Base Change and the Birch-Swinnerton-Dyer Conjecture

M. Ram Murty and V. Kumar Murty*

Dedicated to the memory of Emil Grosswald

1. Introduction.

Let k be a number field and E an elliptic curve defined over k. Let L_{k}(s) denote the L-series of E over k. This is defined as an Euler product:

L_{k}(s) = ∏_{v} L_{v}(s)

where v runs through all the finite places of k and for v coprime to the conductor of E,

L_{v}(s) = (1 − a_{v}Nv^{−s} + Nv^{1−2s})^{−1}
Nv + 1 − a_{v}

represents the number of points of E mod v and Nv is the absolute norm of v. Birch and Swinnerton-Dyer conjecture that L_{k}(s) extends to an entire function, satisfies a suitable functional equation and at s = 1 has a zero of order equal to the rank of the group E(k) of k-rational points of E.

If K is a finite extension of k, it is evident that rank E(K) ≥ rank E(k). This observation motivates several natural questions concerning L_{K}(s). Given that L_{k}(s) extends to an entire function, does it follow that L_{K}(s) extends to an entire function ? This is the first question. As we shall see below, this is related to the problem of base change in the Langlands program. An affirmative answer to this question leads us to inquire whether

ord_{s=1} L_{K}(s) ≥ ord_{s=1} L_{k}(s)

whenever K ⊇ k.

In this paper, we prove the following theorems.

Theorem 1. Let E be an elliptic curve defined over k. Suppose that E has CM and K is a finite extension of k. Then, L_{K}(s) extends to an

* Research of both authors partially supported by grants from NSERC entire function of s. If K is Galois over k or if K is contained in a solvable extension of k, then L_{K}(s)/L_{k}(s) is entire. In particular,

ord_{s=1} L_{K}(s) ≥ ord_{s=1} L_{k}(s).

A classical conjecture of Taniyama predicts that every elliptic curve defined over ℚ occurs as a factor of the Jacobian of the modular curve. For CM elliptic curves, Shimura [11, 12] proved Taniyama’s conjecture. If such is the case in general, one can prove that for E defined over ℚ and K contained in a finite, solvable extension of ℚ that L_{K}(s) extends to an entire function and L_{K}(s)/L_{ℚ}(s) is entire. More generally, denote by A_{k} the adele ring of k and let 𝔄(GL_{2}(A_{k})) be the space of automorphic representations of GL_{2}(A_{k}). Langlands [8] has shown how to attach an L-function L(s, π) for each π ∈ 𝔄(GL_{2}(A_{k})). He established the analytic continuation and the functional equation for each L(s, π). It is suspected that L_{k}(s) (after a suitable translation) is equal to L(s, π) for some cuspidal π ∈ 𝔄(GL_{2}(A_{k})). This we shall call the Taniyama conjecture over k (or the generalised Taniyama conjecture).

Theorem 2. Suppose that E satisfies the Taniyama conjecture over k. If K is contained in a solvable extension of k, then L_{K}(s) extends to an entire function and L_{K}(s)/L_{k}(s) is entire. In particular,

ord_{s=1} L_{K}(s) ≥ ord_{s=1} L_{k}(s).

The theorems are highly reminiscent of the classical Aramata-Brauer theorem that ζ_{K}(s)/ζ_{k}(s) is entire if K/k is a Galois extension. This was later extended to the case when K is contained in a solvable extension of k by Uchida [14] and van der Waall [15]. It is a conjecture of Dedekind that ζ_{K}(s)/ζ_{k}(s) is always entire if K is a finite algebraic extension of k. A similar conjecture can be made here for L_{K}(s)/L_{k}(s) the truth of which would follow from the Langlands program.

There are two key ingredients in the proof of these theorems. The first is the theory of base change as initiated by Saito, Shintani, Langlands [9] and further developed by Arthur-Clozel [1]. As we shall see, the constraints of Theorem 2 are forced upon us by the present state of development of this theory. The second ingredient is a zero-counting formalism implicit in the work of Heilbronn [5] and Stark [13]. This formalism was made more explicit and applied to Artin’s holomorphy conjecture in Kumar Murty [10] and R. Foote and K. Murty [4]. We will begin with the formalism first and then discuss base change. In the final section, we apply these ideas to complete the proofs of the theorems.

2. Formalism.

Our approach applies in a wider context of an L-function formalism which is satisfied by a variety of objects in number theory and algebraic geometry. Let G be a finite group. For every subgroup H of G and complex character ψ of H, we attach a complex number n(H, ψ) satisfying the following properties:

(1) n(H, ψ + ψ′) = n(H, ψ) + n(H, ψ′),

(2) Invariance under induction: n(G, Ind_{H}^{G} ψ) = n(H, ψ).

The formalism can be applied to the case when G is the Galois group of a normal extension K/k and n(H, ψ) is the order of the zero at s = s_{0} of the Artin L-series attached to ψ corresponding to the Galois extension K/K^{H}. It can also be applied the situation when E is an elliptic curve over k and n(H, ψ) corresponds to the order of the zero at s = s_{0} of the “twist” by ψ (see definition below) of L_{K^{H}} (s) in the notation of section 1.

Set

θ_{H} = ∑_{ψ} n(H, ψ)ψ

where the sum is over all irreducible characters ψ of H. Our first step is to show that

Proposition 1. θ_{G}|_{H} = θ_{H}.

Proof.

\begin{aligned}θ_{G}|_{H} & = ∑_{χ} n(G, χ)χ|_{H} \\ & = ∑_{χ} n(G, χ) (∑_{ψ}(χ|_{H}, ψ)ψ) \end{aligned}

where the inner sum is over all irreducible characters of H and the outer sum is over all irreducible characters of G. By Frobenius reciprocity, (χ|_{H}, ψ) = (χ, Ind_{H}^{G} ψ) and so

θ_{G}|_{H} = ∑_{ψ} (∑_{χ} n(G, χ)(χ, Ind_{H}^{G}ψ)) ψ.

But now, by property (1), the inner sum is n(G, Ind_{H}^{G}ψ) which equals n(H, ψ) by property (2). Thus, θ_{G}|_{H} = θ_{H}.

This immediately implies:

Proposition 2. Let reg denote the regular representation of G. Suppose for every cyclic subgroup H of G, we have n(H, ψ) ≥ 0, then

∑_{χ} n(G, χ)^{2} ≤ n(G, reg)^{2}.

Proof. By the orthogonality relations,

(θ_{G}, θ_{G}) = ∑_{χ} n(G, χ)^{2}.

On the other hand,

(θ_{G}, θ_{G}) = \frac{1}{|G|} ∑_{g∈G} |θ_{G}(g)|^{2}.

By Proposition 1,

θ_{G}(g) = θ_{⟨g⟩}(g) = ∑_{ψ} n(⟨g⟩, ψ)ψ(g)

which is bounded by n(G, reg) in absolute value by our hypothesis and property (1). This completes the proof.

Similar reasoning implies

Proposition 3. Let ρ be an arbitrary character of G. Suppose for every cyclic subgroup H of G, and irreducible character ψ of H, we have n(H, ρ|_{H} ⊗ ψ) ≥ 0, then

∑_{χ} n(G, ρ ⊗ χ)^{2} ≤ n(G, ρ ⊗ reg)^{2}.

3. Base change and automorphic induction.

Let K/k be a Galois extension and let G = Gal(K/k). If ρ is an irreducible representation of G, let L(s, ρ, K/k) denote the Artin L-series attached to ρ. We can extend the definition of Artin L-series to an arbitrary representation of G by additivity:

L(s, ρ_{1} ⊕ ρ_{2}, K/k) = L(s, ρ_{1}, K/k)L(s, ρ_{2}, K/k).

If now ψ is a representation of a subgroup H of G, then L(s, ψ, K/K^{H}) is the Artin L-series belonging to the extension K/K^{H} where K^{H} denotes the field fixed by H. A simple calculation shows that Artin L-series are invariant under induction:

L(s, Ind_{H}^{G}ψ, K/k) = L(s, ψ, K/K^{H}).

Recall the Langlands’ reciprocity conjecture [8]: for each ρ, there is π(ρ) ∈ 𝔄(GL_{n}(A_{k})) (n = deg ρ ) so that

L(s, ρ, K/k) = L(s, π).

The natural question is, how does the map ρ ↦ π(ρ) behave under restriction to a subgroup? By the reciprocity conjecture, we should have ρ|_{H} ↦ π(ρ|_{H}) ∈ 𝔄(GL_{n}(A_{K^{H}})). What is L(s, ρ|_{H}, K/K^{H}) ? A standard group theoretic result is

Ind_{H}^{G}(ρ|_{H} ⊗ ψ) = ρ ⊗ Ind_{H}^{G}ψ.

Thus,

L(s, Ind_{H}^{G}(ρ|_{H} ⊗ ψ), K/k) = L(s, ρ ⊗ Ind_{H}^{G}ψ, K/k).

But the former L-function is L(s, ρ|_{H} ⊗ ψ, K/K^{H}) by the invariance of Artin L-series. Therefore, taking ψ = 1,

L(s, ρ|_{H}, K/K^{H}) = L(s, ρ ⊗ Ind_{H}^{G}1, K/k).

But Ind_{H}^{G}1 = reg_{H} is the permutation representation on the cosets of H. This suggests that we make the following definition. Let π ∈ 𝔄(GL_{n}(A_{k})). For each unramified π_{v}, let A_{v} ∈ GL_{n}(ℂ) be the semi-simple conjugacy class attached by Langlands [8]. If in addition, v is unramified in K, define

L_{v}(s, B(π)) = det(1 − A_{v} ⊗ reg_{H}(σ_{v})Nv^{−s})^{−1},

where σ_{v} is the Artin symbol of v. The idea is that there should be a B(π) ∈ 𝔄(GL_{n}(A_{M})), where M = K^{H}, so that the v-factor in the definition of L(s, B(π)) coincides with L_{v}(s, B(π)) as defined above for all but finitely many places v. The problem of base change is to determine when this map exists.

For n = 2, this was done by Langlands [9] He then used these ideas to deal with the tetrahedral case of Artin’s conjecture. The case n = 3 was considered by Flicker [2]. For arbitrary n, it is a recent breakthrough achieved by Arthur and Clozel [1]. Again, the situation is for M/k cyclic.

Now suppose the ψ is a representation of H. Corresponding to ψ there should be a π ∈ 𝔄(GL_{n}(A_{M})) where n = deg ψ. But the invariance of Artin L-series under induction implies that there should be an I(π) ∈ 𝔄(GL_{nr}(A_{k})) (r = [G : H]) so that for almost all places v, the v-factor of L(s, I(π)) coincides with the v-factor of L(s, Ind_{H}^{G}ψ, K/k). This map π ↦ I(π), called the automorphic induction map, is conjectured to exist. Again, Arthur and Clozel [1] showed this exists when M/k is cyclic and arbitrary n. Thus, if M/k is contained in a solvable extension of k, the base change maps and automorphic induction maps exist.

We have used the theory of Artin L-series to motivate the discussion of base change and automorphic induction. We will now use the existence of these maps to prove Theorems 1 and 2. We state the theorem of Arthur and Clozel [1] for future reference. We first need a few definitions.

Denote by M_{r} a generic r by r matrix and by I_{r} the r×r identity matrix. The standard parabolic subgroups of GL_{n} are in one-one correspondence with the partitions of n = n_{1} + ⋯ + n_{r}. The standard parabolic subgroup corresponding to a partition n = n_{1} + ⋯ + n_{r} consists of matrices of the form

( \begin{matrix}M_{n_{1}} & ∗ & ∗ & ∗ \\ & . & ∗ & ∗ \\ & & . & ∗ \\ & & & M_{n_{r}} \end{matrix} )

and any parabolic subgroup is a GL_{n} conjugate of a standard parabolic subgroup. Any parabolic subgroup P has a decomposition (called the Levi decomposition) of the form P = MN where N is the unipotent radical. In the case of the standard parabolic, M and N can be described as consisting of matrices of the form

( \begin{matrix}M_{n_{1}} \\ & . \\ & & . \\ & & & M_{n_{r}} \end{matrix} )  ( \begin{matrix}I_{n_{1}} & ∗ & ∗ & ∗ \\ & . & ∗ & ∗ \\ & & . & ∗ \\ & & & I_{n_{r}} \end{matrix} )

respectively. An automorphic representation π of GL_{n}(A_{k}) is said to be induced from cuspidal if there is a cuspidal unitary representation σ of M(A_{k}) where P = MN is a k-parabolic subgroup of GL_{n} such that

π = Ind _{M(A_{k})N(A_{k})}^{GL_{n}(A_{k})} (σ ⊗ 1).

Of course, a cuspidal automorphic representation of GL_{n}(A_{k}) is (trivially, by definition) induced from cuspidal.

Proposition 4. (Arthur and Clozel [1] )

Let E/F be a cyclic extension and π, Π denote representations which are induced from cuspidal of GL_{n}(A_{F}) and GL_{n}(A_{E}) respectively. Then B(π) and I(Π) exist.

We will also need results concerning the Rankin-Selberg convolution of two L-functions. Let π and σ be two cuspidal, unitary automorphic representations of GL_{n}(A_{k}) and GL_{m}(A_{k}) respectively. Let S be a finite set of primes such that π and σ are unramified outside of S. Form the L-function

L(s, π ⊗ σ) = ∏_{v∉S} det(1 − A_{σ,v} ⊗ A_{π,v} Nv^{−s})^{−1}

where A_{π,v} and A_{σ,v} are the semi-simple conjugacy classes of GL_{n}(ℂ) and GL_{m}(ℂ) attached to π and σ respectively.

Proposition 5. (Jacquet, Piatetski-Shapiro and Shalika [7] ) The Lfunction L(s, π ⊗ σ) extends to a meromorphic function of s.

Proposition 6. (Jacquet [6]) Let E/F be a cyclic extension and suppose π ∈ 𝔄(GL_{n}(A_{F})) and σ ∈ 𝔄(GL_{m}(A_{E})). Then, the Rankin-Selberg Lfunction satisfies the formal identity:

L(s, B(π) ⊗ σ) = L(s, π ⊗ I(σ)).

4. Lemmas.

We record in this section the necessary facts from the theory of elliptic curves that will be needed in the proofs of Theorems 1 and 2.

Lemma 1. (Deuring [2]) Let E be an elliptic curve defined over k. Suppose that E has CM by an order in an imaginary quadratic field F. If k ⊇ F, then L_{k}(s) is the product of two Hecke L-series of k. If k ⊉ F, then L_{k}(s) is equal to a Hecke L-series of kF which is a quadratic extension of k.

Lemma 2. The generalised Taniyama conjecture is true for CM elliptic curves.

Remark. This was proved by Shimura [11, 12] for elliptic curves over ℚ using Weil’s converse theorem.

Proof. A Hecke character of k is an automorphic form of GL_{1}(A_{k}). Thus, the sum of two Hecke characters corresponds to a (reducible) representation of GL_{2}(A_{k}). Suppose that E is defined over k and has CM by an order in F. By Lemma 1, we see by the previous remark that the lemma is true if k ⊇ F. If k ⊉ F, then by Lemma 1, L_{k}(s) equals a Hecke L-series L(s, ϕ) of kF, where ϕ ∈ 𝔄(GL_{1}(A_{kF})). We can apply proposition 4 to the cyclic extension kF/k to obtain I(ϕ) ∈ 𝔄(GL_{2}(A_{k})) and L_{v}(s, ϕ) = L_{v}(s, I(ϕ)) for all but finitely many places. The adjustment at the finite places involves multiplication by functions which are analytic and nonvanishing for Re(s) > 0. By the functional equation for L(s, I(ϕ)) and L(s, ϕ), we deduce the adjustment factor extends to an entire function (satisfying the same functional equation). This completes the proof.

Remark. The argument involving the adjustment at the finite places will be implicit in the remainder of the paper. This lemma could have been established by using the earlier work of Jacquet and Langlands as we are dealing with only quadratic extensions and the theory of GL_{2} applies.

We now recall the definition of Artin-Hecke L-series [16]. Let K/k be a Galois extension with group G. Let L(s, ψ) be a Hecke L-series of k and ρ a representation of G. Weil [16] defined the Artin-Hecke L-series L(s, ψ ⊗ ρ, K/k) as a Euler product over all places v of k, which at the finite unramified places v of k is

det(1 − ψ(v)ρ(σ_{v})Nv^{−s})^{−1}.

Weil proved using Brauer induction:

Lemma 3. (Weil [16] ) Each of these L-series L(s, ψ ⊗ρ, K/k) extends to a meromorphic function and satisfies the identity: if ϕ is a representation of a subgroup H of G, then

L(s, ψ ⊗ Ind_{H}^{G} ϕ, K/k) = L(s, ψ ∘ N_{K^{H}/k} ⊗ ϕ, K/K^{H}).

5. Proof of Theorem 1.

Fix s_{0} ∈ ℂ. Now suppose that K/k is a Galois extension with group G. For each subgroup H of G and complex character ψ of H, let n(H, ψ) be the order of the zero at s = s_{0} of the Artin-Hecke L-series L_{K^{H}} (s, ψ) which is the “twist” by ψ of the (single or the product of two) Hecke L-series L_{K^{H}} (s). Weil [16] proved by using Brauer induction that these L-series are meromorphic. Thus, n(H, ψ) is well-defined. Moreover, properties (1) and (2) are satisfied by them because the Artin-Hecke L-series are invariant under induction. If H is cyclic and ψ is an irreducible character of H, then n(H, ψ) ≥ 0 because the corresponding L-series is a Hecke L-series attached to K^{H} and is therefore entire. By proposition 2, we find

n(G, 1) ≤ n(G, reg).

Since L_{k}(s, reg) = L_{K}(s), we deduce that L_{K}(s)/L_{k}(s) is entire. This completes the proof in the CM case when K/k is Galois.

If now, K is contained in a solvable extension of k, we proceed as follows. We prove the theorem by induction on [K : k]. Let K̃ denote the Galois closure of K in k. Set G = Gal(K̃/k). Let H be the subgroup of G corresponding to K. It suffices to prove the result when H is a maximal subgroup of G because if G ⊇ I ⊇ H and M is the subfield fixed by I, then

L_{K}(s)/L_{k}(s) = L_{K}(s)/L_{M}(s) · L_{M}(s)/L_{k}(s).

The first factor is entire by induction and the second factor is entire since we can take M to correspond to a maximal subgroup of G.

Let A be a minimal normal subgroup. Then, it is elementary abelian and in particular A ≠ 1. Thus, HA = H or G. If HA = H, then A ⊆ H. But then the subfield M fixed by A is normal over k and contains K and so M = K̃, contrary to A ≠ 1. Thus, HA = G. Also, H ∩ A is a normal subgroup of G and also abelian, as it is contained in A. Hence, H ∩ A = 1 as A is minimal.

Let us write

Ind_{H}^{G}1_{H} − 1_{G} = ∑_{χ} m_{χ}χ

where m_{χ} are non-negative integers and χ ranges over irreducible characters of G. Consider

Ind_{H}^{G}(1_{H})|_{A} = ∑_{χ} m_{χ}χ|_{A}.

By Mackey’s theorem,

Ind_{H}^{G}(1_{H})|_{A} = Ind_{H∩A}^{A}(1_{H}|_{H∩A}) = Ind_{1}^{A} 1 = reg_{A} = ∑_{ϕ} ϕ

where ϕ ranges over all irreducible characters of A. Thus, for all χ,

m_{χ} = 0 or 1

and

(χ|_{A}, ϕ) = 0 or 1

for any irreducible character ϕ of A. Now take any such ϕ so that there is a χ so that m_{χ} ≠ 0 and (χ|_{A}, ϕ) = 1. Let T_{ϕ} be the inertia group of ϕ:

T_{ϕ} = {σ ∈ G : ϕ^{σ} = ϕ}.

Of course T_{ϕ} contains A and we can write T_{ϕ} = H_{ϕ}A where H_{ϕ} is contained in H. We can extend ϕ to a character ϕ̃ of T_{ϕ} by setting

ϕ̃(ha) = ϕ(a)

for any h ∈ H_{ϕ} and a ∈ A. Write

Ind_{A}^{T_{ϕ}}(ϕ̃|_{A}) = ∑_{ψ_{i}} m_{i}ψ_{i}

where ψ_{i} run through the irreducible characters of T_{ϕ}. Then (Ind_{A}^{T_{ϕ}} ϕ)|_{A} = [T_{ϕ} : A]ϕ and so ψ_{i}|_{A} is a multiple of ϕ. Also

(ψ_{i}|_{A}, ϕ) = (ψ_{i}, Ind_{A}^{T_{ϕ}} ϕ) = m_{i}.

Hence, restricting both sides of the penultimate formula to A,

[T_{ϕ} : A] = ∑_{i} m_{i}^{2}.

Moreover, the same formula implies that

Ind_{A}^{G}ϕ = ∑_{i} m_{i}Ind_{T_{ϕ}}^{G}ψ_{i}.

Now,

(Ind_{A}^{G} ϕ, Ind_{A}^{G} ϕ) = (ϕ, (Ind_{A}^{G}ϕ)|_{A})

= (ϕ, [T_{ϕ} : A] ∑_{i} ϕ^{g_{i}})

= [T_{ϕ} : A] = ∑_{i} m_{i}^{2}

where in the penultimate step g_{i} run over a set of coset representatives for T_{ϕ} in G. But

∑_{i} m_{i}^{2} = (Ind_{A}^{G} ϕ, Ind_{A}^{G} ϕ) = ∑_{i,j} m_{i}m_{j}(Ind_{T_{ϕ}}^{G}ψ_{i}, Ind_{T_{ϕ}}^{G}ψ_{j}) ≥ ∑_{i} m_{i}^{2}.

Thus, it follows that each Ind_{T_{ϕ}}^{G} ψ_{i} is an irreducible character of G. Since

1 = (χ|_{A}, ϕ) = (χ, Ind_{A}^{G} ϕ) = ∑_{i} m_{i}(χ, Ind_{T_{ϕ}}^{G}ψ_{i})

there is a unique i with m_{i} = 1 and (χ, Ind_{T_{ϕ}}^{G}ψ_{i}) ≠ 0. By the irreducibility, χ = Ind_{T_{ϕ}}^{G} ψ_{i}. But ψ_{i}|A = m_{i}ϕso ψ_{i}(1) = m_{i} = 1. Hence, χ is the induction of a one-dimensional character. Since

L_{K}(s)/L_{k}(s) = ∏_{χ} L(s, π ⊗ χ)^{m_{χ}},

where m_{χ} ≥ 0 where π is the Hecke character (or the sum of two Hecke characters) of k corresponding to K̃. By Lemma 3,

L(s, π ⊗ Ind_{T_{ϕ}}^{G}ψ_{i}) = L(s, π ∘ N_{E^{T_{ϕ}} /k} ⊗ ψ)

and the latter object is a Hecke L-series or a product of two such L-series which is entire. This completes the proof.

6. Proof of Theorem 2.

If Taniyama’s conjecture is true for E over k, then L_{k}(s) equals L(s, π) for some π ∈ 𝔄(GL_{2}(A_{k})). We will first handle the situation when K/k is a solvable extension with group G. Since the base change B(π) of π to K exists, L_{K}(s) is an entire function. The next step is to establish the analogue of Weil’s theorem that the twist by an irreducible representation ρ of G is a meromorphic function. Let

tr ρ = ∑_{i} m_{i}Ind_{H_{i}}^{G} χ_{i}

where H_{i} are abelian subgroups of G and χ_{i} are characters of H_{i}. We have

L(s, π ⊗ ρ) = ∏_{i} L(s, π ⊗ Ind_{H_{i}}^{G} χ_{i}).

Since χ_{i} are abelian characters of H_{i}, they correspond, by class field theory, to automorphic forms on GL_{1}(A_{K^{H_{i}}}). Once this identification is made, we can apply automorphic induction to obtain automorphic forms I(χ_{i}) on GL_{[K^{H_{i}}:k]}(A_{k}). By proposition 5, we know that the Rankin-Selberg convolutions L(s, π ⊗ I(χ_{i})) are meromorphic functions for all values of s. By the properties of the ℓ-adic representation, we know that

L_{K}(s) = ∏_{ρ} L(s, π ⊗ ρ)^{ρ(1)}.

Indeed, if E is an elliptic curve over k and K/k is a finite extension, the L-function of E over K is given by the family of ℓ-adic representations

ρ_{ℓ} : Gal(k̄/K)→Aut(T_{ℓ}(E_{/K}))

where T_{ℓ} denotes the Tate module. Since T_{ℓ}(E_{/K}) = T_{ℓ}(E_{/k}) as Gal(k̄/K)- modules, it follows that ρ_{ℓ} is the restriction of the representation

ψ_{ℓ} : Gal(k̄/k)→Aut(T_{ℓ}(E_{/k})).

Thus,

L(s, ρ_{ℓ}) = L(s, ψ_{ℓ}|_{Gal(k̄/k)}).

But

Ind_{Gal(k̄/K)}^{Gal(k̄/k)}(ψ_{ℓ}|_{Gal(k̄/k)}) = ψ_{ℓ} ⊗ Ind_{Gal(k̄/K)}^{Gal(k̄/k)} 1.

If we suppose that K/k is normal, this is

L(s, ψ_{ℓ} ⊗ reg) = ∏_{ρ} L(s, ψ_{ℓ} ⊗ ρ)^{ρ(1)},

where ρ ranges over the irreducible representations of Gal(K/k). Let H be a subgroup of G and ψ a character of H. Fix s_{0} ∈ ℂ. Define n(H, ψ) to be the order of the zero at s = s_{0} of the L-function

L(s, B(π) ⊗ ψ)

where B(π) is the base change of π to K^{H}. The base change map satisfies

L(s, B(π) ⊗ χ) = L(s, π ⊗ I(χ)).

From this it follows that n(H, ψ) satisfy property (2). Property (1) is immediate. An application of Proposition 2 implies the theorem. This completes the proof if K/k is a solvable Galois extension. The case when k is contained in a solvable extension of k is identical to the one given above in the CM case.

7. Concluding remarks.

The method is capable of further generalization and application. First, an application of Proposition 3 to the Artin L-series L(s, ρ, K/k) corresponding to a Galois representation yields the following result which can be thought of as a generalization of the Aramata-Brauer theorem:

Theorem 3. Let K/k be a Galois extension with group G. Let ρ be an irreducible representation of G and L(s, ρ, K/k) the corresponding Artin L-series. Then,

L(s, ρ ⊗ reg, K/k)/L(s, ρ, K/k)

is entire.

An identical derivation also yields the following theorem:

Theorem 4. Let π be an automorphic cuspidal representation of GL_{n}(A_{k}). If K is contained in a solvable extension of k and B(π) denotes the base change of π to K, then,

L(s, B(π))/L(s, π)

is entire.

In this theorem, if n = 1 and K/k is a Galois extension, then L(s, B(π))/L(s, π) is entire by an application of Lemma 3 instead of Proposition 4. This has arithmetic consequences. For instance, the zeta function of a CM abelian variety over an arbitrary number field is given in terms of Hecke L-series (see Yoshida [17]) and therefore the analogue of Theorem 1 applies to such varieties. There are other varieties for which the zeta functions have been identified. For instance, the Jacobians of the modular curves have zeta function equal to products of L-functions attached to modular forms by a theorem of Shimura. Recent work of Kottwitz identifies the zeta functions of other classes of Shimura varieties. In both instances, the appropriate generalization of Theorem 2 applies.

Acknowledgement. The authors would like to thank L. Clozel and H. Kisilevsky for their suggestions and comments.

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Department of Mathematics, McGill University, Montréal, Canada Department of Mathematics, University of Toronto, Toronto, Canada