ABELIAN VARIETIES WITH COMPLEX MULTIPLICATION (FOR PEDESTRIANS)

J.S. MILNE

Abstract. (June 7, 1998.) This is the text of an article that I wrote and disseminated in September 1981, except that I’ve updated the references, corrected a few misprints, and added a table of contents, some footnotes, and an addendum.

The original article gave a simplified exposition of Deligne’s extension of the Main Theorem of Complex Multiplication to all automorphisms of the complex numbers. The addendum discusses some additional topics in the theory of complex multiplication.

Contents

1. Statement of the Theorem 2 2. Definition of f_{Φ}(τ) 5 3. Start of the Proof; Tate’s Result 7 4. The Serre Group 9 5. Definition of e^{E} 13 6. Proof that e = 1 14 7. Definition of f^{E} 16 8. Definition of g^{E} 18 9. Re-statement of the Theorem 20 10. The Taniyama Group 21 11. Zeta Functions 24 12. The Origins of the Theory of Complex Multiplication for Abelian Varieties

of Dimension Greater Than One 26 13. Zeta Functions of Abelian Varieties of CM-type 27 14. Hilbert’s Twelfth Problem 30 15. Algebraic Hecke Characters are Motivic 36 16. Periods of Abelian Varieties of CM-type 40 17. Review of: Lang, Complex Multiplication, Springer 1983. 41

The main theorem of Shimura and Taniyama (see Shimura 1971, Theorem 5.15) describes the action of an automorphism τ of ℂ on a polarized abelian variety of CM-type and its torsion points in the case that τ fixes the reflex field of the variety. In his Corvallis article (1979), Langlands made a conjecture concerning conjugates of Shimura varieties that (see Milne and Shih 1979) leads to a conjectural description of the action of τ on a polarized abelian variety of CM-type and its torsion points for all τ. Recently (July 1981) Deligne proved the conjecture^{1} (Deligne 1982b). Deligne expresses his result as an identity between two pro-reductive groups, the Taniyama group of Langlands and his own motivic Galois group associated with the Tannakian category of motives of abelian varieties over ℚ of potential CM-type. Earlier (April 1981) Tate gave a more down-to-earth conjecture than that stated in (Milne and Shih 1979) and partially proved his conjecture (Tate 1981).

All footnotes were added in June 1998.

The purpose of these notes is to use Deligne’s ideas to give as direct a proof as possible of the conjecture in the form stated by Tate. It is also checked that the three forms of the conjecture, those in Deligne 1982b, Milne and Shih 1979, and Tate 1981 are compatible. Also, Tate’s ideas are used to simplify the construction of the Taniyama group. In the first three sections, I have followed Tate’s manuscript (1981) very closely, sometimes word-for-word.

These notes are a rough write-up of two of my lectures at the conference on Shimura Varieties, Vancouver, 17–25 August, 1981. In the remaining lectures I described how the result on abelian varieties of CM-type could be applied to give a proof of Langlands’s conjecture on conjugates of Shimura varieties for most^{2} (perhaps all) Shimura varieties.

Notations. We let ℤ̂ = \underleftarrow{lim} ℤ/mℤ and 𝔸_{f} = ℤ̂ ⊗ ℚ. For a number field E, 𝔸_{f,E} = 𝔸_{f} ⊗ E is the ring of finite adèles and 𝔸_{E} = 𝔸_{f,E} × (E ⊗ ℝ) the full ring of adèles. When E is a subfield of ℂ, E^{ab} and E^{al} denote respectively the maximal abelian extension of E in ℂ and the algebraic closure of E in ℂ. Complex conjugation is denoted by ι.

For a number field E, rec_{E}: 𝔸_{E}^{×} → Gal(E^{ab}/E) is the reciprocity law, normalized so that a prime element parameter corresponds to the inverse of the usual (arithmetic) Frobenius: if a ∈ 𝔸_{f,E}^{×} has v-component a prime element a_{v} in E_{v} and w-component a_{w} = 1 for w ≠ v, then rec_{E}(a) = σ^{−1} if σx ≡ x^{ℕ(v)} mod 𝔭_{v}. When E is totally complex, rec_{E} factors into 𝔸_{E}^{×} → 𝔸_{f,E}^{×} ⟶^{r_{E}} Gal(E^{ab}/E). The cyclotomic character χ = χ_{cyc}: Aut(ℂ) → ℤ̂^{×} ⊂ 𝔸_{f}^{×} is the homomorphism such that τζ = ζ^{χ(τ)} for every root of 1 in ℂ. The composite r_{E} ∘χ = Ver_{E/ℚ}, the Verlagerung map Gal(ℚ^{al}/ℚ)^{ab} → Gal(ℚ^{al}/E)^{ab}.

When T is a torus over E, X_{∗}(T) is the cocharacter group Hom_{E^{al}}(𝔾_{m}, T) of T. *Be wary^{3} of signs.*

1. Statement of the Theorem

Let A be an abelian variety over ℂ, and let K be a subfield of End(A)⊗ℚ of degree 2 dim A over ℚ. The representation of K on the tangent space to A at zero is of the form ⊕_{ϕ∈Φ}ϕ with Φ a subset of Hom(K, ℂ). A Riemann form for A is a ℚ-bilinear skew-symmetric form ψ on H_{1}(A, ℚ) such that

1That is, the conjectural description of the action of τ on a polarized abelian variety of CM-type, not Langlands’s conjecture!

2In fact all, see Milne 1983.

3This is universally good advice, but I believe the signs here to be correct.

(x, y) ↦ ψ(x, iy): H_{1}(A, ℝ) × H_{1}(A, ℝ) → ℝ

is symmetric and positive definite. We assume that there exists a Riemann form ψ compatible with the action of K in the sense that

ψ(ax, y) = ψ(x, (ιa)y), a ∈ K, x, y ∈ H_{1}(A, ℚ).

Then K is a CM-field, and Φ is a CM-type on K, i.e., Hom(K, ℂ) = Φ ∪ ιΦ (disjoint union). The pair (A, K ↪ End(A) ⊗ ℚ) is said to be of CM-type (K, Φ). For simplicity, we assume that K ∩ End(A) = O_{K}, the full ring of integers in K.

Let ℂ^{Φ} be the set of complex-valued functions on Φ, and embed K into ℂ^{Φ} through the natural map a ↦ (ϕ(a))_{ϕ∈Φ}. There then exist a ℤ-lattice 𝔞 in K stable under O_{K}, an element t ∈ K^{×}, and an O_{K}-linear analytic isomorphism θ: ℂ^{Φ}/𝔞 → A such that ψ(x, y) = Tr_{K/ℚ}(tx · ιy) where, in the last equation, we have used θ to identify H_{1}(A, ℚ) with 𝔞 ⊗ ℚ = K. The variety is said to be of type (K, Φ; 𝔞, t) relative^{4} to θ. The type determines the triple (A, K ↪ End(A) ⊗ ℚ, ψ) up to isomorphism. Conversely, the triple determines the type up to a change of the following form: if θ is replaced by θ ∘ a^{−1}, a ∈ K^{×}, then the type becomes (K, Φ; a𝔞, \frac{t}{a·ιa}).

Let τ ∈ Aut(ℂ). Then K ↪ End(A) ⊗ ℚ induces a map K ↪ End(τA) ⊗ ℚ, so that τA also has complex multiplication by K. The form ψ is associated with a divisor D on A, and we let τψ be the Riemann form for τA associated with τD. It has the following characterization: after multiplying ψ with a nonzero rational number, we can assume that it takes integral values on H_{1}(A, ℤ); define ψ_{m} to be the pairing A_{m} × A_{m} → μ_{m}, (x, y) ↦ exp(\frac{2πi·ψ(x,y)}{m}); then (τψ)_{m}(τx, τy) = τ(ψ_{m}(x, y)).

In the next section we shall define (following Tate) for each CM-type (K, Φ) a map f_{Φ}: Aut(ℂ) → 𝔸_{f,K}^{×}/K^{×} such that

f_{Φ}(τ) · ιf_{Φ}(τ) = χ(τ)K^{×}, all τ ∈ Aut(ℂ).

We can now state the new main theorem of complex multiplication in the form first appearing (as a conjecture) in Tate 1981.

Theorem 1.1 (Shimura, Taniyama, Langlands, Deligne). Suppose A has type (K, Φ; 𝔞, t) relative to θ: ℂ^{Φ}/𝔞 ⟶^{≈} A. Let τ ∈ Aut(ℂ), and let f ∈ 𝔸_{f,K}^{×} lie in f_{Φ}(τ).

(a) The variety τA has type

(K, τΦ; f𝔞, \frac{tχ(τ)}{f · ιf})

relative to θ′ say.

(b) It is possible to choose θ′ so that

𝔸_{f,K}  \overline{>>} 𝔸_{f,K}/𝔞 ⊗ ℤ̂ ≅ K/𝔞 \overline{^{θ}>} A_{tors}

𝔸_{f,K}^{∨^{f}} \overline{>>} 𝔸_{f,K}/(f𝔞 ⊗ ℤ̂) ≅ K/f𝔞 \overline{>^{θ′}} τA_{tors}^{∨^{τ}}

commutes, where A_{tors} denotes the torsion subgroup of A.

4See Shimura 1971, pp 126–129.

Remark 1.2. Prior to its complete proof, the theorem was known in three^{5} important cases.

(a) If τ fixes the reflex field of (K, Φ), then the theorem becomes the old main theorem of complex multiplication, proved by Shimura and Taniyama (see (2.7) below). This case is used in the proof of the general result.

(b) Tate (1981) proved part (a) of the theorem, and he showed that (b) holds when f is replaced by fe, some e ∈ 𝔸_{f,K_{0}}^{×} with e^{2} = 1, where K_{0} the maximal real subfield of K. We include Tate’s proof of his result, although it is not necessary for the general case.

(c) Shih (1976) proved the theorem under the assumption that there exists an automorphism σ of K of order 2 such that τ(Φ∩Φσ) = Φ∩Φσ and τ(Φ∩ιΦσ) = Φ ∩ ιΦσ for all automorphisms τ of ℂ. As we shall see, his proof is a special case of the general proof.

We now restate the theorem in more invariant form. Let

T A =^{df} \underleftarrow{lim} A_{m}(ℂ) ≅ \underleftarrow{lim}(\frac{1}{m}H_{1}(A, ℤ)/H_{1}(A, ℤ)) = H_{1}(A, ℤ̂)

(limit over all positive integers m), and let

V_{f}A = T A ⊗_{ℤ} ℚ = H_{1}(A, ℚ) ⊗_{ℚ} 𝔸_{f}.

Then ψ gives rise to a pairing

ψ_{f} = \underleftarrow{lim} ψ_{m}: V_{f}A × V_{f}A → 𝔸_{f}(1)

where 𝔸_{f}(1) = (\underleftarrow{lim} μ_{m}(ℂ)) ⊗ ℚ.

Theorem 1.3. Let A have type (K, Φ); let τ ∈ Aut(ℂ), and let f ∈ f_{Φ}(τ).

(a) τA is of type (K, τΦ);

(b) there is an K-linear isomorphism α: H_{1}(A, ℚ) → H_{1}(τA, ℚ) such that (i) ψ(\frac{χ(τ)}{f·ιf}x, y) = (τψ)(αx, αy), x, y ∈ H_{1}(A, ℚ); (ii) the^{6} diagram

V_{f}(A) \overline{^{f}} V_{f}(A)
[figure: τ]
V_{f}(τA)^{^{α⊗1}}

commutes.

Lemma 1.4. The statements (1.1) and (1.3) are equivalent.

5Shimura (1977) investigated the question in some further special cases. After explaining that the action of a general automorphism of ℂ on an elliptic curve of CM-type can be obtained from knowing the actions of complex conjugation and those automorphisms fixing its reflex field, he concludes rather pessimistically that “In the higher-dimensional case, however, no such general answer seems possible.”

6Note that both f ∈ 𝔸_{f,K}^{×} and the K-linear isomorphism α are uniquely determined up to multiplication by an element of K^{×}. Changing the choice of one changes that of the other by the same factor.

Proof. Let θ and θ′ be as in (1.1), and let θ_{1}: K ⟶^{≈} H_{1}(A, ℚ) and θ′_{1}: K ⟶^{≈} H_{1}(τA, ℚ) be the K-linear isomorphisms induced by θ and θ′. Let χ = χ(τ)/f · ιf — it is an element of K^{×}. Then

ψ(θ_{1}(x), θ_{1}(y)) = Tr(tx · ιy)
(τψ)(θ′_{1}(x), θ′_{1}(y)) = Tr(tχx · ιy)

and

𝔸_{f,K} ⟶^{θ_{1}} V_{f}(A)
↓^{f}  ↓^{τ}
𝔸_{f,K} ⟶^{θ′_{1}} V_{f}(τA)

commutes. Let α = θ′_{1} ∘ θ_{1}^{−1}; then

τψ(αx, αy) = Tr(tχθ_{1}^{−1}(x) · ιθ_{1}^{−1}(y)) = ψ(χx, y)

and (on V_{f}(A)),

τ = θ′_{1} ∘ f ∘ θ_{1}^{−1} = θ′_{1} ∘ θ_{1}^{−1} ∘ f = α ∘ f.

Conversely, let α be as in (1.3) and choose θ′_{1} so that α = θ′_{1} ∘ θ_{1}^{−1}. It is then easy to check (1.1).

2. Definition of f_{Φ}(τ)

Let (K, Φ) be a CM-type. Choose an embedding K ↪ ℂ, and extend it to an embedding i: K^{ab} ↪ ℂ. Choose elements w_{ρ} ∈ Aut(ℂ), one for each ρ ∈ Hom(K, ℂ), such that

w_{ρ} ∘ i|K = ρ, w_{ιρ} = ιw_{ρ}.

For example, choose w_{ρ} for ρ ∈ Φ (or any other CM-type) to satisfy the first equation, and then define w_{ρ} for the remaining ρ by the second equation. For any τ ∈ Aut(ℂ), w_{τρ}^{−1}τw_{ρ} ∘ i|K = w_{τρ}^{−1} ∘ τρ|K = i. Thus i^{−1} ∘ w_{τρ}^{−1}τw_{ρ} ∘ i ∈ Gal(K^{ab}/K), and we can define F_{Φ}: Aut(ℂ) → Gal(K^{ab}/K) by

F_{Φ}(τ) = ∏_{ϕ∈Φ} i^{−1} ∘ w_{τϕ}^{−1}τw_{ϕ} ∘ i.

Lemma 2.1. The element F_{Φ} is independent of the choice of {w_{ρ}}.

Proof. Any other choice is of the form w′_{ρ} = w_{ρ}h_{ρ}, h_{ρ} ∈ Aut(ℂ/iK). Thus F_{Φ}(τ) is changed by i^{−1} ∘(∏_{ϕ∈Φ} h_{τϕ}^{−1}h_{ϕ})∘i. The conditions on w and w′ imply that h_{ιρ} = h_{ρ}, and it follows that the inside product is 1 because τ permutes the unordered pairs {ϕ, ιϕ} and so ∏_{ϕ∈Φ} h_{ϕ} = ∏_{ϕ∈Φ} h_{τϕ}.

Lemma 2.2. The element F_{Φ} is independent of the choice of i (and K ↪ ℂ).

Proof. Any other choice is of the form i′ = σ ∘i, σ ∈ Aut(ℂ). Take w′_{ρ} = w_{ρ} ∘σ^{−1}, and then

F′_{Φ}(τ) = ∏ i^{′−1} ∘ (σw_{τϕ}^{−1}τw_{ϕ}σ^{−1}) ∘ i′ = F_{Φ}(τ).

Thus we can suppose K ⊂ ℂ and ignore i; then

F_{Φ}(τ) = ∏_{ϕ∈Φ} w_{τϕ}^{−1}τw_{ϕ} mod Aut(ℂ/K^{ab})

where the w_{ρ} are elements of Aut(ℂ) such that

w_{ρ}|K = ρ, w_{ιρ} = iw_{ρ}.

Proposition 2.3. For any τ ∈ Aut(ℂ), there is a unique f_{Φ}(τ) ∈ 𝔸_{f,K}^{×}/K^{×} such that

(a) r_{K}(f_{Φ}(τ)) = F_{Φ}(τ);

(b) f_{Φ}(τ) · ιf_{Φ}(τ) = χ(τ)K^{×}, χ = χ_{cyc}.

Proof. Since r_{K} is surjective, there is an f ∈ 𝔸_{f,K}^{×}/K^{×} such that r_{K}(f) = F_{Φ}(τ). We have

r_{K}(f · ιf) = r_{K}(f) · r_{K}(ιf)
= r_{K}(f) · ιr_{K}(f)ι^{−1}
= F_{Φ}(τ) · F_{ιΦ}(τ)
= V_{K/ℚ}(τ),

where V_{K/ℚ}: Gal(ℚ^{al}/ℚ)^{ab} → Gal(ℚ^{al}/K)^{ab} is the transfer (Verlagerung) map. As V_{K/ℚ} = r_{K} ∘ χ, it follows that f · ιf = χ(τ)K^{×} mod (Ker r_{K}). The next lemma shows that 1 + ι acts bijectively on Ker(r_{K}), and so there is a unique a ∈ Ker r_{K} such that a · ιa = (f · ιf/χ(τ))K^{×}; we must take f_{Φ}(τ) = f/a.

Lemma 2.4. The kernel of r_{K}: 𝔸_{f,K}^{×}/K^{×} → Gal(K^{ab}/K) is uniquely divisible by all integers, and its elements are fixed by ι.

Proof. The kernel of r_{K} is \overline{K^{×}}/K^{×} where \overline{K^{×}} is the closure of K^{×} in 𝔸_{f,K}^{×}. It is also equal to Ū/U for any subgroup U of O_{K}^{×} of finite index. A theorem of Chevalley (see Serre 1964, 3.5) shows that 𝔸_{f,K}^{×} induces the pro-finite topology on U. If we take U to be contained in the real subfield of K and torsion-free, then it is clear that Ū/U is fixed by ι and (being isomorphic to (ℤ̂/ℤ)^{dim A}) uniquely divisible.

Remark 2.5. A more direct definition of f_{Φ}(τ), but one involving the Weil group, can be found in (7.2).

Proposition 2.6. The maps f_{Φ}: Aut(ℂ) → 𝔸_{f,K}^{×}/K^{×} have the following properties:

(a) f_{Φ}(στ) = f_{τΦ}(σ) · f_{Φ}(τ);

(b) f_{Φ(τ^{−1}|K)}(σ) = τf_{Φ}(σ) if τK = K;

(c) f_{Φ}(ι) = 1.

Proof. Let f = f_{τΦ}(σ) · f_{Φ}(τ). Then

r_{K}(f) = F_{τΦ}(σ) · F_{Φ}(τ) = ∏_{ϕ∈Φ} w_{στϕ}^{−1}σw_{τϕ}w_{τϕ}^{−1}τw_{ϕ} = F_{Φ}(στ)

and f · ιf = χ(σ)χ(τ)K^{×} = χ(στ)K^{×}. Thus f satisfies the conditions that determine f_{Φ}(στ). This proves (a), and (b) and (c) can be proved similarly.

Let E be the reflex field for (K, Φ), so that Aut(ℂ/E) = {τ ∈ Aut(ℂ) | τΦ = Φ}. Then Φ Aut(ℂ/K) =^{df} ∪_{ϕ∈Φ}ϕ · Aut(ℂ/K) is stable under the left action of Aut(ℂ/E), and we write

Aut(ℂ/K)Φ^{−1} = ∪ψ Aut(ℂ/E) (disjoint union);

the set Ψ = {ψ|E} is a CM-type for E, and (E, Ψ) is the reflex of (K, Φ). The map a ↦ ∏_{ψ∈Ψ} ψ(a): E → ℂ factors through K and defines a morphism of algebraic tori Ψ^{×}: E^{×} → K^{×}. The (old) main theorem of complex multiplication states the following: let τ ∈ Aut(ℂ/E), and let a ∈ 𝔸_{f,E}^{×}/E^{×} be such that r_{E}(a) = τ; then (1.1) is true after f has been replaced by Ψ^{×}(a). (See Shimura 1971, Theorem 5.15; the sign differences result from different conventions for the reciprocity law and the actions of Galois groups.) The next result shows that this is in agreement with (1.1).

Proposition 2.7. For any τ ∈ Aut(ℂ/E) and a ∈ 𝔸_{f,E}^{×}/E^{×} such that r_{E}(a) = τ, Ψ^{×}(a) ∈ f_{Φ}(τ).

Proof. Partition Φ into orbits, Φ = ∪_{j}Φ_{j}, for the left action of Aut(ℂ/E). Then Aut(ℂ/K)Φ^{−1} = ∪_{j} Aut(ℂ/K)Φ_{j}^{−1}, and

Aut(ℂ/K)Φ_{j}^{−1} = Aut(ℂ/K)(σ_{j}^{−1} Aut(ℂ/E)) = (Hom_{K}(L_{j}, ℂ) ∘ σ_{j}^{−1}) Aut(ℂ/E)

where σ_{j} is any element of Aut(ℂ) such that σ_{j}|K ∈ Φ_{j} and L_{j} = (σ_{j}^{−1}E)K. Thus Ψ^{×}(a) = ∏ b_{j}, with b_{j} = Nm_{L_{j}/K}(σ_{j}^{−1}(a)). Let

F_{j}(τ) = ∏_{ϕ∈Φ_{j}} w_{τϕ}^{−1}τw_{ϕ} (mod Aut(ℂ/K^{ab})).

We begin by showing that F_{j}(τ) = r_{K}(b_{j}).

The basic properties of Artin’s reciprocity law show that

𝔸_{f,E}^{×} ^{⊂}  > 𝔸_{f,σL_{j}}^{×}  \overline{^{σ_{j}^{−1}} >} 𝔸_{f,L_{j}}^{×}  \overline{^{Nm_{L_{j} /K}} >} 𝔸_{f,K}^{×}

Gal(E^{ab}/E) ∨^{r_{E}}

>^{V_{σ_{j} L_{j} /E}}

σ_{j} Gal(L_{j}^{ab}/L_{j})σ_{j}^{−1} ∨^{r_{σL_{j}}} >^{adσ_{j}^{−1}} Gal(L_{j}^{ab}/L_{j}) ∨^{r_{L_{j}}} restriction > Gal(K^{ab}/K) ∨^{r_{K}}

commutes. Therefore r_{K}(b_{j}) is the image of r_{E}(a) by the three maps in the bottom row of the diagram. Consider {t_{ϕ} | t_{ϕ} = w_{ϕ}σ_{j}^{−1}, ϕ ∈ Φ_{j}}; this is a set of coset representatives for σ_{j} Aut(ℂ/L_{j})σ_{j}^{−1} in Aut(ℂ/E), and so F_{j}(τ) = ∏_{ϕ∈Φ_{j}} σ_{j}^{−1}t_{τϕ}^{−1}τt_{ϕ}σ_{j} = σ_{j}^{−1}V (τ)σ_{j} mod Aut(ℂ/K^{ab}).

Thus r_{K}(Ψ^{×}(a)) = ∏ r_{K}(b_{j}) = ∏ F_{j}(τ) = F_{Φ}(τ). Clearly, Ψ^{×}(a) · ιΨ^{×}(a) ∈ χ(τ)K^{×}, and so this shows that Ψ^{×}(a) ∈ f_{Φ}(τ).

3. Start of the Proof; Tate’s Result

We shall work with the statement (1.3) rather than (1.1). The variety τA has type (K, τΦ) because τΦ describes the action of K on the tangent space to τA at zero. Choose any K-linear isomorphism α: H_{1}(A, ℚ) → H_{1}(τA, ℚ). Then

V_{f}(A) →^{τ} V_{f}(τA) →^{(α⊗1)^{−1}} V_{f}(A)

is an 𝔸_{f,K}-linear isomorphism, and hence is multiplication by some g ∈ 𝔸_{f,K}^{×}; thus (α ⊗ 1) ∘ g = τ.

Lemma 3.1. For this g, we have

(αψ)(\frac{χ(τ)}{g · ιg} x, y) = (τψ)(x, y), all x, y ∈ V_{f}(τA).

Proof. By definition,

(τψ)(τx, τy) = τ(ψ(x, y)) x, y ∈ V_{f}(A)
(αψ)(αx, αy) = ψ(x, y) x, y ∈ V_{f}(A).

On replacing x and y by gx and gy in the second inequality, we find that

(αψ)(τx, τy) = ψ(gx, gy) = ψ((g · ιg)x, y).

As τ(ψ(x, y)) = χ(τ)ψ(x, y) = ψ(χ(τ)x, y), the lemma is now obvious.

Remark 3.2. (a) On replacing x and y with αx and αy in (3.1), we obtain the formula

ψ(\frac{χ(τ)}{g · ιg} x, y) = (τψ)(αx, αy).

(b) On taking x, y ∈ H_{1}(A, ℚ) in (3.1), we can deduce that χ(τ)/g · ιg ∈ K^{×}; therefore g · ιg ≡ χ(τ) mod K^{×}.

The only choice involved in the definition of g is that of α, and α is determined up to multiplication by an element of K^{×}. Thus the class of g in 𝔸_{f,K}^{×}/K^{×} depends only on A and τ. In fact, it depends only on (K, Φ) and τ, because any other abelian variety of type (K, Φ) is isogenous to A and leads to the same class gK^{×}. We define g_{Φ}(τ) = gK^{×} ∈ 𝔸_{f,K}^{×}/K^{×}.

Proposition 3.3. The maps g_{Φ}: Aut(ℂ) → 𝔸_{f,K}^{×}/K^{×} have the following properties:

(a) g_{Φ}(στ) = g_{τΦ}(σ) · g_{Φ}(τ);

(b) g_{Φ(τ^{−1}|K)}(σ) = τg_{Φ}(σ) if τK = K;

(c) g_{Φ}(ι) = 1;

(d) g_{Φ}(τ) · ιg_{Φ}(τ) = χ(τ)K^{×}.

Proof. (a) Choose K-linear isomorphisms α: H_{1}(A, ℚ) → H_{1}(τA, ℚ) and β: H_{1}(τA, ℚ) → H_{1}(στA, ℚ), and let g = (α ⊗ 1)^{−1} ∘ τ and g_{τ} = (β ⊗ 1)^{−1} ∘ σ so that g and g_{τ} represent g_{Φ}(τ) and g_{τϕ}(σ) respectively. Then

(βα) ⊗ 1 ∘ (g_{τ}g) = (β ⊗ 1) ∘ g_{τ} ∘ (α ⊗ 1) ∘ g = στ,

which shows that g_{τ}g represents g_{Φ}(στ).

(b) If (A, K ↪ End(A) ⊗ ℚ) has type (K, Φ), then (A, K →^{τ^{−1}} K → End(A) ⊗ ℚ) has type (K, Φτ^{−1}). The formula in (b) can be proved by transport of structure.

(c) Complex conjugation ι: A → ιA is a homeomorphism (relative to the complex topology) and so induces a K-linear isomorphism ι_{1}: H_{1}(A, ℚ) → H_{1}(A, ℚ). The map ι_{1} ⊗ 1: V_{f}(A) → V_{f}(ιA) is ι again, and so on taking α = ι_{1}, we find that g = 1.

(d) This is proved in (3.2).

Theorem 1.3 (hence also 1.1) becomes true if f_{Φ} is replaced by g_{Φ}. Our task is to show that f_{Φ} = g_{Φ}. To this end we set

e_{Φ}(τ) = g_{Φ}(τ)/f_{Φ}(τ) ∈ 𝔸_{f,K}^{×}/K^{×}.

Proposition 3.4. The maps e_{Φ}: Aut(ℂ) → 𝔸_{f,K}^{×}/K^{×} have the following properties:

(a) e_{Φ}(στ) = e_{τΦ}(σ) · e_{Φ}(τ);

(b) e_{Φ(τ^{−1}|K)}(σ) = τe_{Φ}(σ) if τK = K;

(c) e_{Φ}(ι) = 1;

(d) e_{Φ}(τ) · ιe_{Φ}(τ) = 1;

(e) e_{Φ}(τ) = 1 if τΦ = Φ.

Proof. Statements (a), (b), and (c) follow from (a), (b), and (c) of (2.6) and (3.3), and (d) follows from (3.3d) and (2.3b). The condition τΦ = Φ in (e) means that τ fixes the reflex field of (K, Φ) and, as we observed in §2, the theorems are known to hold in that case, which means that f_{Φ}(τ) = g_{Φ}(τ).

Proposition 3.5. Let K_{0} be the maximal real subfield of K; then e_{Φ}(τ) ∈ 𝔸_{f,K_{0}}^{×}/K_{0}^{×} and e_{Φ}(τ)^{2} = 1; moreover, e_{Φ}(τ) depends only on the effect of τ on K_{0}, and is 1 if τ|K_{0} = id.

Proof. Replacing τ by σ^{−1}τ in (a), we find using (e) that e_{Φ}(τ) = e_{Φ}(σ) if τΦ = σΦ, i.e., e_{Φ}(τ) depends only on the restriction of τ to the reflex field of (K, Φ). From (b) with τ = ι, we find using ιΦ = Φι that e_{ιΦ}(σ) = ιe_{Φ}(σ). Putting τ = ι, then σ = ι, in (a), we find that e_{Φ}(σι) = ιe_{Φ}(σ) and e_{Φ}(ιτ) = e_{Φ}(τ). Since ιτ and τι have the same effect on E, we conclude e_{Φ}(τ) = ιe_{Φ}(τ). Thus e_{Φ}(τ) ∈ (𝔸_{f,K}^{×}/K^{×})^{⟨ι⟩} = 𝔸_{f,K_{0}}^{×}/K_{0}^{×}, where ⟨ι⟩ = Gal(K/K_{0}), and (d) shows that e_{Φ}(τ)^{2} = 1.

Corollary 3.6. Part (a) of (1.1) is true; part (b) of (1.1) becomes true when f is replaced by ef with e ∈ 𝔸_{f,K_{0}}^{×}, e^{2} = 1.

Proof. Let e ∈ e_{Φ}(τ). Then e^{2} ∈ K_{0}^{×} and, since an element of K_{0}^{×} that is a square locally at all finite primes is a square, we can correct e to achieve e^{2} = 1. Now (1.1) is true with f replaced by ef, but e (being a unit) does not affect part (a) of (1.1).

We can now sketch the proof of the Theorems 1.1 and 1.3 — for this, we must prove e_{Φ}(τ) = 1 for all τ. It seems to be essential to prove this simultaneously for all abelian varieties. To do this, one needs to define a universal e, giving rise to all the e_{Φ}. The universal e is a map into the Serre group. In §4 we review some of the theory concerning the Serre group, and in (5.1) we state the existence of e. The proof of (5.1), which requires Deligne’s result (Deligne 1982a) on Hodge cycles on abelian varieties, is carried out in §7 and §8 . The remaining step, proving that e = 1, is less difficult, and is carried out in §6.

4. The Serre Group

Let E be a CM-field. The Serre group corresponding to E is a pair (S^{E}, μ^{E}) comprising a ℚ-rational torus S^{E} and a cocharacter μ^{E} ∈ X_{∗}(S^{E}) defined over E whose weight w^{E} =^{df} −(ι + 1)μ^{E} is defined over ℚ. It is characterized by having the following universal property: for any ℚ-rational torus T and μ ∈ X_{∗}(T) defined over E whose weight is ℚ-rational, there is a unique ℚ-rational homomorphism ρ_{μ}: S^{E} → T such that ρ_{μ} ∘ μ^{E} = μ.

For ρ ∈ Hom(E, ℂ), let [ρ] be the character of the torus E^{×} defined by ρ. Then {[ρ] | ρ ∈ Hom(E, ℂ)} is a basis for X^{∗}(E^{×}), and S^{E} is the quotient of the torus E^{×} with

X^{∗}(S^{E}) = {χ ∈ X^{∗}(E^{×}) | (τ − 1)(ι + 1)χ = 0, all τ ∈ Aut(ℂ)}
X^{∗}(μ^{E}) = ∑ n_{ρ}[ρ] ↦ n_{1}: X^{∗}(S^{E}) → ℤ

because this pair has the universal property dual to that of (S^{E}, μ^{E}). In particular, there is a canonical homomorphism E^{×} → S^{E}, and it is known (cf. Serre 1968, II) that the kernel of the map is the Zariski closure of any sufficiently small subgroup U of finite index in O_{E}^{×}.

When E is Galois over ℚ, the action of σ ∈ Gal(E/ℚ) on E defines an automorphism σ̃ of the torus S^{E}, whose action on characters is

∑ n_{ρ}[ρ] ↦ ∑ n_{ρ}[ρσ] = ∑ n_{ρσ^{−1}}[ρ].

Lemma 4.1. Let E_{0} be the maximal real subfield of E; there is an exact sequence of algebraic tori

1 ⟶ E_{0}^{×} \underrightarrow{( \begin{matrix}incl. \\ Nm & _{E_{0}/ℚ} \end{matrix} )} E^{×} × ℚ^{×} ⟶^{(can.,w^{E})} S^{E} → 1.

Proof. It suffices to show that the sequence becomes exact after the functor X^{∗} has been applied. As

X^{∗}(E_{0}) = {∑ n_{ρ′}[ρ′] | ρ′ ∈ Hom(E_{0}, ℂ)}
X^{∗}(E^{×} × ℚ^{×}) = {∑ n_{ρ}[ρ] + n | ρ ∈ Hom(E, ℂ)}
X^{∗}(S^{E}) = {∑ n_{ρ}[ρ] | n_{ρ} + n_{ιρ} = constant}
X^{∗}(( \begin{matrix}incl. \\ Nm & E_{0}/ℚ \end{matrix} )) = ∑ n_{ρ}[ρ] + n ↦ ∑ n_{ρ}[ρ|E_{0}] + n ∑_{ρ′}[ρ′]
X^{∗}((can., w^{E})) = ∑ n_{ρ}[ρ] ↦ ∑ n_{ρ}[ρ] − (n_{1} + n_{ι})

this is trivial.

Lemma 4.2. The map Nm_{E/ℚ}: E^{×} → ℚ^{×} factors through S^{E}, and gives rise to a commutative diagram

S^{E}  \overline{^{1+ι}} S^{E}
[figure: Nm_{E/ℚ}; −w^{E}]
ℚ^{×}.

Proof. The map X^{∗}(Nm_{E/ℚ}) is n ↦ n ∑[ρ], which clearly factors through X^{∗}(S^{E}) ⊂ X^{∗}(E^{×}). Moreover, the endomorphisms

X^{∗}(−w^{E} ∘ Nm_{E/ℚ}) = (∑ n_{ρ}[ρ] ↦ n_{1} + n_{ι} ⟼ (n_{1} + n_{ι}) ∑ n_{ρ}[ρ])

X^{∗}(1 + ι) = (∑ n_{ρ}[ρ] ↦ ∑ n_{ρ})([ρ] + [ιρ]) ↦ ∑(n_{ρ} + n_{ιρ})[ρ] = (n_{1} + n_{ι}) ∑[ρ]

are equal.

Let E_{1} ⊃ E_{2} be CM-fields. The norm map E_{1}^{×} → E_{2}^{×} induces a norm map Nm_{E_{1}/E_{2}}: S^{E_{1}} → S^{E_{2}} which is the unique ℚ-rational homomorphism such that Nm_{E_{1}/E_{2}} ∘μ^{E_{1}} = μ^{E_{2}}. The following diagram commutes:

1 ⟶ (E_{1})_{0}^{×} ⟶ E_{1}^{×} × ℚ^{×} ⟶ S^{E_{1}} ⟶ 1
↓^{Nm}  ↓^{Nm × id}  ↓^{Nm}
1 ⟶ (E_{2})_{0}^{×} ⟶ E_{2}^{×} × ℚ^{×} ⟶ S^{E_{2}} ⟶ 1.

Remark 4.3. The Serre group can be defined for all fields of finite degree over ℚ. If L contains a CM-field and E is the maximal such subfield, then Nm_{L/K}: S^{L} →^{≈} S^{E}; if not, then Nm_{L/ℚ}: S^{L} →^{≈} S^{ℚ} = ℚ^{×}.

Let (K, Φ) be a CM-type with K ⊂ ℂ. Write T = Res_{K/ℚ}(𝔾_{m}), and define μ_{Φ} ∈ X_{∗}(T) by the condition

[ρ] ∘ μ_{Φ} = \{ \begin{matrix}id & , ρ ∈ Φ \\ 1, & ρ ∉ Φ. \end{matrix}

Thus, μ_{Φ} is the map

\begin{matrix}ℂ^{×} & → & T (ℂ) = (K ⊗_{ℚ} ℂ)^{×} & = & ∏_{ϕ∈Φ} ℂ^{×} × ∏_{ϕ∉Φ} ℂ^{×} \\ z & ↦ & (z, …, z, 1, …, 1). \end{matrix}

The weight of μ_{Φ} is the map induced by x ↦ x^{−1}: ℚ^{×} ↪ K^{×}, which is defined over ℚ, and μ_{Φ} itself is defined over the reflex field of (K, Φ). There is therefore, for any CM-field E containing the reflex field of (K, Φ), a unique ℚ-rational homomorphism ρ_{Φ}: S^{E} → T such that μ_{Φ} = ρ_{Φ} ∘ μ^{E}. From now on, we assume E to be Galois over ℚ.

Lemma 4.4. (a) τμ_{Φ} = μ_{τΦ}, τ ∈ Aut(ℂ).

(b) Let τ ∈ Aut(ℂ) be such that τK = K, so that τ induces an automorphism τ̃ of T; then τ̃ ∘ μ_{Φ} = μ_{Φτ^{−1}}.

Proof. (a) Consider the canonical pairing

⟨·, ·⟩: X^{∗}(T) × X_{∗}(T) → ℤ.

By definition, for ρ ∈ Hom(K, ℂ),

⟨[ρ], μ_{Φ}⟩ = \{ \begin{matrix}1 & if ρ ∈ Φ \\ 0 & otherwise. \end{matrix}

For τ ∈ Aut(ℂ),

⟨[ρ], τμ_{Φ}⟩ = ⟨τ^{−1}[ρ], μ_{Φ}⟩ = ⟨[τ^{−1}ρ], μ_{Φ}⟩,

which equals ⟨[ρ], μ_{τΦ}⟩. (b)

[ρ] ∘ τ̃ ∘ μ_{Φ} = [ρτ] ∘ μ_{Φ} = \{ \begin{matrix}id & if & ρτ ∈ Φ \\ 1 & if & ρτ ∉ Φ. \end{matrix}

Thus (b) is clear.

Proposition 4.5. (a) For any τ ∈ Aut(ℂ), ρ_{Φ} ∘ τ̃^{−1} = ρ_{τΦ}.

(b) If τK = K, then τ̃ ∘ ρ_{Φ} = ρ_{Φτ^{−1}}.

Proof. (a) We shall show that τ̃^{−1} ∘ μ^{E} = τ(μ^{E}); from this it follows that

ρ_{Φ} ∘ τ̃^{−1} ∘ μ^{E} = ρ_{Φ} ∘ (τμ^{E})
= τ(ρ_{Φ} ∘ μ^{E})  (ρ_{Φ} is ℚ-rational)
= τ(μ_{Φ})  (definition of ρ_{Φ})
= μ_{τΦ}  (4.4a),

which implies that ρ_{Φ} ∘ τ^{−1} = ρ_{τΦ}. It remains to show that X^{∗}(τ̃^{−1} ∘ μ^{E}) = X^{∗}(τμ^{E}), but

X^{∗}(τ̃^{−1} ∘ μ^{E}) = X^{∗}(μ^{E}) ∘ X^{∗}(τ̃^{−1}) = (∑ n_{ρ}[ρ] ↦ ∑ n_{ρ}[ρτ^{−1}] ↦ n_{τ})

and

X^{∗}(τμ^{E}) = τ(X^{∗}(μ^{E})) = (∑ n_{ρ}[ρ] ↦ ∑ n_{ρ}[τ^{−1}ρ] ↦ n_{τ}).

(b)

τ̃ ∘ ρ_{Φ} ∘ μ^{E} = τ̃ ∘ μ_{Φ}  (definition of ρ_{Φ})
= μ_{Φτ^{−1}}  (4.4b).

Proposition 4.6. For E_{1} ⊃ E_{2},

S^{E_{1}}

[figure: ρ_{Φ}]

[figure: S^{E_{2}}; ^{Nm^{E_{1}/E_{2}}}; ρ_{Φ}; T]

commutes.

Proof. We have

((ρ_{Φ})_{2} ∘ Nm_{E_{1}/E_{2}}) ∘ μ^{E_{1}} = (ρ_{Φ})_{2} ∘ μ^{E_{2}} = μ_{Φ}.

Proposition 4.7. Let E be a CM-field, Galois over ℚ, and consider all maps ρ_{Φ} for Φ running through the CM-types on E; then ∩ Ker(ρ_{Φ}) = 1.

Proof. We have to show that ∑ Im(X^{∗}(ρ_{Φ})) = X^{∗}(S^{E}); but the left hand side contains ∑_{Φ}[ϕ] for all CM-types on E, and these elements generate X^{∗}(S^{E}).

Proposition 4.8. Let K_{1} ⊃ K_{2} be CM-fields, and let Φ_{1} and Φ_{2} be CM-types for K_{1} and K_{2} respectively such that Φ_{1}|K_{2} = Φ_{2}. Then, for any CM-field E containing the reflex field of (K_{1}, Φ_{1}), the composite of

S^{E} →^{ρ_{Φ_{2}}} K_{2}^{×} ↪ K_{1}^{×}

is ρ_{Φ_{1}}.

Proof. Let i: K_{2}^{×} ↪ K_{1}^{×} be the inclusion map. Then i ∘ μ_{Φ_{2}} = μ_{Φ_{1}} and so i ∘ ρ_{Φ_{2}} ∘ μ^{E} = i ∘ μ_{Φ_{2}} = μ_{Φ_{1}}, which shows that i ∘ ρ_{Φ_{2}} = ρ_{Φ_{1}}.

5. Definition of e^{E}

Proposition 5.1. Let E ⊂ ℂ be a CM-field, Galois over ℚ. Then there exists a unique map e^{E}: Aut(ℂ) → S^{E}(𝔸_{f})/S^{E}(ℚ) such that, for all CM-types (K, Φ) whose reflex fields are contained in E, e_{Φ}(τ) = ρ_{Φ}(e(τ)).

Proof. The existence of e^{E} will be shown in §7 and §8. The uniqueness follows from (4.7) for this shows that there is an injection S^{E} ^{⊂}^{(ρ_{Φ})}> ∏ T_{Φ} where T_{Φ} = Res_{E/ℚ} 𝔾_{m} and the product is over all CM-types on E. Thus

S^{E}(𝔸_{f})/S^{E}(ℚ) ↪ ∏ T_{Φ}(𝔸_{f})/T_{Φ}(ℚ) = ∏ 𝔸_{f,E}^{×}/E^{×},

and so any element a ∈ S^{E}(𝔸_{f})/S^{E}(ℚ) is determined by the set (ρ_{Φ}(a)).

Proposition 5.2. The family of maps e^{E}: Aut(ℂ) → S^{E}(𝔸_{f})/S^{E}(ℚ) has the following properties:

(a) e^{E}(στ) = τ̃^{−1}e^{E}(σ) · e^{E}(τ), σ, τ ∈ Aut(ℂ);

(b) if E_{1} ⊃ E_{2}, then

Aut(ℂ) \overline{^{e^{E_{1}}}} S^{E_{1}}(𝔸_{f})/S^{E_{1}}(ℚ)

[figure: e^{E_{2}}]

S^{E_{2}}(𝔸_{f})/S^{E_{2}}(ℚ)^{^{Nm}}

commutes.

(c) e^{E}(ι) = 1;

(d) e(τ) · ι̃e(τ) = 1, τ ∈ Aut(ℂ);

(e) e^{E}| Aut(ℂ/E) = 1.

Proof. (a) We have to check that ρ_{Φ}(e(στ)) = ρ_{Φ}(τ̃^{−1}e(σ) · e(τ)) for all (K, Φ). But ρ_{Φ}(e^{E}(στ)) = e_{Φ}(στ) and

ρ_{Φ}(τ̃^{−1}e^{E}(σ)e^{E}(τ)) = ρ_{Φ}(τ̃^{−1}e^{E}(σ))ρ_{Φ}(e^{E}(τ)) = ρ_{τΦ}(e^{E}(σ)) · ρ_{Φ}(e^{E}(τ)) = e_{τΦ}(σ)e_{Φ}(τ);

thus the equality follows from (3.4a).

(b) This follows from (4.6) and the definition of e^{E}.

(c) ρ_{Φ}(e^{E}(ι)) = e_{Φ}(ι) = 1 (by 3.4c), and so e^{E}(ι) = 1.

(d) This follows from (3.4d).

(e) Assume τ fixes E; then τΦ = Φ whenever E contains the reflex field of (K, Φ), and so ρ_{Φ}(e^{E}(τ)) = e_{Φ}(τ) = 1 by (3.4e).

Remark 5.3. (a) Define ε^{E}(τ) = e(τ^{−1})^{−1}; then the maps ε^{E} satisfy the same conditions (b), (c), (d), and (e) of (5.2) as e^{E}, but (a) becomes the condition ε^{E}(στ) = σ̃ε^{E}(τ) · ε^{E}(σ): ε^{E} is a crossed homomorphism.

(b) Condition (b) shows that e^{E} determines e^{E′} for all E′ ⊂ E. We extend the definition of e^{E} to all CM-fields E ⊂ ℂ by letting e^{E} = Nm_{E_{1}/E} ∘e^{E_{1}} for any Galois CM-field E_{1} containing E.

(c) Part (d) of (5.2) follows from the remaining parts, as is clear from the following diagram:

[figure: Aut(ℂ); e^{E}; S^{E}(𝔸_{f})/S^{E}(ℚ); 1+ι; S^{E}(𝔸_{f})/S^{E}(ℚ)]

[figure: e^{ℚ[i]}=1; Nm_{E/ℚ}]

[figure: S^{ℚ[i]}(𝔸_{f})/S^{ℚ[i]}(ℚ); ^{Nm^{E/ℚ[i]}}; Nm_{ℚ[i]/ℚ}; S^{ℚ}(𝔸_{f})/S^{ℚ}(ℚ); −w^{E}]

The right hand triangle is (4.2). (We can assume E ⊃ ℚ[i]; S^{ℚ[i]} = ℚ[i]^{×}, S^{ℚ} = ℚ^{×}). In his (original) letter to Langlands (see Deligne 1979), Deligne showed that the difference between the motivic Galois group and the Taniyama group was measured by a family of crossed homomorphisms (e^{E}) having properties (b), (c), and (e) of (5.2). After seeing Tate’s result he used the above diagram to show that his maps e^{E} had the same properties as Tate’s e_{Φ}(τ), namely, e^{E}(τ) · ιe^{E}(τ) = 1, e^{E}(τ)^{2} = 1.

6. Proof that e = 1

We replace e with τ ↦ e(τ^{−1})^{−1}. Proposition 6.1. Suppose there are given crossed homomorphisms e^{E}: Aut(ℂ) →

S^{E}(𝔸_{f})/S^{E}(ℚ), one for each CM-field E ⊂ ℂ, such that

(a) e^{E}(ι) = 1, all E;

(b) e^{E}| Aut(ℂ/E) = 1;

(c) if E_{1} ⊃ E_{2} then

Aut(ℂ/ℚ) \overline{^{e^{E_{1}}}} S^{E_{1}}(𝔸_{f})/S^{E_{1}}(ℚ)

[figure: e^{E_{2}}]

S^{E_{2}}(𝔸_{f})/S^{E_{2}}(ℚ)^{^{Nm^{E_{1}/E_{2}}}}

commutes.

Then e^{E} = 1 — i.e., e^{E}(τ) = 1 for all τ — for all E.

Proof. Clearly, it suffices to show that e^{E} = 1 for all sufficiently large E — in particular, for those that are Galois over ℚ.

The crossed homomorphism condition is that

e(στ) = σ̃e(τ) · e(σ).

Condition (b) implies that e^{E}(τ) = e^{E}(τ′) if τ|E = τ′|E. In particular, e^{E}(ιτ) = e^{E}(τι) for all τ ∈ Aut(ℂ). Since

\{ \begin{matrix}e^{E}(τι) & = τe^{E}(ι) · e^{E}(τ) & = e^{E}(τ) \\ e^{E}(ιτ) & = ιe^{E}(τ) · e^{E}(ι) & = ιe^{E}(τ) \end{matrix}

we conclude that ιe^{E}(τ) = e^{E}(τ).

Lemma 6.2. Assume that E is Galois over ℚ, and let ⟨ι⟩ be the subgroup of Gal(E/ℚ) generated by ι|E.

(a) There is an exact commutative diagram

1 ⟶ ℚ^{×} ⟶ S^{E}(ℚ)^{⟨ι⟩} ⟶ μ_{2}(E_{0}) ⟶^{Nm_{E_{0}/ℚ}} μ_{2}(ℚ)
↓  ↓  ↓  ↓
1 ⟶ 𝔸_{f}^{×} ⟶ S^{E}(𝔸_{f})^{⟨ι⟩} ⟶ μ_{2}(𝔸_{f,E_{0}}) ⟶^{Nm} μ_{2}(𝔸_{f})

where μ_{2}(R) denotes the set of square roots of 1 in a ring R.

(b) The canonical map

H^{1}(⟨ι⟩, S^{E}(ℚ)) → H^{1}(⟨ι⟩, S^{E}(𝔸_{f}))

is injective.

Proof. From (4.1) we obtain cohomology sequences

\begin{aligned}1 & → & E_{0}^{×} & → & E_{0}^{×} × ℚ^{×} & → & S^{E}(ℚ)^{⟨ι⟩} & → & μ_{2}(E_{0}) & \overline{>^{Nm}} & μ_{2}(ℚ) & → & H^{1}(⟨ι⟩, S^{E}(ℚ)) & → & E_{0}^{×}/E_{0}^{×2} \\ 1 & → & 𝔸_{f,E_{0}}^{×} & ∨ & → & 𝔸_{f,E_{0}}^{×} × 𝔸_{f}^{×} & ∨ & → & S^{E}(𝔸_{f})^{⟨ι⟩} & ∨ & → & μ_{2}(𝔸_{f,E_{0}}) & ∨ & \overline{>^{Nm}} & μ_{2}(𝔸_{f}) & ∨ & → & H^{1}(⟨ι⟩, S^{E}(𝔸_{f})) & ∨ & → & 𝔸_{f,E_{0}}^{×}/𝔸_{f,E_{0}}^{×2} & ∨ \end{aligned}

It is easy to extract from this the diagram in (a). For (b), let γ ∈ H^{1}(⟨ι⟩, S^{E}(ℚ)) map to zero in H^{1}(⟨ι⟩, S^{E}(𝔸_{f})). As E_{0}^{×}/E_{0}^{×2} → 𝔸_{f,E_{0}}^{×}/𝔸_{f,E_{0}}^{×2} is injective (an element of E_{0} that is a square in E_{0,v} for all finite primes is a square in E_{0}), we see that γ is the image of ±1 ∈ μ_{2}(ℚ). The map Nm_{E_{0}/ℚ}: μ_{2}(E_{0}) → μ_{2}(ℚ) sends −1 to (−1)^{[E_{0}:ℚ]}. If [E_{0}: ℚ] is odd, it is surjective, and therefore γ = 0. Suppose therefore that [E_{0}: ℚ] is even, and that γ is the image of −1. The assumption that γ maps to zero in H^{1}(⟨ι⟩, S^{E}(𝔸_{f})) then implies that −1 ∈ ℚ_{ℓ} is in the image of Nm: E_{0} ⊗ ℚ_{ℓ} → ℚ_{ℓ} for all ℓ; but this is impossible, since for some ℓ, [E_{0v}: ℚ_{ℓ}] will be even for one (hence all) v dividing ℓ.

Part (b) of the lemma shows that

S^{E}(𝔸_{f})^{⟨ι⟩}/S^{E}(ℚ)^{⟨ι⟩} = (S^{E}(𝔸_{f})/S^{E}(ℚ))^{⟨ι⟩}.

The condition ιe^{E}(τ) = e^{E}(τ) shows that e^{E} maps into the right hand group, and we shall henceforth regard it as mapping into the left hand group.

From part (a) we can extract an exact sequence

1 → 𝔸_{f}^{×}/ℚ^{×} →^{w} S^{E}(𝔸_{f})^{⟨ι⟩}/S^{E}(ℚ)^{⟨ι⟩} → μ_{2}(𝔸_{f,E_{0}})/μ_{2}(E_{0}).

Now assume that E ⊃ ℚ[i], so that E = E_{0}[i]. We show first that the image of e^{E}(τ) in μ_{2}(𝔸_{f,E_{0}})/μ_{2}(E_{0}) is 1. Let ε represent the image; then ε = (ε_{v}), ε_{v} = ±1, and ε itself is defined up to sign. We shall show that, for any two primes v_{1} and v_{2}, ε_{v_{1}} = ε_{v_{2}}. Choose a totally real quadratic extension E′_{0} of E_{0} in which v_{1} and v_{2} remain prime, and let E′ = E′_{0}[i]. Let ε′ represent the image of e^{E′}(τ) in μ_{2}(𝔸_{f,E′_{0}})/μ_{2}(E′_{0}). Then condition (c) shows that Nm_{E′_{0}/E_{0}} ε′ represents the image of e^{E}(τ), and so Nm_{E′_{0}/E_{0}} ε′ = ±ε. But if v′_{i}|v_{i}, then Nm_{E′_{0,v′_{i}}/E_{0,v_{i}}} = 1 for i = 1, 2.

It follows that e^{E} factors through w(𝔸_{f}^{×}/ℚ^{×}). Consider,

E:  1 ⟶ 𝔸_{f}^{×}/ℚ^{×} ⟶^{w} S^{E}(𝔸_{f})/S^{E}(ℚ)
↓^{id}  ↓
ℚ[i]:  1 ⟶ 𝔸_{f}^{×}/ℚ^{×} ⟶^{w} S^{ℚ[i]}(𝔸_{f})/S^{ℚ[i]}(ℚ).

According to (c), e^{E}(τ) maps to e^{ℚ[i]}(τ) under the right hand arrow, which according to (a) and (b), is 1. As e^{E}(τ) lies in 𝔸_{f}^{×}/ℚ^{×}, and the map from there into S^{ℚ[i]}(𝔸_{f})/S^{ℚ[i]}(ℚ) is injective, this shows that e^{E}(τ) = 1.

Remark 6.3. The argument used in the penultimate paragraph of the above proof is that used by Shih 1976, p101, to complete his proof of his special case of (1.1). For the argument in the final paragraph, cf. 5.3c. These two arguments were all that was lacking in the original version Deligne 1979b of Deligne 1982.

7. Definition of f^{E}

We begin the proof of (5.1) by showing that there is a universal f, giving rise to the f_{Φ}.

Let E ⊂ ℂ. The Weil group W_{E/ℚ} of E/ℚ fits into an exact commutative diagram:

1  \overline{>} 𝔸_{E}^{×}/E^{×}  \overline{>} W_{E/ℚ}  \overline{>} Hom(E, ℂ) \overline{>} 1
∥
1 \overline{>} Gal(E^{ab}/E)^{∨ ∨^{rec^{E}}}  \overline{>} Hom(E^{ab}, ℂ) ∨ ∨  \overline{>} Hom(E, ℂ) \overline{>} 1

(see Tate 1979). Assume that E is totally imaginary. Then E_{∞}^{×}E^{×} ⊂ Ker(rec_{E}), and so we can divide out by this group and its image in W_{E/ℚ} to obtain the exact commutative diagram:

1  \overline{>} 𝔸_{f,E}^{×}/E^{×}  \overline{>} W_{f,E/ℚ}  \overline{>} Hom(E, ℂ) \overline{>} 1
∥
1 \overline{>} Gal(E^{ab}/E)^{∨ ∨^{r_{E}}}  \overline{>} Hom(E^{ab}, ℂ) ∨ ∨  \overline{>} Hom(E, ℂ) \overline{>} 1

Assume now that E is a CM-field Galois over ℚ. The cocharacter μ^{E} is defined over E, and gives rise to a map μ^{E}(R): R^{×} → S^{E}(R) for any E-algebra R. Choose elements w_{σ} ∈ W_{E/ℚ}^{f}, one for each σ ∈ Hom(E, ℂ), such that

w_{σ}|E = σ, w_{ισ} = ι̃w_{σ} all σ,

where ι̃ maps to ι ∈ Hom(E^{ab}, ℂ) (cf. §2). Let τ ∈ Aut(ℂ) and let τ̃ ∈ W_{f,E/ℚ} map to τ|E^{ab}. Then w_{τσ}^{−1} ∘ τ̃ ∘ w_{σ} ∈ 𝔸_{f,E}, and we define

f(τ) = ∏_{σ∈Hom(E,ℂ)}(σ^{−1}μ^{E})(w_{τσ}^{−1}τ̃w_{σ}) mod S^{E}(E).

Thus f is a map Aut(ℂ) → S^{E}(𝔸_{f,E})/S^{E}(E).

Proposition 7.1. Let (K, Φ) be a CM-type whose reflex field is contained in E, and let T = Res_{K/ℚ} 𝔾_{m}. Identify T (𝔸_{f})/T (ℚ) with a subgroup of T (𝔸_{f,E})/T (E). Then

ρ_{Φ}(f(τ)) = f_{Φ}(τ).

Proof. Because of (4.8), it suffices to show this with K = E.

Lemma 7.2. With the above notations,

f_{Φ}(τ) = ∏_{ϕ∈Φ} w_{τϕ}^{−1} ∘ τ̃ ∘ w_{ϕ} mod E^{×}.

Proof. Let f′ denote the right hand side. Then r_{E}(f′) = F_{Φ}(τ) (obviously), and the same argument as in the proof of (2.3) shows that f′ · ιf′ = χ(τ).

We now assume that E = K, E/ℚ Galois. Write i for the map T (ℚ) → T (E) induced by ℚ ↪ E; then, for any ρ ∈ Hom(E, ℂ) and a ∈ T (ℚ) = E^{×}, [ρ](i(a)) = ρa. Thus [ρ](i(f_{Φ}(τ))) = ρf_{Φ}(τ) = f_{Φρ^{−1}}(τ) by (2.6b). On the other hand,

[ρ](ρ_{Φ}(f(τ))) = [ρ] ∏_{σ} ρ_{Φ} ∘ (σ^{−1}μ^{E})(w_{τσ}^{−1}τ̃w_{σ})
= [ρ] ∏_{σ} σ^{−1}(ρ_{Φ} ∘ μ^{E})(w_{τσ}^{−1}τ̃w_{σ})
= [ρ](∏_{σ} σ^{−1}μ_{Φ}(w_{τσ}^{−1}τ̃w_{σ}))
= ∏_{σ}([ρ] ∘ μ_{σ^{−1}Φ})(w_{τσ}^{−1}τ̃w_{σ}) ( by 4.4a)
= ∏_{σ such that ρ∈σ^{−1}Φ} w_{τσ}^{−1}τ̃w_{σ}
= ∏_{σ∈Φρ^{−1}} w_{τσ}^{−1}τ̃w_{σ}
= f_{Φρ^{−1}}(τ).

Corollary 7.3. (a) f(τ) depends only on E and τ; we have therefore defined

maps f^{E}: Aut(ℂ) → S^{E}(𝔸_{f,E})/S^{E}(E), one for each CM-field (Galois over ℚ);

(b) f^{E}(στ) = τ̃^{−1}f^{E}(σ) · f^{E}(τ), σ, τ ∈ Aut(ℂ);

(c) if E_{1} ⊃ E_{2}, then

Aut(ℂ) \overline{^{e^{E_{1}}}} S^{E_{1}}(𝔸_{f})/S^{E_{1}}(ℚ)

[figure: e^{E_{2}}]

S^{E_{2}}(𝔸_{f})/S^{E_{2}}(ℚ)^{}

commutes;

(d) f^{E}(ι) = 1;

(e) f^{E}(τ) · ĩf^{E}(τ) = w^{E}(τ)^{−1}; (f) σf^{E}(τ) = f^{E}(τ) for all σ ∈ Gal(E/ℚ).

Proof. (a) f(τ) is the unique element of S^{E}(𝔸_{f,E})/S^{E}(E) such that ρ_{Φ}(f(τ)) = f_{Φ}(τ) for all (K, Φ). (Cf. the proof of the uniqueness of e^{E} in (5.1).)

(b), (c), (d), (e). These are proved as (a), (b), (c), (d) of (5.2).

(f).ρ_{Φ}(σf^{E}(τ)) = σ(ρ_{Φ}(f^{E}(τ))) = σf_{Φ}^{E}(τ) = f_{Φ}^{E}(τ).

Remark 7.4. Let w̄_{σ} ∈ W_{f,E/ℚ} be such that

w̄_{σ}|E = σ, w̄_{σι} = w̄_{σ}ι̃.

Then (Langlands 1979; Milne and Shih 1982a), b̄(τ) is defined by

b̄(τ) = ∏_{σ∈Gal(E/ℚ)}σμ^{E}(w̄_{σ}τ̃w̄_{στ}^{−1}) mod S^{E}(E).

Let w_{σ} = w̄_{σ^{−1}}^{−1}; then w_{σ}|E = σ and w_{ισ} = ĩw_{σ}; moreover,

b̄(τ^{−1})^{−1} = ∏_{σ∈Gal(E/ℚ)}σμ^{E}(w_{σ^{−1}τ}^{−1} τ̃w_{σ^{−1}}) = f^{E}(τ).

Thus, in the notation of Milne and Shih 1982a, 2.9, f^{E}(τ) = β̄(τ).

8. Definition of g^{E}

We complete the proof of (5.1) by showing that there is a universal g giving rise to all g_{Φ}. For simplicity, we shall assume that E is Galois over ℚ — for a non Galois field, g^{E} can be defined as the norm of the element from the Galois closure.

Proposition 8.1. Let E ⊂ ℂ be a CM-field. There exists a unique map g^{E}: Aut(ℂ) → S^{E}(𝔸_{f,E})/S^{E}(E) with the following property: for any CM-type (K, Φ) whose reflex field is contained in E,

ρ_{Φ}(g^{E}(τ)) = g_{Φ}(τ)

in T (𝔸_{f,E})/T (E), where T = Res_{K/ℚ}(𝔾_{m}).

Proof. The uniqueness follows from (4.7). For the existence, we need the notion of a Hodge cycle.

For any variety X over ℂ, write H^{s}(X, ℚ)(r) = H^{s}(X, (2πi)^{r}ℚ) (cohomology with respect to the complex topology). A Hodge cycle on A is an element s ∈ H^{2p}(A^{k}, ℚ)(p), some p, k, that is of type (p, p), i.e., under the embedding (2πi)^{p}ℚ ↪ ℂ, s maps into H^{p,p} ⊂ H^{2p}(X, ℂ). Recall that H^{r}(A^{k}, ℚ) = ⋀^{r}(⊕^{k}H_{1}(A, ℚ)^{∨}), and so GL(H_{1}(A, ℚ)) acts by transport of structure on H^{r}(A^{k}, ℚ). The Mumford-Tate group MT (A) of A is the largest ℚ-rational algebraic subgroup of GL(H_{1}(A, ℚ)) such that MT (A)(ℚ) is the set of α ∈ GL(H_{1}(A, ℚ)) for which there exists a ν(α) ∈ ℚ^{×} such that αs = ν(α)^{p}s for any Hodge cycle s on A (of type (p, p)).

Lemma 8.2. Assume A is of CM-type (K, Φ), where the reflex field of (K, Φ) is contained in E. Then the image of ρ_{Φ}: S^{E} → K^{×} ⊂ GL(H_{1}(A, ℚ)) is equal to MT (A).

Proof. Cf. Deligne 1982a, 3.4.

Write H^{2p}(A^{k}, 𝔸_{f})(p) = H^{2p}(A^{k}, ℚ)(p)⊗𝔸_{f}. Then there is a canonical isomorphism

H^{2p}(A^{k}, 𝔸_{f})(p) →^{≈} ⋀^{r}(⊕^{k}V_{f}(A)^{∨})

and so the action of Aut(ℂ) on V_{f}(A) gives rise to an action on H^{2p}(A^{k}, 𝔸_{f})(p). We shall need to use the following important result fo Deligne.

Theorem 8.3. Let s ∈ H^{2p}(A^{k}, ℚ)(p) be a Hodge cycle on A, and let s_{f} be the image of s in H^{2p}(A^{k}, 𝔸_{f})(p); then for any τ ∈ Aut(ℂ) there exists a Hodge cycle s_{1} on τA whose image in H^{2p}(A^{k}, 𝔸_{f})(p) is τs_{f}.

Proof. See Deligne 1982a.

The cycle s_{1} of the theorem is uniquely determined, and will be written τs.

Proposition 8.4. With the notations of (8.2), there exists a K-linear isomorphism α: H_{1}(A, E) →^{≈} H_{1}(τA, E) such that α(s) = ν(α)^{p}τ(s) for all Hodge cycles s on A (of type (p, p)).

Proof. For any ℚ-algebra R, let

P (R) = {α: H_{1}(A, R) →^{≈} H_{1}(τA, R) | α(s) = ν(α)^{p}τ(s), all s}.

Then P (R) is either empty or is a principal homogeneous space over MT (A)(R). Thus P is either the empty scheme or is a principal homogeneous space over MT (A). The existence of τ: H_{1}(A, 𝔸_{f}) → H_{1}(τA, 𝔸_{f}) in P (𝔸_{f}) shows that the latter is true. It therefore corresponds to an element of H^{1}(ℚ, MT (A)). But MT (A)_{E} ≈ 𝔾_{m} ×⋯× 𝔾_{m}, and so H^{1}(E, MT (A)) = 0 by Hilbert’s Theorem 90.

Both (8.2) and (8.4) obviously also apply to products of abelian varieties of CMtype. Let A = ∏ A_{Φ}, where Φ runs through the CM-types on E and A_{Φ} is of type (E, Φ). Then ρ: S^{E} →^{≈} MT (A). Choose α as in (8.4). Then

V_{f}(A) ⊗ E →^{τ} V_{f}(τA) ⊗ E →^{(α⊗1)^{−1}} V_{f}(A) ⊗ E

is an 𝔸_{f,E}-linear isomorphism and sends a Hodge cycle s of type (p, p) to ν^{p}s, some ν ∈ 𝔸_{f,E}^{×}. Therefore it is multiplication by an element g ∈ MT (A)(𝔸_{f,E}) = S^{E}(𝔸_{f,E}). The class g(τ) of g in S^{E}(𝔸_{f,E})/S^{E}(E) has the properties required for (8.1).

The map g: Aut(ℂ) → S^{E}(𝔸_{f,E})/S^{E}(E) has the same properties as those listed for f in (7.3). In particular, g(τ) is fixed by Gal(E/ℚ). Set

e(τ) = \frac{g(τ)}{f(τ)}.

Then e(τ) ∈ (S^{E}(𝔸_{f,E})/S^{E}(E))^{Gal(E/ℚ)}, and it remains to show that it lies in S^{E}(𝔸_{f})/S^{E}(ℚ) — the next proposition completes the proof.

Proposition 8.5. e(τ) lies in S^{E}(𝔸_{f})/S^{E}(ℚ).

Proof. There is a cohomology sequence

0 → S^{E}(ℚ) → S^{E}(𝔸_{f}) → (S^{E}(𝔸_{f,E})/S^{E}(E))^{Gal(E/ℚ)} → H^{1}(ℚ, S^{E}).

Thus, we have to show that the image γ of e(τ) in H^{1}(ℚ, S^{E}) is zero. But H^{1}(ℚ, S^{E}) ↪ ∏_{ℓ,∞} H^{1}(ℚ_{ℓ}, S^{E}), as follows easily from (4.1), and the image of e(τ) in H^{1}(ℚ_{ℓ}, S^{E}) is obviously zero for all finite ℓ. It remains to check that the image of γ in H^{1}(ℝ, S^{E}) is zero. Let

T = {a ∈ ∏_{CM-types on E} E^{×} | a · ιa ∈ ℚ^{×}} ( torus over ℚ).

Lemma 8.6. The image of γ in H^{1}(ℚ, T) is zero.

Proof. In the proof of (3.6) it shown that the image of e in T (𝔸_{f,E})/T (E) lifts to an element ε ∈ T (𝔸_{f}). The image of γ in H^{1}(ℚ, T) is represented by the cocycle σ ↦ σε − ε = 0.

Lemma 8.7. The map H^{1}(ℝ, S^{E}) → H^{1}(ℝ, T) is injective.

Proof. There is a norm map a ↦ a·ιa: T ↠ 𝔾_{m}, and we define ST and SMT (A) to make the rows in

1 \overline{>} SMT (A) \overline{>} MT (A) \overline{>} 𝔾_{m}  \overline{>} 1
∥
1  \overline{>} ST^{∨ ∩}  \overline{>} T^{∨ ∩}  \overline{>} 𝔾_{m}  \overline{>} 1

exact. (Here A = ∏ A_{Φ}.) This diagram gives rise to an exact commutative diagram

ℝ^{×} \overline{>} H^{1}(ℝ, SMT) \overline{>} H^{1}(ℝ, MT) \overline{>} 0
∥
ℝ^{×}  \overline{>} H^{1}(ℝ, ST) ∨  \overline{>} H^{1}(ℝ, T) ∨  \overline{>} 0.

Note that ST (and hence SMT) are anisotropic over ℝ; hence, H^{1}(ℝ, SMT) = SMT (ℂ)_{2} and H^{1}(ℝ, ST) = ST (ℂ)_{2}, and so H^{1}(ℝ, SMT) ↪ H^{1}(ℝ, SMT). The five-lemma now completes the proof.

See also Milne and Shih, 1982b, §5.

Remark 8.8. It seems to be essential to make use of Hodge cycles, and consequently Shimura varieties (which are used in the proof of (8.3)), in order to show the e_{Φ}(τ) have the correct functorial properties. Note that Shih (1976) also needed to use Shimura varieties to prove his case of the theorem.

9. Re-statement of the Theorem

The following statement of the main theorem of complex multiplication first appeared (as a conjecture) in Milne and Shih 1979.

Theorem 9.1. Let A be an abelian variety of CM-type (K, Φ); let τ ∈ Aut(ℂ), and let f ∈ f(τ). Then

(a) τA is of type (K, τΦ); (b) there is an K-linear isomorphism α: H_{1}(A, E) → H_{1}(τA, E) where E is the

reflex field of (K, Φ), such that

(i) α(s) = ν(α)^{p}τ(s), for all Hodge cycles s on A, where ν(α) ∈ ℚ^{×} and 2p is the degree of s;

(ii)

V_{f}(A) ⊗ E \overline{^{ρ_{Φ}(f)}} V_{f}(A) ⊗ E

[figure: τ]

V_{f}(τA) ⊗ E^{^{α⊗1}}
commutes (note that ρ_{Φ}(f) ∈ 𝔸_{f,K⊗E}^{×}).

Proof. The theorem is true (by definition) if f(τ) is replaced by g(τ), but we have shown that g(τ) = f(τ).

Remark 9.2. Let T be a torus such that

MT (A) ⊂ T ⊂ {a ∈ K^{×} | a · ιa ∈ ℚ^{×}}

and let h be the homomorphism defining the Hodge structure on H_{1}(A, ℝ). Then the Shimura variety Sh(T, {h}) is, in a natural way, a moduli scheme, and the (new) main theorem of complex multiplication gives a description of the action of Aut(ℂ) on Sh(T, {h}) (see Milne and Shih 1982, §6).

Remark 9.3. Out of his study of the zeta functions of Shimura varieties, Langlands (1979) was led to a conjecture concerning the conjugates of Shimura varieties. The conjecture is trivial for the Shimura varieties associated with tori, but in Milne and Shih 1982b it is shown that for groups of symplectic similitudes the conjecture is equivalent to (9.1). It is also shown (ibid.) that the validity of the conjecture for a Shimura variety Sh(G, X) depends only on (G^{der}, X^{+}). Thus (ibid.) similar methods to those used in Deligne 1979a can be used to prove Langlands’s conjecture for exactly those Shimura varieties for which Deligne proves the existence of canonical models in that article.

10. The Taniyama Group

By an extension of Gal(ℚ^{al}/ℚ) by S^{E} with finite-adèlic splitting, we mean an exact sequence

1 → S^{E} → T^{E} →^{π^{E}} Gal(ℚ^{al}/ℚ) → 1

of pro-algebraic groups over ℚ (Gal(ℚ^{al}/ℚ) is to be regarded as a constant proalgebraic group) together with a continuous homomorphism sp^{E}: Gal(ℚ^{al}/ℚ) → T^{E}(𝔸_{f}) such that sp^{E} ∘ π^{E} = id. We always assume that the action of Gal(ℚ^{al}/ℚ) on S^{E} given by the extension is the natural action. Assume E ⊂ ℂ is Galois over ℚ, and a CM-field.

Proposition 10.1. (a) Let (T^{E}, sp^{E}) be an extension of Gal(ℚ^{al}/ℚ) by S^{E} with finite-adèlic splitting. Choose a section a^{E}: Gal(ℚ^{al}/ℚ) → (T^{E})_{E} that is a morphism of pro-algebraic groups. Define h(τ) ∈ S^{E}(𝔸_{f,E})/S^{E}(E) to be the class of sp^{E}(τ) · a^{E}(τ)^{−1}.

(i) h(τ) is well-defined; (ii) σh(τ) = h(τ), σ ∈ Gal(E/ℚ); (iii) h(τ_{1}τ_{2}) = h(τ_{1}) · τ̃_{1}h(τ_{2}), τ_{1}, τ_{2} ∈ Gal(ℚ^{al}/ℚ); (iv) h lifts to a continuous map h′: Gal(ℚ^{al}/ℚ) → S^{E}(𝔸_{f,E}) such that the map

(τ_{1}, τ_{2}) ↦ d_{τ_{1},τ_{2}} =^{df} h′(τ_{1}) · τ̃_{1}h′(τ_{2}) · h′(τ_{1}τ_{2})^{−1} is locally constant.

(b) Let h: Gal(ℚ^{al}/ℚ) → S^{E}(𝔸_{f,E})/S^{E}(E) be a map satisfying conditions (i), (ii), (iii), (iv); then h arises from a unique extension of Gal(ℚ^{al}/ℚ) by S^{E} with finite-adèlic splitting.

Proof. Easy; see Milne and Shih 1982a, §2.

Let S = \underleftarrow{lim} S^{E}, where E runs through the CM-fields E ⊂ ℂ that are Galois over ℚ. By an extension of Gal(ℚ^{al}/ℚ) by S with finite-adèlic splitting, we mean a projective system of extensions of Gal(ℚ^{al}/ℚ) by S^{E} with finite-adélic splitting, i.e., a family

1 ⟶ S^{E_{1}} ⟶ T^{E_{1}} ⟶ Gal(ℚ^{al}/ℚ) ⟶ 1
↓^{Nm}^{E_{1}/E_{2}}  ↓^{Nm}^{E_{1}/E_{2}}  ↓^{id}
1 ⟶ S^{E_{2}} ⟶ T^{E_{2}} ⟶ Gal(ℚ^{al}/ℚ) ⟶ 1
T^{E_{1}}(𝔸_{f})
[figure: sp^{E_{1}}]
Gal(ℚ^{al}/ℚ)
[figure: sp^{E_{2}}]
T^{E_{2}}(𝔸_{f})^{^{Nm^{E_{1}/E_{2}}}}

of commutative diagrams.

Theorem 10.2. Let T_{1} and T_{2} be two extensions of Gal(ℚ^{al}/ℚ) by S with finiteadèlic splittings. Assume:

(a) for each E, and i = 1, 2, there exists a commutative diagram

1  \overline{>} S^{E}  \overline{>}_{E}T_{i}^{E}  \overline{>} Gal(ℚ^{al}/ℚ)  \overline{>} 1
∥
1  \overline{>} S^{E}  \overline{>} M^{E} ∨ \overline{>} Gal(ℚ^{al}/E)^{ab} ∨ ∨  \overline{>} 1

compatible with the finite-adèlic splittings, where _{E}T^{E} is the inverse image of Gal(ℚ^{al}/ℚ) in T^{E} and the lower row is the extension constructed by Serre (1968, II).

(b) for each τ ∈ Gal(ℚ^{al}/ℚ), π_{1}^{−1}(τ) ≈ π_{2}^{−1}(τ) as principal homogeneous spaces over S^{E}.

(c) sp^{E}(ι) ∈ T_{i}^{E}(ℚ), i = 1, 2.

Then there is a unique family of isomorphisms ϕ^{E}: T_{1}^{E} → T_{2}^{E} making the following diagrams commute:

1 ⟶ S^{E} ⟶ T_{1}^{E} ⟶ Gal(ℚ^{al}/ℚ) ⟶ 1
↓^{id}  ↓^{ϕ^{E}}  ↓^{id}
1 ⟶ S^{E} ⟶ T_{2}^{E} ⟶ Gal(ℚ^{al}/ℚ) ⟶ 1

T_{1}^{E_{1}} ⟶^{Nm_{E_{1}/E_{2}}} T_{1}^{E_{2}}
↓^{ϕ^{E_{1}}}  ↓^{ϕ^{E_{2}}}
T_{2}^{E_{2}} ⟶^{Nm_{E_{1}/E_{2}}} T_{2}^{E_{2}}

T_{1}^{E}(𝔸_{f}) ⟵^{sp_{1}^{E}} Gal(ℚ^{al}/ℚ)
↓  ↓^{id}
T_{2}^{E}(𝔸_{f}) ⟵^{sp_{2}^{E}} Gal(ℚ^{al}/ℚ)

Proof. Let (h_{1}^{E}) and (h_{2}^{E}) be the families of maps corresponding as in (10.1a) to T_{1} and T_{2}. The hypotheses of the theorem imply that the family (e^{E}), where e^{E} = h_{1}^{E}/h_{2}^{E}, satisfies the hypotheses of (6.1). Thus h_{1}^{E} = h_{2}^{E} for all E, and we apply (10.1b).

Definition 10.3. The extension corresponding to the family of maps (f^{E}) (rather, τ ↦ f^{E}(τ^{−1})^{−1}) defined in (7.3) is called the Taniyama group.

Remark 10.4. In (1982b), Deligne proves the following:

(a) let T′ be the group associated with the Tannakian category of motives over ℚ generated by Artin motives and abelian varieties of potential CM-type; then T′ is an extension of Gal(ℚ^{al}/ℚ) by S with finite-adèlic splitting in the sense defined above. (From a more naive point of view, T′ is the extension defined by the maps (g^{E}) of §8.)

(b) Theorem 10.2, by essentially the same argument as we have given in §6, except expressed directly in terms of the extensions rather than cocycles.

These two results combine to show that the motivic Galois group is isomorphic to the explicitly constructed Taniyama group (as extensions with finite-adèlic splitting). This can be regarded as another statement of the (new) main theorem of complex multiplication.

Note however that without the Taniyama group, Deligne’s result says very^{7} little. This is why I have included Langlands as one of the main contributors^{8} to the proof of (1.1) even though he never explicitly considered abelian varieties with complex multiplication (and neither he nor Deligne explicitly considered a statement like (1.1)).

7It is only a uniqueness result: it says that there is at most extension consistent with the theorem of Shimura and Taniyama; Langlands wrote down an explicit extension with this property.

8Probably Shih and Tate should also be included.

11. Zeta Functions

Lemma 11.1. There exists a commutative diagram

\begin{aligned}T^{E}(ℂ) & <^{sp_{∞}^{E}} & W_{ℚ} \\ T^{E}(ℚ) & ^{^{∪} ∧} & \overline{>} & Gal(ℚ^{al}/ℚ) & ∨ \end{aligned}

where W_{ℚ} is the Weil group of ℚ and T is the Taniyama group.

Proof. Easy; see Milne and Shih 1982a, 3.17.

Theorem 11.2. Let A be an abelian variety over ℚ of potential CM-type (K, Φ). Let E be a CM-field containing the reflex field of (K, Φ). Then there exists a representation ρ: T^{E} → Aut(H_{1}(A_{ℂ}, ℚ)) such that

(a) ρ_{f} =^{df} ρ ∘ sp^{E}: Gal(ℚ^{al}/ℚ) → Aut(V_{f}(A)) describes the action of Gal(ℚ^{al}/ℚ) on V_{f}(A);

(b) L(s, A/ℚ) = L(s, ρ_{∞}) where ρ_{∞} = ρ ∘ sp_{∞}^{E} is a complex representation of W_{E}.

Proof. The existence of ρ is obvious from the interpretation of T as the motivic Galois group M (see 10.4a) or, more naively, as the extension corresponding to (g^{E}(τ^{−1})^{−1}).

Remark 11.3. The proof of (11.2) does not require the full strength of Deligne’s results, and in fact is proved by Deligne (1979b). Subsequently Yoshida (1981) found another proof that L(s, A/ℚ) = L(s, ρ_{∞}) for some complex representation ρ_{∞} of W_{ℚ}. When Gal(ℚ^{al}/ℚ) stabilizes K ⊂ End(A_{ℚ^{al}}) ⊗ ℚ, this last result was proved independently by Milne (1972) (all primes) and Shimura (1971) (good primes only).

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Deligne, P., Hodge cycles on abelian varieties, 1982a. In: Deligne et al.1982, pp 9–100.

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ABELIAN VARIETIES WITH COMPLEX MULTIPLICATION (FOR PEDESTRIANS) 25

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Shimura, G., Arithmetic Theory of Automorphic Functions, Princeton U. P., 1971. Shimura, G., On the zeta-function of an abelian variety with complex multiplica-

tion, Annals of Math. 94 (1971), 504–533.

Tate, J., Number theoretic background, in Automorphic forms, representations and L-functions (Proc. Sympos.Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, 3–26, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979.

Tate, J., On conjugation of abelian varieties of CM-type, 8pp, April 1981. Yoshida, H., Abelian varieties with complex multiplication and representations of

the Weil groups. Annals of Math. 114 (1981), 87–102.

September 19, 1981.

Addendum (June 1998)

The sections of the Addendum are largely independent.

12. The Origins of the Theory of Complex Multiplication for Abelian Varieties of Dimension Greater Than One

On this topic, one can not do better than to quote Weil’s commentary (Œuvres, Vol II, pp 541–542) on his articles in the Proceedings^{9} of the International Symposium on Algebraic Number Theory, held in Tokyo and Nikko, September 8–13, 1955.

Comme contribution au colloque, j’apportais quelques idées que je croyais neuves sur l’extension aux variétés abéliennes de la théorie classique de la multiplication complexe. Comme chacun sait, Hecke avait eu l’audace, stupéfiante pour l’époque, de s’attaquer à ce problème dès 1912; il en avait tiré sa thèse, puis avait poussé son travail assez loin pour découvrir des phénomènes que lui avaient paru inexplicables, après quoi il avait abandonné ce terrain de recherche dont assurément l’exploration était prématurée. En 1955, à la lumière des progrès effectués en géométrie algébrique, on pouvait espérer que la question était mûre.

Elle l’était en effet; à peine arrivé à Tokyo, j’appris que deux jeunes japonais venaient d’accomplir sur ce même sujet des progrès décisifs. Mon plaisir à cette nouvelle ne fut un peu tempéré que par ma crainte de n’avoir plus rien à dire au colloque. Mais il apparut bientôt, d’abord que Shimura et Taniyama avaient travaillé indépendamment de moi et même indépendamment l’un de l’autre, et surtout que nos résultats à tous trois, tout en ayant de larges parties communes, se complétaient mutuellement. Shimura avait rendu possible la réduction modulo 𝔭 au moyen de sa théorie des intersections dans les variétés définies sur un anneau local (Am. J. of Math. 77 (1955), pp. 134–176); il s’en était servi pour l’étude de variétés abéliennes à multiplication complexe, bien qu’initialement, à ce qu’il me dit, il eût plutôt eu en vue d’autres applications. Taniyama, de son côté, avait concentré son attention sur les fonctions zêta des variétés en question et principalement des jacobiennes, et avait généralisé à celles-ci une bonne partie des résultats de Deuring sur le cas elliptique. Quant à ma contribution, elle tenait surtout à l’emploi de la notion de “variété polarisée”; j’avais choisi ce terme, par analogie avec “variétés orientées” des topologues, pour désigner une structure supplémentaire qu’on peut mettre sur une variété complète et normale quand elle admet un plongement projectif. Faute de cette structure, la notion de modules perd son sens.

Il fut convenu entre nous trois que je ferais au colloque un exposé général ([Weil 1956b]) ésquissant à grands traits l’ensemble des résultats obtenus, exposé qui servirait en même temps d’introduction aux communications de Shimura et de Taniyama; il fut entendu aussi que par la suite ceux-ci rédigeraient le tout avec des démonstrations détaillées.

9This is the same conference where Taniyama gave his somewhat enigmatic statement of the Taniyama conjecture.

Leur livre a paru en 1961 sous le titre Complex multiplication of abelian varieties and its application to number theory (Math. Soc. of Japan, Tokyo); mais Taniyama était mort tragiquement en 1958, et Shimura avait dû l’achever seul.

D’autre part, tout en restant loin des résultats de Taniyama sur les fonctions zêta des variétés “de type CM” (comme on dit à présent), j’avais aperc.u le rôle que devaient jouer dans cette théorie certains caractères de type (A_{0})”, ainsi que les caractères à values 𝔓-adiques qu’ils permettent de définir (cf. [Weil 1955b], p6). Je trouvai là une première explication du phénomène qui avait le plus étonné Hecke; il consiste en ce que, dès la dimension 2, les modules et les points de division des variétés de type CM définissent en géneral des extensions abéliennes, non sur le corps de la multiplication complexe, mais sur un autre qui lui est associé. Ce sujet a été repris et plus amplement développé par Taniyama (J. Math. Soc. Jap. 9 (1957), pp. 330–366); cf. aussi [Weil 1959].

As mentioned in the text, the first theorem extending the Main Theorem of Complex Multiplication to automorphisms not fixing the reflex field was that of Shih 1976. This theorem of Shih was used in Milne and Shih 1981 to give an explicit description of the involution defined by complex conjugation on the points of Shimura variety whose reflex field is real (Conjecture of Langlands 1979, p 234).

Apparently, it was known to Grothendieck, Serre, and Deligne in the 1960s that the conjectural theory of motives had as an explicit consequence the existence of a Taniyama group — these ideas inform the presentation in Chapters I and II of Serre 1968 — but they were unable to construct such a group. It was not until 1977, when Langlands’s efforts to understand the conjugates of Shimura led him to define his cocycles, that the group could be constructed (and it was Deligne who recognized that Langlands’s cocycles answered the earlier problem).

The rest of the story is described in the text of the article.

13. Zeta Functions of Abelian Varieties of CM-type

In this section I explain the elementary approach (Milne 1972), not using the theorems in the first part of this article, to the zeta function of abelian varieties of CM-type.

First some terminology: For abelian varieties A and B over a field k, Hom(A, B) denotes the group of homomorphisms A → B defined over k, and Hom^{0}(A, B) = Hom(A, B) ⊗_{ℤ} ℚ. Similar notations are used for endomorphisms. An abelian variety over k is simple if it contains no nonzero proper abelian subvariety defined over k, and it is absolutely simple if it is simple^{10} over k^{al}. An abelian variety A over a field k of characteristic zero is said to be of CM-type if its Mumford-Tate group is a torus. Thus, A is of CM-type if, for each simple isogeny factor B of A_{k^{al}}, End^{0}(B) is a CMfield of degree 2 dim B over ℚ. For an abelian variety A over a number field k ⊂ ℂ and finite prime v of k, the polynomial

10An older terminology, based on Weil’s Foundations, uses “simple” where we use “absolutely simple”, see, for example, Lang 1983 or Shimura 1998.

P_{v}(A, T) = det(1 − F_{v}T |V_{ℓ}(A)^{I_{v}})

where ℓ is any prime number different from the characteristic of the residue field at v, I_{v} is the inertia group at a prime v′|v, and F_{v} is a Frobenius element in the quotient of the decomposition group at v′ by I_{v} — it is known that P_{v}(A, T) is independent of the choice of ℓ, v′, and F_{v}. Finally, the zeta function of A is

ζ(A, s) = ∏_{v} \frac{1}{P_{v}(ℕ(v)^{−s})}

where v runs over all finite primes of k and ℕ(v) is the order of the residue field at v. Clearly ζ(A, s) depends only on the isogeny class of A, and if A is isogenous to A_{1} × ⋯ × A_{m}, then ζ(A, s) = ∏_{i=1}^{m} ζ(A_{i}, s).

Case that all endomorphisms of A are defined over k. In this subsection, A is an abelian variety of CM-type such that End(A) = End(A_{k^{al}}). Because ζ(A, s) depends only on the isogeny class of A, we may suppose that A is isotypic, i.e., that it is isogenous to a power of a simple abelian variety. Then there exists a CM-field K ⊂ End^{0}(A) of degree 2 dim A over ℚ.

The tangent space T to A is a finite-dimensional vector space over ℚ on which both k and K act. Since K acts k-linearly, the actions commute. An element α ∈ k^{×} defines an automorphism of T viewed as K-vector space, whose determinant we denote ψ_{0}(α). Then ψ_{0}: k^{×} → K^{×} is a homomorphism. Let 𝕀_{k} denote the group of idèles of k.

Theorem 13.1. There exists a unique homomorphism

ε: 𝕀_{k} → K^{×}

such that

(a) the restriction of ε to k^{×} is ψ_{0};

(b) the homomorphism ε is continuous, in the sense that its kernel is open in 𝕀_{k};

(c) there is a finite set S of primes of k, including those where A has bad reduction, such that for all finite primes v ∉ S, ε maps any prime element at v to F_{v}.

Proof. This is a restatement of the Theorem of Shimura and Taniyama (1961, p148) — see Serre and Tate 1968, Theorem 10.

There is a unique continuous homomorphism χ: 𝕀_{k} → (K ⊗_{ℚ} ℝ)^{×} that is trivial on k^{×} and coincides with ε on the group 𝕀_{k}^{∞} of idèles whose infinite component is 1 (ib. p513). For each σ: K → ℂ, let χ_{σ} be the composite

𝕀_{k} →^{χ} (K ⊗_{ℚ} ℝ)^{×} →^{σ⊗1} ℂ^{×}.

It is continuous and trivial on k^{×}, that is, it is a Hecke character in the broad sense (taking values in ℂ^{×} rather than the unit circle).

Theorem 13.2. The zeta function of A,

ζ(A, s) = ∏_{σ:K↪ℂ}L(s, χ_{σ}).

Proof. This is proved in Shimura and Taniyama 1961 except for the factors corresponding to a finite set of primes, and for all primes in Serre and Tate 1968.

General Case. We now explain how to extend these results to abelian varieties that are of CM-type, but whose endomorphisms are not defined over the given field of definition.

Let k be a field of characteristic zero, and let A be an abelian variety over a finite extension k′ of k. The restriction of scalars Res_{k′/k} A of A to k is the variety A_{∗} over k representing the functor of k-algebras, R ↦ A(R ⊗_{k} k′). For any finite Galois extension k̄ of k containing k′, there is a canonical isomorphism

P: A_{∗k̄} →^{≈} ∏_{σ∈Hom_{k}(k′,k̄)} σA.

Lemma 13.3. Let k be a number field, and let A_{∗} be the abelian variety over k obtained by restriction of scalars from an abelian variety A over a finite extension k′ of k. Then ζ(A_{∗}, s) = ζ(A, s).

Proof. It is immediate from the definition of A_{∗} that V_{ℓ}(A_{∗}) is the Gal(k^{al}/k)- module induced from the Gal(k^{al}/k′)-module V_{ℓ}(A). This implies the statement. (See Milne 1972, Proposition 3.)

Lemma 13.4. Let A be an abelian variety over a field k, and let k′ be a finite Galois extension of k of degree m and Galois group G. Suppose that there exists a ℚsubalgebra R ⊂ End^{0}(A_{k′}) such that R^{G} is a field and [R: R^{G}] = m. Then Res_{k′/k} A_{k′} is isogenous to A^{m}.

Proof. Let α_{1}, …, α_{m} be an R^{G}-basis for R over R^{G}, and let ϕ: A_{k′}^{m} → A_{k′}^{m} be the homomorphism (σ_{i}α_{j})_{1≤i,j≤m}, where G = {σ_{1}, …, σ_{m}}. Then ϕ is an isogeny. When we identify the second copy of A_{k′}^{m} with ∏ σ_{i}A_{k′} and compose ϕ with P^{−1}, we obtain an isogeny A_{k′}^{m} → A_{∗k′} that is invariant under G, and hence defined over k (ibid. Theorem 3).

Example 13.5. Let A be a simple abelian variety over a field k. Let R be the centre of End^{0}(A_{k^{al}}), and let k′ be the smallest field containing k and such that all elements of R are over defined over k′. Then A, k′, and R satisfy the hypotheses of Lemma 13.4 (ibid. p186), and so A^{m} is isogenous to (A_{k′})_{∗}. Hence, when k is a number field

ζ(A, s)^{m} = ζ(A_{k′}, s).

Example 13.6. Let A be an abelian variety over a number field k that, over ℂ, becomes of CM-type (K, Φ) for some field K. Assume that K is stable under the action of Gal(k^{al}/k) on End^{0}(A_{k^{al}}), and let k′ be the smallest field containing k such that all elements of K are defined over k′. Then A, k′, and K satisfy the hypotheses of Lemma 13.4, and so

ζ(A, s)^{m} = ζ(A_{k′}, s) =^{(13.2)} ∏_{σ∈Σ} L(s, χ_{σ}), Σ = Hom(K, ℂ).

In this case, we can improve the result. The group G = Gal(k′/k) acts faithfully on K, and a direct calculation shows that L(s, χ_{σ∘τ}) = L(s, χ_{σ}) for all σ ∈ Σ and τ ∈ G.

Therefore,

∏_{σ∈Σ} L(s, χ_{σ}) = ( ∏_{σ∈Σ/G} L(s, χ_{σ}))^{m}.

We can take an mth root, and obtain

ζ(A, s) = ∏_{σ∈Σ/G} L(s, χ_{σ}).

Now we consider the general case. Let A be an abelian variety of CM-type over a number field k. As noted earlier, we may suppose A to be simple. Then ζ(A, s)^{m} = ζ(A_{k′}, s) where k′ is the smallest field containing k over which all endomorphisms in the centre of End^{0}(A_{k^{al}}) are defined. Replacing A/k with A_{k′}/k′, we may suppose that the endomorphisms in the centre of End^{0}(A_{k^{al}}) are defined over k, and we may again suppose that A is simple. Then A_{k^{al}} is isotypic, and, for example, if it is simple, we can apply (13.5) to obtain the zeta function of A.

14. Hilbert’s Twelfth Problem

This asks for

… those functions that play for an arbitrary algebraic number field the role that the exponential function plays for the field of rational numbers and the elliptic modular functions play for an imaginary quadratic number field.

The classical result, referred to by Hilbert, can be stated as follows: for any quadratic imaginary field E, the maximal abelian extension of E is obtained by adjoining to it the moduli of elliptic curves and their torsion points with complex multiplication by E.

As Weil observed (see §12), in dimension > 1, the moduli of abelian varieties of CMtype and their torsion points generate abelian extensions, not of the field of complex multiplication, but of another field associated with it — the latter is now called the reflex field. In principle, the theory of Shimura and Taniyama allows one to list the abelian varieties of CM-type whose reflex field is contained in a given CM-field E, and to determine the extensions of E obtained from the moduli. However, the results in the published literature are unsatisfactory — for example, they don’t give a good description of the largest abelian extension of a field obtainable in this fashion (see Shimura and Taniyama 1961, Chapter IV; Shimura 1962; Shimura 1998, Chapter IV). Thus, the next theorem is of considerable interest.

Theorem 14.1 (Wei 1993, 1994). Let E be a CM-field. Let F be the maximal totally real subfield of E, and let H be the image of Gal(F^{ab}/F · ℚ^{ab}) in Gal(E^{ab}/E) under the Verlagerung map

Gal(ℚ^{al}/F)^{ab} → Gal(ℚ^{al}/E)^{ab}.

Then the field obtained by adjoining to E the moduli of all polarized abelian varieties of CM-type (and their torsion points) with reflex field contained in E is

M_{E} = E^{ab}^{H}

The theorem is proved by combining the three lemmas below.

Let T be a torus over ℚ and μ a cocharacter of T. We are only interested in pairs (T, μ) satisfying the conditions:

(a) T is split by a CM-field; equivalently, for all automorphisms τ of ℂ, the actions of τι and ιτ on X^{∗}(T) agree;

(b) the weight −μ − ιμ of μ is defined over ℚ.

Let (T, μ) be a pair satisfying (a) and (b). Its reflex field E(T, μ) is the field of definition of μ — because of (a), E(T, μ) is a subfield of a CM-field. Let E ⊃ E(T, μ). On applying Res_{E/ℚ} (Weil restriction) to the homomorphism μ: 𝔾_{m} → T_{E} and composing with the norm map, we obtain a homomorphism N(T, μ):

Res_{E/ℚ} 𝔾_{m} ⟶^{Res_{E/ℚ} μ} Res_{E/ℚ} T_{E} ⟶^{Norm_{E/ℚ}} T

For any ℚ-algebra R, this gives a homomorphism

(E ⊗_{ℚ} R)^{×} → T (R).

Let \overline{T (ℚ)} be the closure of T (ℚ) in T (𝔸_{f}). The reciprocity map

r(T, μ): Gal(E^{ab}/E) → T (𝔸_{f})/\overline{T (ℚ)}

is defined as follows: let τ ∈ Gal(E^{ab}/E), and let t ∈ 𝔸_{E}^{×} be such that rec_{E}(t) = τ; write t = t_{∞} · t_{f} with t_{∞} ∈ (E ⊗_{ℚ} ℝ)^{×} and t_{f} ∈ (E ⊗_{ℚ} 𝔸_{f})^{×}; then

r(T, μ)(τ) {df \atop =} N(T, μ)(t_{f}) mod \overline{T (ℚ)}.

Lemma 14.2. Let E be a CM-field, and let H be as in the statement of the theorem. Then

H = ⋂ Ker(r(T, μ))

where (T, μ) runs over the pairs satisfying (a) and (b) and such that E(T, μ) ⊂ E.

Proof. There is a universal such pair, namely, (S^{E}, μ^{E}), and so

⋂ Ker r(T, μ) = Ker r(S^{E}, μ^{E}).

Because S^{E} has no ℝ-split subtorus that is not already split over ℚ, S^{E}(ℚ) is closed in S^{E}(𝔸_{f}). Thus, to prove the lemma, one must show that H is the kernel of

r(S^{E}, μ^{E}): Gal(E^{ab}/E) → S^{E}(𝔸_{f})/S^{E}(ℚ).

This can be done by direct calculation (Wei 1994, Theorem 2.1).

For any CM-field K with CM-type Φ, we obtain a pair (K^{×}, μ_{Φ}) satisfying (a) and (b) (see §4).

Lemma 14.3. Let E be a CM-field, and let H be as above. Then

H = ⋂ Ker r(K^{×}, μ_{Φ})

where the intersection is over all CM-types (K, Φ) with reflex field contained in E.

Proof. For each (K, Φ) with reflex field contained in E, we obtain a homomorphism ρ_{Φ}: S^{E} → K^{×} (see §4), and (cf. the preceding proof) it suffices to show that ⋂ Ker ρ_{Φ} = 1. But X^{∗}(S^{E}) is generated by the CM-types Ψ on E, and Ψ occurs in ρ_{Φ} for Φ the reflex of Ψ (ibid. 1.5.1).

Lemma 14.4. Let (A, i) be an abelian variety over ℂ of CM-type (K, Φ), and let E be the reflex field of (K, Φ). The field of moduli of (A, i) and its torsion points is (E^{ab})^{H(Φ)} where H(Φ) is the kernel of r(K^{×}, Φ).

Proof. This is (yet another) restatement of the Theorem of Shimura and Taniyama.

In fact, (ibid.) for a CM-field E, the following fields are equal:

(a) the fixed field of H;

(b) the field generated over E by the fields of moduli of all CM-motives and their torsion points with reflex field contained in E;

(c) the field generated over E by the fields of moduli of the CM-motive and its torsion points defined by any faithful representation of S^{E};

(d) the field generated over E by the fields of moduli of the polarized abelian varieties and their torsion points of CM-type with reflex field contained in E;

Moreover, for some Siegel modular variety and special point z, this is the field generated by the values at z of the E-rational modular functions on the variety (ib. 3.3.2; see also the next section).

Special Values of Modular Functions. ^{11}

Abelian class field theory classifies the abelian extensions of a number field k, but does not explain how to generate the fields. In his Jugendtraum, Kronecker suggested that the abelian extensions of ℚ can be generated by special values of the exponential function, and that the abelian extensions of an imaginary quadratic number field can be generated by special values of elliptic modular functions. This idea of generating abelian extensions using special values of holomorphic functions was taken up by Hilbert in his twelfth problem, where he suggested “finding and discussing those functions that play the part for any algebraic number field corresponding to that of the exponential function for the field of rational numbers and of the elliptic modular functions for imaginary quadratic number fields.”

Here we explain how the theory of Shimura varieties allows one to define a class of modular functions naturally generalizing that of the elliptic modular functions, and that it allows one to identify the fields generated by the special values of the functions as the fields of moduli of CM-motives.

Modular functions over ℂ. To define a Shimura variety, one needs a reductive group G over ℚ and a G(ℝ)-conjugacy class X of homomorphism 𝕊 → G_{ℝ} satisfying the following conditions:

SV1: for each h ∈ X, the Hodge structure on the Lie algebra 𝔤 of G defined by

Ad ∘h: 𝕊 → GL(𝔤_{ℝ}) is of type {(−1, 1), (0, 0), (−1, 1)}; SV2: for each h ∈ X, adh(i) is a Cartan involution on G_{ℝ}^{ad}; SV3: the adjoint group G^{ad} of G has no factor defined over ℚ whose real points

11This subsection is a manuscript of mine dated May 6, 1993.

form a compact group, and the identity component of the centre of G splits over a CM-field.

The condition (SV1) implies that the restriction of h to 𝔾_{m} ⊂ 𝕊 is independent of h ∈ X. We denote its reciprocal by w_{X}: 𝔾_{m} → G_{ℂ}, and call it the weight of the Shimura variety. The weight is always defined over a totally real number field, and we shall be especially interested in Shimura varieties for which it is defined over ℚ.

Consider a pair (G, X) satisfying the Axioms (SV1-3). The set X has a canonical G(ℝ)-invariant complex structure for which the connected components are isomorphic to bounded symmetric domains.

For each compact open subset K of G(𝔸_{f}),

Sh_{K}(G, X) {df \atop =} G(ℚ)\X × G(𝔸_{f})/K

is a finite disjoint union of quotients of the connected components of X by arithmetic subgroups of G^{ad}(ℚ)^{+}, say

Sh_{K}(G, X) = ⋃ Γ_{i}\X_{i}.

For K sufficiently small, each space Γ_{i}\X_{i} will be a complex manifold, and, according to Baily and Borel (1966), it has a natural structure of a quasi-projective variety over ℂ. Hence Sh_{K}(G, X) is an algebraic variety over ℂ, and the Shimura variety Sh(G, X) is the projective system of these varieties, or (what amounts to the same thing) the limit of the system, together with the action of G(𝔸_{f}) defined by the rule:

[x, a] · g = [x, ag], x ∈ X, a, g ∈ G(𝔸_{f}).

A rational function f on Sh_{K}(G, X) is called an automorphic function over ℂ when dim X > 0. Such a function defines (for each i) a meromorphic function f_{i} on each X_{i} invariant under Γ_{i}. Conversely a family (f_{i}) of invariant meromorphic functions defines an automorphic function f provided each f_{i} is “meromorphic at infinity” (this condition is automatically satisfied except when X_{i} has dimension 1).

When the weight w_{X} of the Shimura variety is defined over ℚ, we shall call the automorphic functions modular functions. Classically this name is reserved for functions on Shimura varieties that are moduli variety for abelian varieties, but it is known that most Shimura varieties with rational weight are moduli varieties for abelian motives, and it is hoped that they are all moduli varieties for motives, and so our nomenclature is reasonable. This class of functions is the most natural generalization of the class of elliptic modular functions.

Note that it doesn’t yet make sense to speak of the algebraic (much less arithmetic) properties of the special values of modular functions, because, for example, the product of a modular function with a complex number is again a modular function.

Example 14.5. Let G = GL_{2} and let X be the G(ℝ)-conjugacy class of homomorphism 𝕊 → GL_{2,ℝ} containing the homomorphism

a + ib ↦ (\begin{matrix}a & −b \\ b & a \end{matrix}) .

The map h ↦ h(i)·i identifies X with {z ∈ ℂ | ℜ(z) ≠ 0}, and in this case Sh_{K}(G, X) is a finite union of elliptic modular curves over ℂ. If K = GL_{2}(ℤ̂), then the field of modular functions on Sh_{K}(G, X) is ℂ[j].

Example 14.6. Let T be a torus over ℚ split by a CM-field, and let μ ∈ X_{∗}(T). Define h: 𝕊 → T_{ℝ} by h(z) = μ(z) · \overline{μ(z)}. Then (T, \{h\}) defines a Shimura variety.

Remark 14.7. Shimura varieties have been studied for 200 years^{12}...Gauss, Picard, Poincaré, Hilbert, Siegel, Shimura,... The axiomatic definition given above is due to Deligne (except that he doesn’t require that the identity component of the centre split over a CM-field). The name is due^{13} to Langlands.

Special points. A point x ∈ X is said to be special if there exists a torus T ⊂ G (this means T is rational over ℚ), such that Im(h_{x}) ⊂ T_{ℝ}. By a special pair (T, x) in (G, X) we mean a torus T ⊂ G together with a point x ∈ X such that h_{x} factors through T_{ℝ}.

Example 14.8. In the Example 2, the special points correspond to points z ∈ ℂ\ℝ such that [ℚ[z]: ℚ] = 2. For such a z, the choice of a ℚ-basis for E =_{df} ℚ[z] determines an embedding ℚ[z]^{×} ↪ GL_{2}(ℚ), and hence an embedding T =_{df} (𝔾_{m})_{E/ℚ} ↪ GL_{2}. The map h_{z} factors through T_{ℝ} ↪ GL_{2,ℝ}.

Modular functions defined over number fields. To a torus T defined over ℚ and a cocharacter μ of T defined over a number field E, we attach a reciprocity map

r(T, μ): Gal(E^{ab}/E) ⟶ T (𝔸_{f})/T (ℚ)^{−}

as in (Milne 1992, p164)^{14}. The reflex field E(G, X) is defined to be the field of definition of the G(ℂ)-conjugacy class of homomorphisms 𝔾_{m} → 𝔾_{ℂ} containing μ_{x} for x ∈ X. It is a number field, and is a subfield of a CM-field. Hence it is either itself a CM-field or is totally real.

By a model of Sh(G, X) over a subfield k of ℂ, we mean a scheme S over k endowed with an action of G(𝔸_{f}) (defined over k) and a G(𝔸_{f})-equivariant isomorphism Sh(G, X) → S ⊗_{k} ℂ. We use this isomorphism to identify Sh(G, X)(ℂ) with S(ℂ).

Theorem 14.9. There exists a model of Sh(G, X) over E(G, X) with the following property: for all special pairs (T, x) ⊂ (G, X) and elements a ∈ G(𝔸_{f}), the point [x, a] is rational over E(T, x)^{ab} and τ ∈ Gal(E(T, x)^{ab}/E(T, x)) acts on [x, a] according to the rule:

τ[x, a] = [x, ar(τ)], where r = r(T, μ_{x}).

The model is uniquely determined by this condition up to a unique isomorphism.

The model in the theorem is said to be canonical.

Remark 14.10. For Shimura varieties of PEL-type, models over number fields were constructed by Mumford and Shimura (and his students Miyake and Shih). That they satisfy condition in the theorem follows from the theorem of Shimura and Taniyama. Shimura defined the notion of a canonical model more generally, and proved the existence in one interesting case where the weight is not defined over ℚ.

12This seems to be an exaggeration.

13Earlier Shimura curves had been so named by Ihara.

14Better, see above

Deligne modified Shimura’s definition, proved that the canonical model is unique (if it exists) (1971), and showed that it exists for all Shimura varieties of abelian type (1979a). In 1981 Borovoi suggested using a trick of Piateski-Shapiro to extend the proof to the remaining cases, and this was carried out by Milne in 1982 (Milne 1983).

Write Sh(G, X)_{E} for the model in the theorem, and for any k ⊃ E, write Sh(G, X)_{k} for Sh(G, X)_{E} ⊗_{E} k.

For a connected variety V over a field k, the field of rational functions on V is a subfield of the field of rational functions on V ⊗_{k} ℂ. We say that a modular function f on Sh_{K}(G, X) is rational over a subfield k of ℂ if it arises from a rational function on Sh_{K}(G, X)_{k}.

Let x ∈ X be special, say Im(h_{x}) ⊂ T_{ℝ}. Then the field of definition of μ_{x} is written E(x) — it is the reflex field of (T, h_{x}), and is a finite extension of E(G, X).

The fields generated by special values of modular functions. Let V be a connected algebraic variety over a field k. A point P ∈ V (k^{al}) is a morphism Spec k^{al} → V — we also use P to denote the image of the map. It corresponds to a k-homomorphism O_{V,P} → k^{al}. This homomorphism factors through O_{V,P} /𝔪_{P}, and hence its image in k^{al} is a subfield of k^{al}, which we denote k[P].

For any open affine neighbourhood U of P, the field of rational functions k(V) on V is the field of fractions of k[U]. For f = g/h ∈ k(V), we can speak of

f(P) {df \atop =} g(P)/h(P) ∈ k^{al}

whenever f does not have a pole at P, i.e., when h ∉ 𝔪_{𝔓}.

Lemma 14.11. With the above notations,

k[P] = ⋃ k[f(P)]

where the union runs over the f ∈ k(V) without a pole at P (i.e., over f ∈ O_{V,P}).

Proof. We may replace V with an open affine neighbourhood, and embed V in 𝔸^{n}. Then k[P] is the field generated by the coordinates (a_{1}, …, a_{n}) of P. Clearly, for any rational function f(X_{1}, ⋯, X_{n}) with coordinates in k, f(a_{1}, …, a_{n}) ∈ k[P] (if it is defined). Conversely, k[P] = ⋃ k[f(a_{1}, …, a_{n})] where f runs through the polynomials in X_{1}, …, X_{n}.

For a number field k, let k_{c} be the subfield of k^{al} corresponding to

⋂ Ker(r(T, μ))

where (T, μ) runs over the pairs (T, μ) consisting of a torus T split by a CM-field and μ is a cocharacter of T whose weight −μ − ιμ is defined over ℚ. (Equivalently over the pairs (T, μ) consisting of a torus T over ℚ and a cocharacter μ of T satisfying the Serre condition^{15}.)

Theorem 14.12. Let k be an algebraic number field. For any Shimura variety Sh(G, X) such that E(G, X) ⊂ k, modular function f on Sh_{K}(G, X) rational over k, and special point x of X such that E(x) ⊂ k, f(x) ∈ k_{c} (if it is defined, i.e., f doesn’t have a pole at x). Moreover, if k contains a CM-field, then k_{c} is generated by these special values.

15This is the condition (σ − 1)(ι + 1)μ = 0 = (ι + 1)(σ − 1)μ.

Proof. Let (T, x) be a special pair in (G, X). I claim that T splits over a CM-field. To prove this, it suffices to show that the action ι on X^{∗}(T) (or even X^{∗}(T) ⊗ ℚ) commutes with that of τ, for all τ ∈ Gal(ℚ^{al}/ℚ). But

X^{∗}(T) ⊗ ℚ = X^{∗}(T′) ⊗ ℚ ⊕ X^{∗}(G^{ab})

where T′ = T/Z(G) (use that G → G^{ad} × G^{ab} is an isogeny). By assumption (SV3), X^{∗}(G^{ab}) splits over a CM-field, and it follows from (SV2) that ι acts as -1 on X^{∗}(T′) and hence commutes with everything. From Theorem 5, it is clear that k([x, 1]) is fixed by Ker(r(T, μ_{x})) and so is contained in k_{c}. From the lemma, this implies that f(x) ∈ k_{c} for all f.

Before proving the converse, we need a construction. Let E be a CM-field, with maximal totally real subfield F. Let N be the kernel of

(𝔾_{m})_{E/ℚ} ⟶ S^{E}.

It is a subgroup of (𝔾_{m})_{F/ℚ}, and hence is contained in the centre of GL_{2,F}, and we define G = GL_{2,F} /N. The choice of a basis for E as an F -space determines an inclusion (𝔾_{m})_{E/ℚ} ↪ GL_{2,F}, and hence an inclusion S^{E} ↪ G. Let X be the G(ℝ) conjugacy class of the composite

𝕊 ⟶^{h_{can}} S^{E} ⟶ G.

Then Sh(G, X) is a Shimura variety of dimension [F: ℚ] with weight defined over ℚ and whose reflex field is ℚ.

On applying this construction to the largest CM-field contained in k, we obtain a Shimura variety Sh(G, X) containing Sh(S^{k}, h_{can}), where S^{k} = S^{E} is the Serre group for k. The statement is now (more-or-less) obvious.

Nonabelian solutions to Hilbert’s Twelfth Problem. By applying the new Main Theorem of Complex Multiplication (Theorem 9.1) in place of the original, one obtains explicit non-abelian extensions of number fields (Milne and Shih 1981, §5).

15. Algebraic Hecke Characters are Motivic

16 Weil’s Hecke characters of type A_{0} are now called algebraic Hecke characters. In this section, I show that they are all motivic (and explain what this means).

Algebraic Hecke characters. In this subsection, I explain the description of algebraic Hecke characters given in Serre 1968.

Notations: ℚ^{al} is the algebraic closure of ℚ in ℂ; K is a fixed CM-field, Σ = Hom(K, ℚ^{al}) = Hom(K, ℂ), and 𝕀 = 𝕀_{∞} × 𝕀_{f} is the group of idèles of K. For a finite extension k′/k, (𝔾_{m})_{k′/k} is the torus over k obtained from 𝔾_{m/k′} by restriction of scalars.

I define an algebraic Hecke character to be a continuous homomorphism χ: 𝕀 → ℚ^{al}^{×} such that

(a) χ = 1 on 𝕀_{∞}; (b) the restriction of χ to K^{×} ⊂ 𝕀 is given by an algebraic character of the torus

16This section is the notes of a seminar talk.

(𝔾_{m})_{K/ℚ}.

Condition (b) means that there exists a family of integers (n_{σ})_{σ∈Σ} such that χ(x) = ∏ σ(x)^{n_{σ}} for all x ∈ K^{×} ⊂ 𝕀. Condition (a) means that χ factors through 𝕀 → 𝕀_{f}. Thus, there is a one-to-one correspondence between algebraic Hecke characters and continuous homomorphisms 𝕀_{f} → ℚ^{al×} satisfying the analogue of (b) (restriction to K^{×} ⊂ 𝕀_{f}) (cf. the definition in Harder and Schappacher).

Let χ be a Hecke character.

The character χ admits a modulus. Let 𝔪 be a modulus for K. Because K has no real primes, 𝔪 can be regarded as an integral ideal ∏_{v} 𝔭_{v}^{m_{v}}. Define

W_{𝔪} = ∏_{v|∞} K_{v}^{×} × ∏_{v|𝔪}(1 + 𝔭̂_{v}^{m_{v}}) × ∏ U_{v}

(as in my class field theory notes, Milne 1997, about V.4.6). The W_{𝔪}’s are open subgroups of 𝕀 and any neighbourhood of 1 containing 𝕀_{∞} contains a W_{𝔪}. Let V be a neighbourhood of 1 in ℂ^{×} not containing any subgroup ≠ 1. Because χ is continuous and 1 on 𝕀_{∞}, χ(W_{𝔪}) ⊂ V for some 𝔪, and hence χ(W_{𝔪}) = 1. Such an 𝔪 will be called a modulus for χ. If 𝔪 is a modulus for χ and 𝔪|𝔪′, then 𝔪′ is also a modulus for χ.

The infinity type of χ. Let ℤ^{Σ} be the free abelian group generated by Σ, with τ ∈ Gal(ℚ^{al}/ℚ) acting by τ(∑_{σ∈Σ} n_{σ}σ) = ∑ n_{σ}(τ ∘ σ). The character group X^{∗}((𝔾_{m})_{K/ℚ}) = ℤ^{Σ}, and so χ|K^{×} = ∑ n_{σ}σ for some n_{σ} ∈ ℤ. The element ∑_{σ∈Σ} n_{σ}σ is called the infinity type of χ.

Let U_{𝔪,1} = K^{×} ∩ W_{𝔪} — this is a subgroup of finite index in the units U of K defined by congruence conditions at the primes dividing 𝔪. If 𝔪 is a modulus for χ, χ = 1 on U_{𝔪,1}, and this implies that n_{σ} + nσ̄ = constant, independent of σ (apply the Dirichlet unit theorem).

The Serre group. Let Ξ be the group of infinity types, i.e.,

Ξ = {∑_{σ∈Σ} n_{σ}σ | n_{σ} + nσ̄ = constant} ⊂ ℤ^{Σ}.

It is a free ℤ-module of finite rank on which Gal(ℚ^{al}/ℚ) acts, and we define the Serre group S^{K} to be the torus over ℚ with character group Ξ. Thus, for any field L ⊂ ℚ^{al},

S^{K}(L) = Hom(X^{∗}(S^{K}), ℚ^{al})^{Gal(ℚ^{al}/L)}.

Because X^{∗}(S^{K}) ⊂ X^{∗}((𝔾_{m})_{K/ℚ}), S^{K} is a quotient of (𝔾_{m})_{K/ℚ}. The map on ℚrational points K^{×} → S^{K}(ℚ) sends x ∈ K^{×} to the map ξ ↦ ξ(x), ξ ∈ Ξ.

Serre’s extension. I claim that there exists a modulus 𝔪 such that U_{𝔪,1} is contained in Ker(ξ) for all ξ ∈ Ξ. Indeed, in order for ξ = ∑ n_{σ}σ to lie in Ξ, its restriction to the totally real subfield F of K must be a power of the norm. Thus all ξ = 1 on some subgroup U of index at most 2 in U_{F}. But U_{F} is of finite index in U_{K} (Dirichlet unit theorem again), and so U has finite index in U_{K}. An old theorem of Chevalley states that every subgroup of finite index in U_{K} is a congruence subgroup, i.e., contains U_{𝔪,1} for some 𝔪.

From now on, 𝔪 will denote a modulus with this property; thus the canonical map K^{×} → S^{K}(ℚ) factors through K^{×}/U_{𝔪,1}.

Recall (e.g., Milne 1997, V.4.6) that 𝕀/W_{𝔪} · K^{×} = C_{𝔪}, the ray class group with modulus 𝔪 (= I^{S(𝔪)}/i(K_{𝔪,1})). In particular, it is finite. There is an exact sequence

1 → K^{×}/U_{𝔪,1} → 𝕀/W_{𝔪} → C_{𝔪} → 1.

Serre shows that there is a canonical exact sequence of commutative algebraic groups over ℚ

1 → S^{K} → T_{𝔪} → C_{𝔪} → 1

(here C_{𝔪} is regarded as a finite constant algebraic group) for which there is a commutative diagram

1 → K^{×}/U_{𝔪,1} → 𝕀/W_{𝔪} → C_{𝔪} → 1
↓ ↓ ε ∥
1 → S^{K}(ℚ) → T_{𝔪}(ℚ) → C_{𝔪} → 1.

Moreover, there is a natural one-to-one correspondence between the algebraic Hecke characters χ of K admitting 𝔪 as a modulus and the characters of T_{𝔪} as an algebraic group. [The proofs of these statements are straightforward.] The algebraic Hecke character corresponding to a character χ of T_{𝔪} is the composite

𝕀 → 𝕀/W_{𝔪} ⟶^{ε}T_{𝔪}(ℚ^{al}) → ℚ^{al×}.

From now on, I’ll define an algebraic Hecke character to be a character of T_{𝔪} for some 𝔪. Its infinity type is its restriction to S^{K}. The Dirichlet characters are the Hecke characters with trivial infinity type (and hence factor through C_{𝔪}).

Warning! Our notations differ from those of Serre—in particular, he switches the S and the T.

The ℓ-adic representation. One checks that the two maps

α_{ℓ}: 𝕀 ⟶^{proj} (K ⊗_{ℚ} ℚ_{ℓ})^{×} → S^{K}(ℚ_{ℓ}) → T_{𝔪}(ℚ_{ℓ})

and

ε: 𝕀 → T_{𝔪}(ℚ)

coincide on K^{×}. Therefore, ε_{ℓ} =^{df} ε · α_{ℓ}^{−1}: 𝕀 → T_{𝔪}(ℚ_{ℓ}) factors through 𝕀/K^{×}𝕀_{∞}, and hence through Gal(K^{ab}/K) — thus ε_{ℓ} is a continuous homomorphism

Gal(Kab/K) → T_{𝔪}(ℚ_{ℓ}).

The Hecke character in the usual sense. The same argument with ℓ replaced by ∞ gives a homomorphism ε_{∞}: 𝕀 → T_{𝔪}(ℝ) that is not usually trivial on the connected component of 𝕀. Its composite with any character of T_{𝔪} defined over ℂ is a Hecke character in the usual (broad) sense: continuous homomorphism 𝕀 → ℂ^{×} trivial on K^{×}.

Motivic Hecke Characters. Let k be a subfield of ℂ. An abelian variety A over k is said to be of CM-type if there exists a product of fields E ⊂ End^{0}(A_{ℂ}) such that H_{B}^{1}(A, ℚ) is a free E-module of rank 1. It is said to have CM over k if E ⊂ End^{0}(A). It is possible to choose E so that it is stabilized by the Rosati involution of some polarization of A, which implies that it is a product of CM-fields.

Let A be an abelian variety with CM by E over K. Then Gal(K^{al}/K) acts on V_{ℓ}(A) by E ⊗_{ℚ} ℚ_{ℓ}-linear maps. But V_{ℓ}(A) is a free E ⊗_{ℚ} ℚ_{ℓ}-module of rank one, so this action defines a homomorphism

ρ_{ℓ}: Gal(K^{ab}/K) → (E ⊗_{ℚ} ℚ_{ℓ})^{×} ⊂ GL(V_{ℓ}(A)).

The main theorem of Shimura-Taniyama theory can be stated^{17} as follows:

15.1. For 𝔪 sufficiently large, there exists a unique homomorphism χ: T_{𝔪} → (𝔾_{m})_{E/ℚ} of tori such ρ_{ℓ} = χ(ℚ_{ℓ}) ∘ ε_{ℓ}.

An embedding σ of E into ℚ^{al×}, defines a character χ_{σ} of T_{𝔪}, which (by definition) is an algebraic Hecke character. Such characters are certainly motivic.

The infinity type of a Hecke character arising in this way is a CM-type on K, i.e., n_{σ} ≥ 0, n_{σ} + nσ̄ = 1, and Casselman showed that conversely every Hecke character with infinity type a CM-type arises in this fashion.

More generally, I discussed motives of type M = (A, e), A an abelian variety of CM-type, e^{2} = e, e ∈ C^{g}(A × A)/∼ (algebraic classes of codimension g = dim A modulo numerical equivalence). Such an M has an endomorphism ring and Betti and étale cohomology groups, and so one can make the same definitions as for A. Note that M = (A, e) may have CM over k without A having CM over k. The analogue of (15.1) holds. A Hecke character arising from such a motive, or the product of such a character with a Dirichlet character, will be called motivic.

If we assume the Hodge conjecture, then every algebraic Hecke character is motivic. After a theorem of Deligne (1982a), we no longer need to assume the Hodge con-

jecture, but at the cost of replacing e with an absolute Hodge class.

The proof. (that all algebraic Hecke characters are motivic). The CM-motives discussed above over a field k form a category CM(k) that looks like the category of representations of an algebraic group: it is ℚ-linear, abelian, has a tensor product, duals, and every object has rank equal to a nonnegative integer. The theory of Tannakian categories then shows that it is the category of representations of a pro-algebraic group. (We are using absolute Hodge classes to define motives.)

What is the pro-algebraic group? When k = ℂ, one sees easily that it S = \underleftarrow{lim} S^{K} (projective limit over the CM-subfields of ℚ^{al}). Hint: The abelian varieties of CMtype over ℂ are classified up to isogeny by CM-types, and S^{K} is generated by the CM-types on K.

When k = ℚ^{al}, the group is again S (base change ℚ^{al} → ℂ gives an equivalence of categories of CM-motives).

When k = ℚ, the general theory tells us it is an extension

1 → S → T → Gal(ℚ^{al}/ℚ) → 1.

17Cf. 13.1.

Following Langlands, we call T the Taniyama group.

Deligne, Grothendieck, and Serre knew in the 1960s that the general theory predicted the existence of such an extension, but couldn’t guess what it was. (Although he doesn’t say so, these ideas must have suggested to Serre his interpretation of algebraic Hecke characters.) In the late 1970s, in trying to understand the zeta functions of Shimura varieties, Langlands wrote down some cocycles, which Deligne recognized should give the above extension. He verified they do by proving that there is only one such extension having certain natural properties shared by both extensions.

When k = K ⊂ ℚ^{al}, the group attached to the category of CM-motives over K is the subextension

1 → S → T^{K} → Gal(ℚ^{al}/K) → 1

of the above extension. Thus, to give a CM-motive over K is to give a representation of T^{K} on a finite-dimensional ℚ-vector space.

From Langlands’s description of this extension, one sees that, for any 𝔪, there is a canonical map (see 10.2a) of extensions:

1 → S → T^{K} → Gal(ℚ^{al}/K) → 1
↓  ↓  ↓
1 → S^{K} → T_{𝔪} →  C_{𝔪}  → 1

Let E be a CM-field. A homomorphism T_{𝔪} → (𝔾_{m})_{E/ℚ} defines by composition a representation T^{K} → (𝔾_{m})_{E/ℚ} ↪ GL(E′) where E′ = E regarded as a ℚ-vector space. Therefore a Hecke character χ defines a CM-motive M(χ) with CM by E over K. The motive M(χ) is related to χ as in 15.1, and so χ is motivic.

16. Periods of Abelian Varieties of CM-type

Deligne’s theorem (Deligne 1978, Deligne 1982a) allows one to define a category of CM-motives over any field of characteristic zero (Deligne and Milne 1982, §6).

Let M be a simple CM-motive over ℚ^{al} ⊂ ℂ. Then End(M) is a CM-field K. The Betti realization H_{B}(M) of M is a vector space of dimension 1 over K, and the de Rham realization H_{dR}(M) is free of rank 1 over K ⊗_{ℚ} ℚ^{al}. For σ: K ↪ ℚ^{al}, let H_{dR}(M)_{σ} denote the ℚ^{al}-subspace of H_{dR}(M) on which x ∈ K acts as σ(x) ∈ ℚ^{al}. Then H_{dR}(M) being free of rank 1 means that each H_{dR}(M)_{σ} has dimension 1 and

H_{dR}(M) = ⊕_{σ:K↪ℚ^{al}}H_{dR}(M)_{σ}.

Let e be a nonzero element of H_{B}(M), and let ω_{σ} be a nonzero element of H_{dR}(M)_{σ}. Under the canonical isomorphism

H_{B}(M) ⊗_{ℚ} ℂ → H_{dR}(M) ⊗_{ℚ^{al}} ℂ

e maps to a family (e_{σ}), e_{σ} ∈ H_{dR}(M)_{σ} ⊗_{ℚ^{al}} ℂ. Define p(M, σ) ∈ ℂ by the formula

p(M, σ) · e_{σ} = ω_{σ}.

When regarded as an element of ℂ^{×}/ℚ^{×}, p(M, σ) is independent of the choices of e and of ω_{σ} — the p(M, σ) are called the periods of M. Clearly, p(M, σ) depends only on the isomorphism class of M.

Let S be the Serre group — it is the projective limit of the Serre groups S^{E} for E a CM-field contained in ℂ. It is the protorus over ℚ whose character group X^{∗}(S) consists of locally constant functions ϕ: Gal(ℚ^{cm}/ℚ) → ℤ such that ϕ(τ) + ϕ(ιτ) is independent of τ. The Betti fibre functor defines an equivalence from the category of CM-motives over ℚ^{al} to the category of finite-dimensional representations of S. Thus the set of simple isomorphism classes of CM-motives over ℚ^{al} is in natural one-to-one correspondence with the set of Gal(ℚ^{al}/ℚ)-orbits in X^{∗}(S). Let ϕ ∈ X^{∗}(S), and let M(ϕ) be the CM-motive corresponding to ϕ. The endomorphism algebra of M(ϕ) is K = ℚ^{alH}, where H is the stabilizer of ϕ in Gal(ℚ^{al}/ℚ). Thus, for each ϕ ∈ X^{∗}(S) and coset representative for H in Gal(ℚ^{al}/ℚ), we obtain a period

p(ϕ, σ) =^{df} p(M(ϕ), σ).

Any relation among the ϕ’s yields an isomorphism among the motives, and hence a relation among the periods. When ϕ is taken to be a CM-type, then p(ϕ, σ) is the period of an abelian variety of CM-type. Thus, we see that Deligne’s theorem (Deligne 1978) yields an array of relations among the periods of abelian varieties of CM-type. (See Deligne’s talk at the Colloq., École Polytech., Palaiseau, 1979 (Deligne 1980); also Shimura’s talk at the same conference (Shimura 1980).)

In this context, one should also mention Blasius 1986.

17. Review of: Lang, Complex Multiplication, Springer 1983.

The ^{18} theory of complex multiplication for elliptic curves describes how an automorphism of ℂ acts on an elliptic curve with complex multiplication and its torsion points. As a consequence, when the curve is defined over a number field, one obtains an expression for its zeta function in terms of Hecke L-series. The theory was generalized to abelian varieties in so far as it concerned automorphisms fixing the reflex field by Shimura, Taniyama, and Weil in the fifties. As a consequence, when the abelian variety is defined over a number field containing the reflex field, they obtained an expression for its zeta function (except for finitely many factors) in terms of Hecke L-series. A thorough account of this is given in Shimura and Taniyama (1961). Improvements are to be found in Shimura 1971 (Sections 5.5 and 7.8). Serre and Tate (1968) extended the result on the zeta function to all the factors, and computed the conductor of the variety. Serre (1968), Chapters 1 and 2, re-interpreted some of this work in terms of algebraic tori. In 1977 Langlands made a conjecture concerning Shimura varieties which was shown to have as a corollary a description of how every automorphism of ℂ acts on an abelian variety with complex multiplication and its torsion points, and in 1981 Deligne proved the corollary (Deligne et al. 1982). Since this gives an expression for the zeta function of such a variety over any number field in terms of Weil L-series, it completes the generalization to abelian varieties of the basic theory of complex multiplication for elliptic curves.

The first four chapters of the Lang’s book are devoted to the same material as that in (the sections of) the works of Shimura and Taniyama, Shimura, and Serre and Tate cited above: the analytic theory of abelian varieties with complex multiplication, the reduction of abelian varieties, the main theorem of complex multiplication, and zeta functions. Lang’s account is less detailed but probably more readable than his sources. For example, whereas Shimura and Taniyama’s discussion of reduction is painfully detailed (they, like the author, use the language of Weil’s Foundations), that of the author is brief and sketchy. The result of Serre and Tate on the conductor is not included and, in the statement of the main theorem, it is unnecessarily assumed that the abelian variety is defined over a number field.

18This is the author’s version of MR 85f:11042.

Chapter 5 discusses fields of moduli and the possibility of descending abelian varieties with complex multiplication to smaller fields (mainly work of Shimura), and Chapter 6 introduces some of the algebraic tori associated with abelian varieties having complex multiplication and uses them to obtain estimates for the degrees of the fields generated by points of finite order on the varieties.

The final chapter (based on a manuscript of Tate)^{19} gives the most down-to-earth statement of the new main theorem of complex multiplication (the corollary of Langlands’s conjecture) and includes part of the proof (but, unfortunately, only the more technical, less illuminating, part). Zeta functions are not discussed in this general context.

The exposition is very clear in parts, but in others it is marred by carelessness. For example, in Chapter 3, the definition of 𝔞-multiplication is incorrect (the universal property is not universal), in the proof of (3.1) it is nowhere shown that the reduction of an 𝔞-multiplication is an 𝔞-multiplication, and in the proof of the Main Theorem 6.1 it is not possible to write the idèle s in the way the author claims on p. 82 under his assumptions.

In summary, this book will be useful, in much the same way as a good lecture course, for someone wishing to obtain a first understanding of the subject, but for a more complete and reliable account it will be necessary to turn to the original sources mentioned in this review. James Milne (1-MI)

Additional References

Baily, W., and Borel, A., Compactification of arithmetic quotients of bounded symmetric domains. Ann. of Math. (2) 84, 1966, pp 442–528.

Blasius, Don, On the critical values of Hecke L-series, Ann. of Math. (2) 124 (1986), no. 1, 23–63.

Deligne, Pierre, Travaux de Shimura, in Séminaire Bourbaki, 23ème année (1970/71), Exp. No. 389, 123–165. Lecture Notes in Math., 244, Springer, Berlin, 1971.

Deligne, P., Cycles de Hodge sur les variétés abéliennes, 4pp, 8th January, 1978. Deligne, Pierre, Cycles de Hodge absolus et périodes des intégrales des variétés

abéliennes, Mém. Soc. Math. France (N.S.) 1980, no. 2, 23–33.

Deligne, P., Letter to Tate, 8th October 1981.

Deligne, P., and Milne, J., Tannakian categories. In: Deligne et al. 1982, pp 101–228.

Milne, J., The action of an automorphism of C on a Shimura variety and its special points. Arithmetic and geometry, Vol. I, 239–265, Progr. Math., 35, Birkhuser Boston, Boston, Mass., 1983.

19This is what the book says, but in fact §4 of Chapter 7 is essentially a translation of Deligne 1981. In the same chapter, Lang (p175) credits a theorem of Deligne and Langlands to Deligne alone and (p163, p171) a theorem of mine to Deligne (Langlands’s conjecture; joint with Shih for Shimura varieties of abelian type and with Borovoi in general).

Milne, J., The points on a Shimura variety modulo a prime of good reduction. In: The zeta functions of Picard modular surfaces, 151–253, Univ. Montral, Montreal, PQ, 1992.

Milne, J., Class Field Theory, 222 pages, 1997. Available at

www.math.lsa.umich.edu/∼jmilne/.

Milne, J., and Shih, Kuang-yen, Automorphism groups of Shimura varieties and reciprocity laws, Am. J. Math. 103 (1981), 911–935.

Serre, J.-P., and Tate, J., Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492–517.

Shimura, Goro, On the class-fields obtained by complex multiplication of abelian varieties, Osaka Math. J. 14 (1962), 33–44.

Shimura, G., On abelian varieties with complex multiplication, Proc. LMS 34 (1977), 65–86.

Shimura, G., The periods of abelian varieties with complex multiplication and the special values of certain zeta functions, Mém. Soc. Math. France (N.S.) 1980, no. 2, 103–106.

Shimura, G., Abelian Varieties with Complex Multiplication and Modular Functions, Princeton, 1998.

Shimura, G., and Taniyama, Y., Complex multiplication of abelian varieties and applications to number theory, Math. Soc. Japan, Tokyo, 1961.

Wei, W., Weil numbers and generating large field extensions. Thesis, University of Michigan, 1993.

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  jmilne@umich.edu, www.math.lsa.umich.edu/∼jmilne/.