L-FUNCTIONS FOR CM ABELIAN VARIETIES

TREVOR ARNOLD

The purpose of these notes is to determine the L-function of an abelian variety A over a number field L_{0} when A obtains CM over a finite extension L/L_{0}. This Lfunction can be realized as a product of L-functions of Grössencharaktere of a minimal such L, at least under the assumption that the image of a CM field in the geometric endomorphism ring of A is stable under the action of G = Gal(L/L_{0}). This includes as an important special case those A that are absolutely simple. The main references are Milne [4] and Lang [3], though we will be following [4] more closely, as Milne’s treatment is somewhat simpler (at the expense, perhaps, of greater abstraction).

To establish some notation: if L is a field, we denote by \overline{L} a separable closure of L and by G_{L} the absolute Galois group Gal(\overline{L}/L) of L. If L is a global, resp. non-archimedean local, field and v is a, resp. the, non-archimedean valuation of L, then we denote by π_{v} a choice of uniformizer at v. In this situation, we denote by Nv the size of the residue field k(v) at v.

1. The CM case

In this section, we use the Main Theorem of complex multiplication to determine the L-function of a CM abelian variety.

1.1. CM characters. Let (A, ι) be a CM abelian variety of type (F, Φ) defined over a number field L. Define

ρ : G_{L} ⟶ A_{F,f}^{×}

to be the (continuous) homomorphism given the action on the total Tate module V_{f}(A). Denote by

N_{Φ} : A_{K}^{×} ⟶ A_{F}^{×}

the reflex norm and set η = (N_{Φ} ∘ N_{L/K})^{−1} : A_{L}^{×} → A_{F}^{×}. In this context, the Main Theorem of complex multiplication is: Theorem 1. There is a unique homomorphism α : A_{L}^{×} → F^{×}, continuous for the discrete topology on F^{×}, such that, if we define ψ = αη, then the diagram

(∗)  \xymatrix{A_{L}^{×} \ar[r]^{ψ} \ar[d]_{rec} & A_{F}^{×} \ar[d]^{pr} \\ G_{L}^{ab} \ar[r]_{ρ} & A_{F,f}^{×}}

commutes. Proposition 2. The homomorphism ψ : A_{L}^{×} ⟶ A_{F}^{×} in Theorem 1 is trivial on L^{×} ⊆ A_{L}^{×}.

Proof. If x ∈ L^{×}, then ψ(x)_{fin} = 1 by the commutativity of (∗). Note that η(x) ∈ F^{×}, since N_{Φ}|_{K^{×}} carries K^{×} into F^{×}. Thus, ψ(x) ∈ F^{×} and hence ψ(x) = 1. □

Consider the topological product

A_{F}^{×} = A_{F,f}^{×} × F_{∞}^{×} ⊆ ∏_{p}(F ⊗ Q_{p})^{×} × F_{∞}^{×},

where F_{∞} = F ⊗R ≅ ∏_{ϕ∈Φ} C_{ϕ}. Accordingly, we can write each s ∈ A_{F}^{×} as (s_{f}, s_{∞}), where s_{f} = (s_{p})_{p} and s_{∞} = (s_{ϕ})_{ϕ}. For any rational prime p, we define

ψ_{p} : A_{L}^{×} ⟶ (F ⊗ Q_{p})^{×} : x ↦ ψ(x)_{p}.

This map factors as

A_{L}^{×} ⟶^{rec} G_{L}^{ab} ⟶^{ρ} (F ⊗ Q_{p})^{×}

and, by (∗), ψ_{p} thereby gives the action of G_{L} on the p-adic Tate module T_{p}(A).

If τ is any embedding of F into C, then we define the Grössencharakter ψ_{τ} : A_{L}^{×} → C^{×} by the composition

A_{L}^{×} ⟶^{ψ_{∞}} F_{∞}^{×} = (F ⊗ R)^{×} ⟶^{τ⊗1} C^{×}.

Exactly one of τ or \overline{τ} is equal to an element ϕ ∈ Φ, so we can give an equivalent definition of ψ_{τ} by the formula ψ_{τ} (x) = τ(ψ(x)_{ϕ}).

Theorem 3. Let v be a prime of L. Then α, resp. ψ_{τ} for all τ : F ↪ C, is unramified at v if and only if A has good reduction at v. In case this holds, ι(α(π_{v})) reduces at v to the Frobenius endomorphism Fr_{v} of the v-fiber A_{k(v)}.

Proof. Choose any rational prime ℓ such that v ∤ ℓ. By definition, α is unramified at v if and only if ψ_{ℓ}, resp. ψ_{τ} for all τ, is unramified at v, since η(L_{v}^{×})_{ℓ} = 1, resp. η(L_{v}^{×})_{τ} = 1 for all τ. That this is equivalent to A being unramified at v follows from the fact that ψ_{ℓ} gives the action of G_{L} on the ℓ-adic Tate module of A. The statement that ι(α(π_{v})) reduces to Fr_{v} at v can be seen from (∗). □

1.2. L-functions. We pause here to recall some generalities regarding L-functions of Grössencharaktere and abelian varieties. Let M be a number field and let χ : A_{M}^{×} → C^{×} be a Grössencharakter of M. For all places v of M, we define the local L-function

L_{v}(χ, T) = \begin{cases}(1 − χ(π_{v})T)^{−1} & χ unramified at v, \\ 1 & χ ramified at v. \end{cases}

The global L-function of χ is given by L(χ, s) = ∏_{v} L_{v}(χ, Nv^{−s}). This Euler product converges absolutely and uniformly in some right half-plane, has analytic continuation to all of C (provided χ isn’t a power of the adelic norm), and satisfies a functional equation of standard type.

Similarly, if A is any abelian variety defined over M, for all places v of M we define the local L-function

L_{v}(A/M, T) = det(1 − Fr_{v} T | T_{ℓ}(A)^{I_{v}})^{−1},

where ℓ is any rational prime such that v ∤ ℓ. As before we define the global L-function of A over M by L(A/M, s) = ∏_{v} L_{v}(A/M, Nv^{−s}). By the Riemann Hypothesis for abelian varieties over finite fields, this product converges in some right half-plane, but the analytic continuation and functional equation of this Lfunction is only known in a few special cases (some of which are proved in these notes).

We are now in a position to identify the L-function of a CM abelian variety as the product of the L-functions of its associated Grössencharaktere. In order to compute the local factors at the bad primes, however, we will need the following result of Serre-Tate: Theorem 4. If (A, ι) is a CM abelian variety defined over L, then for any place v of L at which A has bad reduction and for any rational prime ℓ not divisible by v, T_{ℓ}(A)^{I_{v}} = 0.

Proof. See Remark (2) on p. 504 of Serre-Tate [7] or Brian’s notes [2, Thm. 6.5]. □

Theorem 5. If (A, ι) is a CM abelian variety defined over a number field L, then

L(A/L, s) = ∏_{τ:F ↪C} L(ψ_{τ} , s),

where ψ_{τ} is the Grössencharakter defined above Theorem 3.

Proof. We check that the local L-functions agree at all primes v of L. By Theorems 3 and 4, we only need to check at places v where A has good reduction, or equivalently where ψ_{τ} is unramified for all τ. Since V_{ℓ}(A) is a free F ⊗ Q_{ℓ}-module of rank 1 on which Fr_{v} acts as α(π_{v}) ∈ F, the characteristic polynomial of the action of Fr_{v} on V_{ℓ}(A) is the same as the characteristic polynomial of the action of α(π_{v}) on F ⊗ C ≅ ∏_{τ} C_{τ}, which is precisely

∏_{τ}(X − τ(α(π_{v}))) = ∏_{τ}(X − ψ_{τ} (π_{v})).

From this it is clear that L_{v}(A/L, T) = ∏_{τ} L_{v}(ψ_{τ} , T). □

Remark. No ψ_{τ} is a power of the adelic norm ∥ · ∥_{L}. Indeed, otherwise, by Thm. 3 the Riemann Hypothesis for abelian varieties over finite fields would show that ψ_{τ} = ∥ · ∥L^{\frac{1}{2}}. If v is any of the infinitely many degree 1 primes of L, then this would imply that ψ_{τ} (π_{v})^{2} = p, where p is the rational prime over which v lies. But ψ_{τ} (π_{v}) ∈ τ(F^{×}), so F contains a square root of infinitely many rational primes, a contradiction.

2. The potential CM case

This section will be devoted to generalizing Theorem 5 to situations where A does not necessarily have complex multiplication defined over a given field of definition for A.

2.1. Restriction of scalars. We will need to discuss some generalities concerning the behavior of the restriction of scalars functor on abelian varieties.

Let L_{0} be an arbitrary field and suppose L/L_{0} be a finite separable field extension. Fix an L_{0}-embedding L ↪ \overline{L}_{0}, so we get an associated open subgroup G_{L} ⊆ G_{L_{0}}. If A is an abelian variety defined over L, then the restriction of scalars functor Res_{L/L_{0}} A : Sch/L_{0} → Ab defined by X ↦ A(X ⊗_{L_{0}} L) is represented by an L_{0}-scheme, which we still denote by Res_{L/L_{0}} A or (for ease of notation) by A_{∗}. For the existence of the restriction of scalars without the affineness restriction imposed in Nick’s talk [6, Thm. 2.1], see the Néron Models book [1, §7.6 Thm. 4]. As in Nick’s talk [6, Thm. 2.5(1)], we have a canonical isomorphism

P : A_{∗,\overline{L}_{0}} ⟶^{∼} ∏_{σ:L↪\overline{L}_{0}} A_{\overline{L}_{0}}^{σ}

induced by the isomorphism \overline{L}_{0} ⊗_{L_{0}} L ≅ ∏_{σ}(\overline{L}_{0})_{σ} (both products are taken over embeddings σ : L ↪ \overline{L}_{0} over L_{0}). This shows in particular that A_{∗} is an abelian variety.

On \overline{L}_{0}-points, we have an isomorphism

A_{∗}(\overline{L}_{0}) ⟶^{∼} Z[G_{L_{0}}] ⊗_{Z[G_{L}]} A(\overline{L}_{0})

of G_{L_{0}}-modules defined by (x_{σ}) ↦ ∑_{σ} σ̃ ⊗ σ^{−1}x_{σ}, where σ̃ is any lift of σ to G_{L_{0}}. The inverse maps sends σ ⊗ x to (0, …, σx, …, 0) (all entries 0 except in the σ|_{L} spot). This shows in particular that

V_{ℓ}(A_{∗}) ≅ Q_{ℓ}[G_{L_{0}}] ⊗_{Q_{ℓ}[G_{L}]} V_{ℓ}(A)

for all rational primes ℓ, i.e., V_{ℓ}(A_{∗}) ≅ Ind_{L}^{L_{0}} V_{ℓ}(A) as representations of G_{L_{0}}. Proposition 6. With notation as above,

L(A_{∗}/L_{0}, s) = L(A/L, s).

Proof. Using the description via induced modules, one shows that

∏_{w|v} L_{w}(A/L, Nw^{−s}) = L_{v}(A/L_{0}, Nv^{−s})

for all primes v of L_{0} by a tedious calculation. See Neukirch [5, Thm. VII.10.4(iv)] for the details. □

2.2. L-functions, bis. Now assume that A is an abelian variety defined over a number field L_{0} such that (A_{\overline{L}_{0}}, ι) has CM of type (F, Φ). We assume that ι(F) ⊆ End_{\overline{L}_{0}}^{0}(A) is stable under the action of G_{L_{0}}. (This assumption holds in particular when A is absolutely simple, as in this case ι(F) = End_{\overline{L}_{0}}^{0}(A).) Let L ⊆ \overline{L}_{0} be the smallest extension of L_{0} such that ι(F) ⊆ End_{L}^{0}(A), so that L/L_{0} is a finite Galois extension. Set G = Gal(L/L_{0}). This Galois group acts on End_{L}^{0}(A) and hence on F via the embedding ι. We denote the fixed field of this action on F by F_{0}, so ι(F_{0}) = ι(F) ∩ End_{L_{0}}^{0}(A). Note that by definition of L and Artin’s lemma, we have that [F : F_{0}] = [L : L_{0}] and the action of G on F gives an identification G ≅ Gal(F/F_{0}). The set Σ = {τ : F ↪ C} has a natural action of G on the right. Proposition 7. In the situation above, there is an L_{0}-isogeny

A^{n} ∼ Res_{L/L_{0}}(A_{L}),

where n = [L : L_{0}] = [F : F_{0}].

Proof. We are free to replace A by an isogenous abelian variety, so we may assume without loss of generality that ι(O_{L_{0}}) ⊆ End_{L_{0}}^{0}(A). Let O be the order of L satisfying ι(O) = ι(O_{L}) ∩ End_{L}^{0}(A), which is a projective O_{L_{0}}-module of generic rank n. As the canonical morphism A → Res_{L/L_{0}}(A_{L}) is O_{L_{0}}-linear (this can be checked on \overline{L}_{0}-points, e.g.), it extends to an O-linear morphism

β : O ⊗_{O_{L_{0}}} A ⟶ Res_{L/L_{0}}(A_{L}).

It suffices to prove that this is an isogeny, so we show that the tangent map at the identity

T_{0}β : F ⊗_{F_{0}} T_{0}A ⟶ T_{0}(A_{L}) = L ⊗_{L_{0}} T_{0}A

is an isomorphism. By our identification G ≅ Gal(F/F_{0}) and the Normal Basis Theorem, F, resp. L, is a free module of rank 1 over F_{0}[G], resp. L_{0}[G]. One sees from the definitions that the map T_{0}β is G-equivariant. Therefore, it is enough to show that the associated map (T_{0}β)^{G} between G-invariant subspaces is an isomorphism, which is clear, since (T_{0}β)^{G} is the identity map on T_{0}A. □

Lemma 8. For all τ ∈ Σ and g ∈ G, we have L(ψ_{τ} , s) = L(ψ_{τg}, s).

Proof. We first check that gη|_{L^{×}} = ηg|_{L^{×}} for all g ∈ G. The map η^{−1} = N_{Φ}∘N_{L/K}, viewed as a map from L^{×} to F^{×}, is given by η^{−1}(x) = det_{F} (x | T_{0}A), and if x : T_{0}A → T_{0}A has matrix M with respect to some F -basis {t_{i}} of T_{0}A, then gx has matrix gM with respect to the F -basis {gt_{i}}. Thus, η(gx)^{−1} = det(gM) = g(det M) = g(η(x)^{−1}).

We now claim that α also commutes with the action of G, i.e., gα = αg for all g ∈ G. This is clear from the fact that both gα and αg are each equal the unique continuous character A_{L}^{×} → F^{×} which is equal to gη = ηg on L^{×} and sends π_{w} to the unique endomorphism of A reducing to Fr_{gw} at gw for primes w of L at which A has good reduction.

Let v be a prime of L_{0}. The above shows that for two primes w and w′ of L lying over the same prime v of L_{0}, ψ_{τ} is unramified at w if and only if it is unramified at w′. (This can also be seen from the fact that A is defined over L_{0} and hence has good reduction at w if and only if it has good reduction at w′.) If these equivalent conditions hold, then, for all g ∈ G, we have

ψ_{τg}(π_{w}) = τg(α(π_{w})) = τ(α(π_{gw})),

so

∏_{w|v} L_{w}(ψ_{τ} , T) = ∏_{w|v} L_{w}(ψ_{τg}, T)

since g permutes the primes over v. □

Theorem 9. In the above situation,

L(A/L_{0}, s) = ∏_{τ∈Σ/G} L(ψ_{τ} , s).

Proof. Let n = |G| = [L : L_{0}] = [F : F_{0}] and choose a prime v of L_{0}. For ease of notation, set A_{∗} = Res_{L/L_{0}}(A_{L}). We compute:

L_{v}(A/L_{0}, Nv^{−s})^{n} = L_{v}(A_{∗}/L_{0}, Nv^{−s})  by Prop. 7
= ∏_{w|v} L_{w}(A_{L}/L, Nw^{−s})  by Prop. 6
= ∏_{w|v} ∏_{τ∈Σ} L_{w}(ψ_{τ} , Nw^{−s})  by Thm. 5
= ∏_{w|v} ∏_{τ∈Σ/G} L_{w}(ψ_{τ} , Nw^{−s})^{n}  by Lemma 8

It follows that

L_{v}(A/L_{0}, Nv^{−s}) = ∏_{w|v} ∏_{τ∈Σ/G} L_{w}(ψ_{τ} , Nw^{−s}),

since both sides are inverses of polynomials in Nv^{−s} with constant term 1. □

References

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3. S. Lang, Complex multiplication, Springer-Verlag, 1983.

4. J. Milne, On the arithmetic of abelian varieties, Invent. Math. 17 (1972), 177–190.

5. J. Neukirch, Algebraic number theory, Springer-Verlag, 1999.

6. N. Ramsey, Weil restriction and adelic points, VIGRE CM seminar, 2005.

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