut_{F −alg}(F ⊗ K^{ab})′_{1}(Q) = id_{F} ⊗ Gal(K^{ab}/Q),  (2.4.2.5)

by Proposition 1.1.6(iv).

(2.4.3) As f̃ extends f (and the restriction of b̃ to Gal(K^{ab}/Q)^{2} satisfies 1.7.5), there is a commutative diagram of affine group schemes over Q with exact rows:

1 ⟶ _{K}𝒮 ⟶^{i} _{K}𝒯 ⟶^{π}  Gal(K^{ab}/Q)  ⟶ 1
∥  ↓  ↓^{(id_{F} ⊗−)}
1 ⟶ _{K}𝒮 \underrightarrow{_{ı}} _{K} 𝒯̃ \underrightarrow{_{π}} Aut_{F −alg}(F ⊗ K^{ab})′_{1} ⟶ 1.

Moreover, there is a commutative diagram of groups

K𝒯 (Q̂) ⟵^{sp}  Gal(K^{ab}/Q)
↓  ↓^{(id_{F} ⊗−)}
K 𝒯̃(K̂) \underleftarrow{_{sp}} Aut_{F −alg}(F ⊗ K^{ab})_{1}

such that π ∘ sp = id, π̃ ∘ s̃p = id. As

∀u ∈ Γ_{Q} ∀g ∈ Aut_{F −alg}(F ⊗ K^{ab})_{1} ^{u}s̃p(g) = ^{u}( b(g)α̃(g)) = c̃_{u}(g)^{u}b̃(g) α̃([u|_{F}]g) =
= b̃([u|_{F}]g) α̃([u|_{F}]g) = s̃p([u|_{F}]g),

the map s̃p is Γ_{Q}-equivariant. As [u|_{F}] depends only on the image of u in Gal(F/Q), it follows that the image of s̃p is contained in _{K} 𝒯̃(F̂), and that s̃p is Gal(F/Q)-equivariant.

(2.4.4) Proposition 2.3.4 implies that the pull-backs of _{K′} 𝒯̃ to Aut_{F −alg}(F ⊗ \overline{Q})′_{1} (for varying K′ ⊃ K) give rise to an extension of Aut_{F −alg}(F ⊗ \overline{Q})′_{1} by 𝒮. These extensions for varying F are again compatible; they give rise to an extension of affine group schemes over Q,

1 ⟶ 𝒮 ⟶ 𝒯̃ ⟶ \underrightarrow{lim}_{F} Aut_{F −alg}(F ⊗ \overline{Q})′_{1} ⟶ 1,

whose pull-back to Γ_{Q} coincides with (1.7.6.1). The direct limit is taken with respect to the transition maps id_{F′} ⊗_{F} − (for F ⊆ F′).

(2.4.5) It would be of interest to give an “abstract” definition of 𝒯̃ along the lines of [De]. As observed in 2.2.5, it is the group Aut_{F −alg}(F ⊗ \overline{Q})_{0} rather than Aut_{F −alg}(F ⊗ \overline{Q})_{1} which has a geometric significance, which means that one should rather consider the subgroup scheme 𝒯̃_{0} ⊂ 𝒯̃ sitting in the exact sequence

1 ⟶ 𝒮 ⟶ 𝒯̃_{0} ⟶ \underrightarrow{lim}_{F} Aut_{F −alg}(F ⊗ \overline{Q})′_{0} ⟶ 1.

References

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[Mi] J.S. Milne, Canonical models of (mixed) Shimura varieties and automorphic vector bundles, in: Automorphic Forms, Shimura Varieties and L-functions, Vol. I, Perspect. in Math. 10, Academic Press, Boston, 1990, pp. 283–414.

[Mi-Sh] J.S. Milne, K. Shih, Langlands’s construction of the Taniyama group, in: Hodge Cycles, Motives and

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