On Zagier’s Conjecture for

Base Changes of Elliptic Curves

François Brunault

Received: June 15, 2012 Revised: December 21, 2012

Communicated by Stephen Lichtenbaum

Abstract. Let E be an elliptic curve over Q, and let F be a finite abelian extension of Q. Using Beilinson’s theorem on a suitable modular curve, we prove a weak version of Zagier’s conjecture for L E_{F} , 2 , where E_{F} is the base change of E to F.

2010 Mathematics Subject Classification: 11G40, 11G55, 19F27 Keywords and Phrases: Elliptic curves, L-functions, elliptic dilogarithm, Zagier’s conjecture, regulators, Beilinson’s conjecture

Introduction

Zagier conjectured in [19] very deep relations between special values of zeta functions at integers, special values of polylogarithms at algebraic arguments and K-theory. While the original conjectures concerned the Dedekind zeta function of a number field and Artin L-functions, theoretical and numerical results by many authors suggested an extension of these conjectures to elliptic curves (see [20] for an historical account). A precise formulation for elliptic curves over number fields was given by Wildeshaus in [17]. The conjecture on L E, 2 , where E is an elliptic curve over Q, was proved by Goncharov and Levin in [11]. In this article, we prove an analogue of Goncharov and Levin’s result for the base change of E to an arbitrary abelian number field. Let E be an elliptic curve defined over Q. Let F \overline{Q} be a finite abelian extension of Q, and let E_{F} be the base change of E to F. The L-function L E_{F} , s admits a factorization _{χ G} L E χ, s , where G is the group of \overline{Q}^{}- valued characters of G Gal F Q . Each factor L E χ, s has an analytic continuation to C with a simple zero at s 0. The functional equation relates L E_{F} , 2 with the leading term of L E_{F} , s at s 0.

Let D_{E} (resp. J_{E}) be the Bloch elliptic dilogarithm (resp. its “imaginary” cousin) on E C C Z τZ (see §2-3 for the definitions). These functions induce linear maps on the free abelian group Z E C , and thus on Z E \overline{Q} after fixing an embedding ι \overline{Q} C. Let Z E \overline{Q} ^{G_{F}} denote the subgroup of divisors fixed by G_{F} Gal \overline{Q} F . It carries a natural action of G. The main theorem of this article can be stated as follows.

Theorem 1. There exists a divisor ℓ Z E \overline{Q} ^{G_{F}} such that for every character χ G, the following identity holds

L^{}E χ, 0 _{Q^{}} ;  ;  ;  ;  ;  ; \frac{1}{π} _{σ G} χ σ D_{E} ℓ^{σ}  if χ is even, \frac{1}{πℑ τ} _{σ G} χ σ J_{E} ℓ^{σ} if χ is odd.  (1)

Using the Dedekind-Frobenius formula for group determinants, we deduce from Theorem 1 the following result. Let us denote the elements of G by σ_{1}, …, σ_{d} (resp. σ_{1}, \overline{σ_{1}}, …, σ_{d 2}, \overline{σ_{d 2}} if F is real (resp. complex), with d F Q .

Corollary (Weak version of Zagier’s conjecture for L E_{F} , 2 ). Let ℓ Z E \overline{Q} ^{G_{F}} be a divisor satisfying the identities (1) of Theorem 1. For any i, define ℓ_{i} ℓ^{σ_{i}^{1}} . If F is real, then we have

L E_{F} , 2 _{Q^{}}π^{d} det D_{E} ℓ_{i}^{σ_{j}}  _{1 i,j d} .  (2)

If F is complex, then we have

L E_{F} , 2 _{Q^{}}\frac{π^{d}}{ℑ τ ^{d 2}} det D_{E} ℓ_{i}^{σ_{j}}  _{1 i,j d 2} det J_{E} ℓ_{i}^{σ_{j}}  _{1 i,j d 2} . (3)

Remarks. 1. Wildeshaus’s formulation of the conjecture [17, Conjecture, Part 2, p. 366] uses Kronecker double series instead of D_{E} and J_{E}. The link between these objects is classical (see the proof of Prop. 6). We have chosen to formulate our results in terms of D_{E} and J_{E} because these functions are easier to compute numerically and make apparent the distinction according to the parity of χ.

2. Because of the definition of ℓ_{i}, the determinant appearing in (2) is a group determinant, indexed by G. In fact, the eigenvalues of the matrix D_{E} ℓ_{i}^{σ_{j}} are precisely the sums _{σ G} χ σ D_{E} ℓ^{σ} appearing in Theorem 1. This is an algebraic counterpart of the factorization of the L-value of E_{F} as a product of twisted L-values.

3. The divisor ℓ produced by Theorem 1 satisfies Goncharov and Levin’s conditions [11, (2)-(4)]. Following [20], let _{E F} Z E \overline{Q} ^{G_{F}} be the group of divisors satisfying these conditions. The strong version of Zagier’s conjecture predicts that if F is real (resp. complex), then for any divisors ℓ_{1}, …, ℓ_{d} _{E F} (resp. ℓ_{1}, …, ℓ_{d 2} _{E F}), the right-hand side of (2) (resp. (3)) is a rational multiple of L E_{F} , 2 (possibly equal to zero). As in the case F Q, this strong conjecture is beyond the reach of current technology.

In order to prove Theorem 1, we prove a weak version of Beilinson’s conjecture for the special value L^{d} E_{F} , 0 (see §3 for the definition of the objects involved in the following theorem).

Theorem 2. There exists a subspace _{E F} H_{Z}^{2} E_{F} , Q 2 such that R_{E F} reg_{E F} _{E F} is a Q-structure of H^{1} E_{F} C , R ^{}and

det R_{E F} L^{d} E_{F} , 0 det H^{1} E_{F} C , Q ^{}.  (4)

We prove Theorem 2 by using Beilinson’s theorem on a suitable modular curve. More precisely, we make use of a result of Schappacher and Scholl [15] on the (non geometrically connected) modular curve X_{1} N _{F}, where N is the conductor of E. We therefore need to work in the adelic setting. We establish a divisibility statement in the Hecke algebra of X_{1} N _{F} in order to get the desired result for E_{F}. The methods used in this article are of inexplicit nature and do not give rise, in general, to explicit divisors. However, Theorem 1 and its corollary can be made explicit in the particular case of the elliptic curve E X_{1} 11 and the maximal real subfield F Q ζ_{11} ^{}inside the cyclotomic field Q ζ_{11} . In fact, we may take for ℓ a divisor on the cuspidal subgroup of E. The tools for proving this are Kato’s explicit version of Beilinson’s theorem for the modular curve X_{1} N _{Q ζ_{m}}, the work of the author [3], as well as a technique used by Mellit [13] to get new relations between values of the elliptic dilogarithm. We hope to give soon an expanded account of this example. The organization of the article is as follows. In §1, we recall well-known facts about L E_{F} , s . In §2 and §3, we recall the definition of the regulator map and we compute it for E_{F} (Prop. 9). In §4, we explain the adelic setting for modular curves. In §5, we prove the divisibility we need in the Hecke algebra (Prop. 16). Finally, we give in §6 the proofs of the main results. We conclude with some remarks and a conjecture in the case F Q is not abelian. Acknowledgements. I would like to thank Anton Mellit for the very inspiring discussions which led to the discovery of the example alluded to above, which in turn motivated all the results presented here.

1 The L-function of the base change

By the Kronecker-Weber theorem, we have F Q ζ_{m} for some m 1, so that G is a quotient of Z mZ ^{}and G can be identified with a subgroup of the Dirichlet characters modulo m. Let f _{n 1} a_{n}q^{n} S_{2} Γ_{0} N be the newform associated to E. For any χ G, define L E χ, s L f χ, s , where f χ is the unique newform of weight 2 whose p-th Fourier coefficient is a_{p}χ p for every prime p Nm. The L-function of E_{F} has the following description.

Proposition 3. The following identity holds :

L E_{F} , s _{χ G}L f χ, s .  (5)

Proof. Let ρ ρ_{ℓ} _{ℓ} be the compatible system of 2-dimensional ℓ-adic representations of G_{Q} attached to f by Deligne [5]. By modularity L E_{F} , s L ρ _{G_{F}} , s . Using Artin’s formalism for L-functions, we have

L ρ _{G_{F}} , s L Ind_{G_{F}}^{G_{Q}} ρ _{G_{F}} , s .  (6)

If 1_{G_{F}} denotes the trivial representation of G_{F}, we have

Ind_{G_{F}}^{G_{Q}} ρ _{G_{F}} Ind_{G_{F}}^{G_{Q}} 1_{G_{F}} Res_{G_{Q}}^{G_{F}} ρ
Ind_{G_{F}}^{G_{Q}} 1_{G_{F}} ρ  (7)

_{χ G}ρ χ.

(Here we chose embeddings \overline{Q} \overline{Q_{ℓ}}.) Finally, since an irreducible ℓ-adic representation of G_{Q} is determined by the traces of all but finitely many Frobenius elements, the compatible system associated to f χ is ρ χ, so that L ρ χ, s L f χ, s for any χ G.

Proposition 4. We have L E_{F} , 2 _{Q^{}}π^{2d}L^{d} E_{F} , 0 , where L^{d} E_{F} , 0 denotes the d-th derivative at s 0.

Proof. Since each L f χ, s has a simple zero at s 0, we get

\frac{L^{d} E_{F} , 0}{d!} _{χ G}L^{}f χ, 0 ,  (8)

Let N_{f χ} be the level of the newform f χ. Putting Λ f χ, s N_{f χ}^{s 2} 2π ^{s}Γ s L f χ, s , we have [7, §5]

Λ f χ, s w_{f χ} Λ f \overline{χ}, 2 s  s C  (9)

where w_{f χ} is the pseudo-eigenvalue of f χ with respect to the Atkin-Lehner involution of level N_{f χ} . Note that (9) implies w_{f χ} w_{f \overline{χ}} 1. Letting w χ G w_{f χ} , we have

w^{2} _{χ G}w_{f χ} w_{f \overline{χ}} 1  (10)

so that w  1. Moreover Λ f χ, 0  L^{}f χ, 0 and Λ f \overline{χ}, 2

N_{f \overline{χ}} 4π^{2} L f \overline{χ}, 2 . Taking the product over χ G yields the result.

2 The regulator map on Riemann surfaces

In this section, we recall the definition of the regulator map on compact Riemann surfaces [8, §1], and its computation in the case of elliptic curves. Let X be a compact connected Riemann surface, and X be its field of meromorphic functions. For any f, g X ^{}, consider the 1-form

η f, g log f darg g log g darg f .  (11)

For any f X 0, 1 , the differential form η f, 1 f is exact on X f^{1} 0, 1, . More precisely η f, 1 f d D f , where D is the Bloch-Wigner dilogarithm function [18]. Let K_{2} X be the Milnor K_{2}-group associated to X . The regulator map on X is the unique linear map

reg_{X} K_{2} X H^{1} X, R  (12)

such that for any f, g X ^{}and any holomorphic 1-form ω on X, we have

X reg_{X} f, g ω \frac{1}{2π} _{X} η f, g ω.  (13)

The map reg_{X} is well-defined by exactness of η f, 1 f and Stokes’ theorem. The construction of reg_{X} easily extends to the case where X is compact but not connected. Indeed, put X _{i 1}^{r} X_{i} where X_{1}, …, X_{r} are the connected components of X. Then K_{2} X _{i} K_{2} X_{i} as well as H^{1} X, R _{i} H^{1} X_{i}, R , and we define reg_{X} to be the direct sum of the maps reg_{X_{i}} for 1 i r. Let us recall the classical computation of the regulator map on a complex torus [1, §4]. Let E_{τ} C Z τZ with τ C, ℑ τ 0. The map z exp 2iπz induces an isomorphism E_{τ} C^{}q^{Z}, where q exp 2iπτ . Let D_{q} E_{τ} R be the Bloch elliptic dilogarithm, defined by D_{q} x _{n}^{}D xq^{n} for any x C^{}. We will also use the function J_{q} E_{τ} R, which is defined as follows. Let J C^{}R be the function defined by J x log x log 1 x if x 1, and J 1 0. Following [18], we put

J_{q} x _{n 0}^{}J xq^{n} _{n 1}^{}J x^{1}q^{n}  \frac{1}{3} log^{2} q B_{3} \frac{log x}{log q}  x C^{}(14)

where B_{3} X^{3} \frac{3}{2}X^{2} \frac{X}{2} is the third Bernoulli polynomial. The function J_{q} is well-defined since J x J \frac{1}{x} log^{2} x and B_{3} X 1 B_{3} X 3X^{2}. Both functions D_{q} and J_{q} extend to linear maps Z E_{τ} R, by setting D_{q} _{i} n_{i} P_{i} _{i} n_{i}D_{q} P_{i} and similarly for J_{q}.

Definition 5. For any f, g  E_{τ} ^{}with divisors div f  _{i} m_{i} P_{i} and

div g _{j} n_{j} Q_{j} , the divisor β f, g Z E_{τ} is defined by

β f, g _{i,j} m_{i}n_{j} P_{i} Q_{j} .  (15)

The following classical result expresses the regulator map on E_{τ} in terms of D_{q} and J_{q}.

Proposition 6. For any f, g E_{τ} ^{}, we have

E_{τ} η f, g dz D_{q} iJ_{q} β f, g .  (16)

Proof. We have _{E_{τ}} η f, g dz \frac{ℑ τ ^{2}}{π} K_{2,1,τ} β f, g by [1, §4.3] and [6, (6.2)], where K_{2,1,τ} is the linear extension of the following Eisenstein-Kronecker series on E_{τ} :

K_{2,1,τ} z _{\begin{matrix}λ Z τZ \\ λ 0 \end{matrix}} \frac{exp \frac{2iπ}{τ \overline{τ}} z\overline{λ} \overline{z}λ}{λ^{2}\overline{λ}}  z C Z τZ .  (17)

The result now follows from the formula \frac{ℑ τ ^{2}}{π} K_{2,1,τ} D_{q} iJ_{q}, for which we refer to [2, Thm 10.2.1] and [18, §2, p. 616].

3 The regulator map on the base change

Let X be a connected (but not necessarily geometrically connected) smooth projective curve over Q. Its function field Q X embeds into X C , so we get a natural map K_{2} Q X K_{2} X C . Let c denote the complex conjugation on X C . For any f, g Q X ^{}, we have c^{}η f, g η f, g , so that (12) induces a map

K_{2} Q X H^{1} X C , R ^{},  (18)

where ^{}denotes the 1 -eigenspace of c^{}. Let K_{2} X be the Quillen algebraic K_{2}-group associated to X. Recall that the motivic cohomology group H_{}^{2} X, Q 2 K_{2}^{2} X is defined as the second Adams eigenspace of K_{2} X Q. The exact localization sequence in K-theory yields a canonical injective map K_{2} X Q K_{2} Q X Q which is compatible with the Adams operations, so that in fact K_{2}^{2} X K_{2} X Q. The integral subspace H_{Z}^{2} X, Q 2 H_{}^{2} X, Q 2 is the image of the map K_{2} Q K_{2} X Q for any proper regular model Z of X (see [16] for a definition in a more general setting). Tensoring (18) with Q and restricting to the integral subspace gives the Beilinson regulator map on X :

reg_{X} H_{Z}^{2} X, Q 2 H^{1} X C , R ^{}.  (19)

Note that the real vector space H^{1} X C , R ^{}admits the natural Q-structure H_{X} H^{1} X C , Q ^{}. Any finite morphism φ X Y between smooth projective curves over Q induces maps φ^{}K_{2} Y K_{2} X and φ_{}K_{2} X K_{2} Y , the latter being defined by K_{2} X ^{}K_{2}^{}X ^{φ_{}}K_{2}^{}Y ^{}K_{2} Y . It is known that φ^{}Q and φ_{}Q preserve the integral subspaces [16, Thm 1.1.6(i)]. Moreover, the Beilinson regulator maps associated to X and Y are compatible with φ^{}and φ_{}(this can be seen at the level of Riemann surfaces). Let us return to our elliptic curve E. Fix an isomorphism E C E_{τ} which is compatible with complex conjugation, and let q exp 2iπτ . Let D_{E} and J_{E} be the real-valued functions on E C induced by D_{q} and J_{q} respectively^{1}. The space H^{1} E C , Q ^{}is generated by the 1-form η^{}, with

η^{}dz d\overline{z}  and  η^{}\frac{dz d\overline{z}}{τ \overline{τ}} .  (20)

Lemma 7. Let f, g C E ^{}and ℓ β f, g . We have

reg_{E C} f, g \frac{1}{2π} D_{E} ℓ η^{}\frac{J_{E} ℓ}{2ℑ τ} η^{}.  (21)

Proof. Put reg_{E C} f, g a^{}η^{}a^{}η^{}with a^{}, a^{}R. Taking the wedge product with dz and integrating over E C yields

E C reg_{E C} f, g dz a^{}2iℑ τ a^{}.  (22)

Using (13) with Prop. 6 and identifying the real and imaginary parts gives the lemma.

Let Σ be the set of embedding of F into C. We consider E_{F} E _{Spec Q} Spec F as a scheme over Spec Q, so that E_{F} C is the disjoint union of d copies of E C . In particular

H^{1} E_{F} C , R _{ψ Σ}H^{1} E C , R  (23)

and H^{1} E_{F} C , Q decomposes accordingly. The group G acts from the right on E_{F}. This induces a left action of G on H^{1} E_{F} C , Q . For any character χ G, consider the idempotent

e_{χ}  \frac{1}{G}_{σ G}χ σ σ \overline{Q} G .  (24)

It acts on H^{1} E_{F} C , \overline{Q} R . For any ψ Σ, let η^{}ψ be the 1-form η^{} sitting in the ψ-component of (23). Note that the embedding ι \overline{Q} C induces a distinguished element ι Σ. Define

η_{χ}  ;  ;  ;  ;  ;  ; e_{χ} η^{}ι if χ is even, e_{χ} η^{}ι if χ is odd.  (25)

1The lattice Z τZ is uniquely determined by E, and q is a well-defined real number such that 0 q 1. But the pair D_{E}, J_{E} is defined only up to sign (choosing an isomorphism E C E_{τ} amounts to specifying an orientation of E R ).

Documenta Mathematica 18 (2013) 395–412

Lemma 8. If c E_{F} C E_{F} C is the map induced by complex conjugation on Spec C, then c^{}η_{χ} η_{χ}.

Proof. For any ψ Σ, we have c^{}η^{}ψ η^{}\overline{ψ} . It follows that

c^{}η_{χ} \frac{1}{G}_{σ G}χ σ c^{}σ η^{χ 1} ι

\frac{χ 1}{G} _{σ G}χ σ \overline{σ} η^{χ 1} ι .

Since χ 1 χ σ χ \overline{σ} , we get the result.

The map β induces a linear map F E ^{}F E ^{}Z E \overline{Q} ^{G_{F}}, which we still denote by β. The following proposition computes explicitly the regulator map associated to E_{F}.

Proposition 9. Let γ F E ^{}F E ^{}and ℓ β γ . For any χ G, we have e_{χ} reg_{E F} γ μ_{χ} ℓ η_{χ}, where μ_{χ} ℓ \overline{Q} R is given by

μ_{χ} ℓ  ;  ;  ;  ;  ;  ; \frac{1}{2π} _{σ G} χ σ D_{E} ℓ^{σ}  if χ is even, \frac{1}{4πℑ τ} _{σ G} χ σ J_{E} ℓ^{σ} if χ is odd.  (26)

Proof. Put r reg_{E F} γ . By Lemma 7, the ψ-component of r is

r_{ψ}  \frac{1}{2π} D_{E} ψ ℓ η^{}ψ \frac{J_{E} ψ ℓ}{2ℑ τ} η^{}ψ .  (27)

Since e_{χ} r and η_{χ} belong to the same G-eigenspace, it suffices to compare their ι-components. By definition, we have η_{χ} _{ι} \frac{1}{G}η^{χ 1} . Moreover

e_{χ} r _{ι} \frac{1}{G}_{σ G}χ σ σ r _{ι} \frac{1}{G}_{σ G}χ σ r_{ι σ}  (28)
\frac{1}{2π G}_{σ G}χ σ D_{E} ℓ^{σ} η^{}\frac{J_{E} ℓ^{σ}}{2ℑ τ} η^{}.  (29)

But D_{E} \overline{P} D_{E} P and J_{E} \overline{P} J_{E} P for any P E C , so that the terms involving J_{E} (resp. D_{E}) cancel out if χ is even (resp. odd).

4 Modular curves in the adelic setting

Let A_{f} be the ring of finite adèles of Q. For any compact open subgroup K GL_{2} A_{f} , there is an associated smooth projective modular curve \overline{M}_{K} over Q. For example X N \overline{M}_{K N} and X_{1} N \overline{M}_{K_{1} N}, where

K N ker GL_{2} Z GL_{2} Z NZ  (30)
K_{1} N  g GL_{2} Z ; g \begin{matrix} &  \\ 0 & 1 \end{matrix}  mod N .  (31)

The Riemann surface \overline{M}_{K} C can be identified with the compactification of GL_{2} Q 𝔥^{}GL_{2} A_{f} K. The set of connected components of \overline{M}_{K} C is in bijection with { \atop Z^{}}det K . For any g GL_{2} A_{f} , we have an isomorphism g \overline{M}_{K} ^{}\overline{M}_{g^{1} Kg} over Q, which is given on the complex points by τ, h τ, hg . For any compact open subgroups K^{}K of GL_{2} A_{f} , we have a finite morphism π_{K^{},K} \overline{M}_{K^{}}\overline{M}_{K}. The Hecke algebra _{K} is the space of functions K GL_{2} A_{f} K \overline{Q} with finite support, equipped with the convolution product [4]. It acts on H^{1} \overline{M}_{K} C , \overline{Q} and Ω^{1} \overline{M}_{K} \overline{Q}. Let T_{K} T_{\overline{M}_{K}} be the image of _{K} in End\overline{Q} Ω^{1} \overline{M}_{K} \overline{Q} . Let

, H^{1} \overline{M}_{K} C , R ^{}Ω^{1} \overline{M}_{K} R R  (32)

be the perfect pairing induced by Poincaré duality. For any T _{K}, we have T η, ω η, T^{}ω , where T^{}_{K} is defined by T^{}g T g^{1} , so that the action of _{K} on H^{1} \overline{M}_{K} C , \overline{Q} R ^{}factors through T_{K}. Following [15, 1.1.1], let _{K} K_{2} \overline{M}_{K} Q be the subspace of Beilinson elements, and let

K _{K^{}K} π_{K^{},K} _{}_{K^{}}K_{2} \overline{M}_{K} Q.  (33)

Schappacher and Scholl [15, 1.1.2] proved that _{K} H_{Z}^{2} \overline{M}_{K}, Q 2 and that reg_{\overline{M}_{K}} _{K} is a Q-structure of H^{1} \overline{M}_{K} C , R ^{}whose determinant with respect to the natural Q-structure H_{\overline{M}_{K}} is given by the leading term of L h^{1} \overline{M}_{K} , s at s 0. In the following, we assume K _{p} K_{p}, where K_{p} a compact open subgroup of GL_{2} Q_{p} . The Hecke algebra then decomposes as a restricted tensor product

K  _{p}^{}_{K_{p}}. For any prime p, let T p  _{K} (resp. T p, p  _{K}) be the

characteristic function of K \begin{matrix}ϖ_{p} & 0 \\ 0 & 1 \end{matrix} K (resp. K \begin{matrix}ϖ_{p} & 0 \\ 0 & ϖ_{p} \end{matrix} ), where ϖ_{p} A_{f}^{}

has component p at the place p, and 1 elsewhere. Let T p (resp. T p, p ) be the image of T p (resp. T p, p ) in T_{K}. When K needs to be specified, we write T p _{K} or T p _{\overline{M}_{K}}.

For any integer M 1, we let _{K}^{M} _{K} be the subalgebra generated by the K_{p} for p M. We use the notation T_{K}^{M} for the corresponding subalgebra of T_{K}.

Lemma 10. If K M K then T_{K}^{M} is in the center of T_{K}.

Proof. For any prime p M, we have K_{p} GL_{2} Z_{p} and by Satake the map \overline{Q} T, S, S^{1} _{K_{p}} given by T T p and S T p, p is an isomorphism. In particular _{K_{p}} is contained in the center of _{K}, whence the result.

Let U_{F}  { \atop Z^{}}denote the preimage of Gal Q ζ_{m} F  Z mZ ^{}under the

natural map { \atop Z^{}}Z mZ ^{}(note that U_{F} does not depend on m). For any compact open subgroup K GL_{2} A_{f} with det K { \atop Z^{}}, let

K_{F}  k K; det k U_{F} .  (34)

Let pr A_{f}^{}{ \atop Z^{}}be the projection associated to the decomposition A_{f}^{}

Q_{0} { \atop Z^{}}.

Definition 11. Let γ GL_{2} A_{f} G be the composite morphism

GL_{2} A_{f} ^{det} A_{f}^{}^{pr} { \atop Z^{}}\frac{Z}{mZ} ^{}G.  (35)

Note that there is an exact sequence

1 K_{F} K ^{γ _{K}} G 1.  (36)

The sequence (36) induces a right action of G on \overline{M}_{K_{F}}, and thus a left action of G on Ω^{1} \overline{M}_{K_{F}} . Moreover, the curve \overline{M}_{K_{F}} can be identified with \overline{M}_{K} F as a curve over Q, and we have a bijection

\overline{M}_{K_{F}} C ^{}G \overline{M}_{K} C  (37)
τ, g γ g , τ, g .

The action of G on \overline{M}_{K_{F}} C corresponds via (37) to the action by translation on the first factor of G \overline{M}_{K} C . Now let us consider the case K K_{1} N , so that \overline{M}_{K_{F}} X_{1} N _{F}. By the previous discussion, the image of G in End Ω^{1} X_{1} N _{F} \overline{Q} is contained in T_{X_{1} N F} . In order to ease notations, let T_{X_{1} N F}^{Nm} End Ω^{1} X_{1} N _{F} \overline{Q}. Let G be the subalgebra of T_{X_{1} N F} generated by and G.

Lemma 12. The algebra G is commutative.

Proof. Note that K Nm K_{1} N _{F}, so is commutative and commutes with G by Lemma 10. Since G is abelian, the result follows.

Since Ω^{1} X_{1} N _{F} Ω^{1} X_{1} N F, we can define the base change morphism ν_{F} End Ω^{1} X_{1} N End Ω^{1} X_{1} N _{F} by ν_{F} T T id_{F}. For any α Z mZ ^{}, let σ_{α} be its image in G.

Lemma 13. For any prime p Nm, we have

ν_{F} T p _{X_{1} N} T p _{X_{1} N F} σ_{p} G  (38)
ν_{F} T p, p _{X_{1} N} T p, p _{X_{1} N F} σ_{p}^{2} G.  (39)

Proof. Let g \begin{matrix}ϖ_{p} & 0 \\ 0 & 1 \end{matrix} and K K_{1} N g^{1}K_{1} N g K_{1} N K_{0} p . Note

that det K { \atop Z^{}}. Consider the following correspondence

\xymatrix{ & \overline{M}_{K} \ar[dl]_{α} \ar[dr]^{β} \\ X_{1} N \ar@{-->}[rr]^{T p _{X_{1} N}} &  & X_{1} N}  (40)

where α π_{K,K_{1} N} and β g^{1} π_{K,g^{1} K_{1} N g} π_{gKg^{1} ,K_{1} N} g^{1}. Then T p _{X_{1} N} β_{}α^{}on Ω^{1} X_{1} N . Similarly T p _{X_{1} N F} is defined by

\xymatrix{ & \overline{M}_{K_{F}} \ar[dl]_{α_{F}} \ar[dr]^{β_{F}} \\ X_{1} N _{F} \ar@{-->}[rr]^{T p _{X_{1} N F}} &  & X_{1} N _{F}}  (41)

where α_{F} is the natural projection and β_{F} is induced by g^{1}. Using the identification \overline{M}_{K_{F}} \overline{M}_{K} F and the description (37) of the complex points, we obtain α_{F} α id_{F} and β_{F} β γ g^{1} . Since γ g σ_{p^{1}} , we get T p _{X_{1} N F} ν_{F} T p _{X_{1} N} σ_{p} _{}and thus (38). The proof of (39) is similar.

5 A divisibility in the Hecke algebra

In this section we define and study a projection associated to E_{F} using the Hecke algebra of X_{1} N _{F}. Let φ X_{1} N E be a modular parametrization of the elliptic curve E, and let φ_{F} X_{1} N _{F} E_{F} be the base change of φ to F. Consider the map e_{F} \frac{1}{deg φ_{F}} φ_{F} ^{}φ_{F} _{}on Ω^{1} X_{1} N _{F} .

Lemma 14. We have e_{F}^{2} e_{F} and e_{F} G.

Proof. The first equality follows from φ_{F} _{}φ_{F} ^{}deg φ_{F}. We have e_{F} ν_{F} e where e \frac{1}{deg φ}φ^{}φ_{}End_{Q} Ω^{1} X_{1} N . The image of e is the Q-vector space generated by ω_{f} 2iπf z dz. Since f is a newform of level N, the Atkin-Lehner-Li theory implies that e T_{X_{1} N}^{Nm}. The result now follows from Lemma 13.

The space Ω \begin{matrix}lim \\ \end{matrix}_{K} Ω^{1} \overline{M}_{K} \overline{Q} has a natural GL_{2} A_{f} -action and decomposes as a direct sum of irreducible admissible representations Ω_{π} of GL_{2} A_{f} . For any K we have Ω^{K} Ω^{1} \overline{M}_{K} \overline{Q}. Let Π K be the set of such π satisfying Ω_{π}^{K} 0 . By [12, p. 393], we have

Ω^{1} \overline{M}_{K} \overline{Q} _{π Π K} Ω_{π}^{K}  (42)

where each Ω_{π}^{K} is a simple T_{K}-module. In particular T_{K} is a semisimple algebra. By Lemma 10, the algebra is contained in the center of T_{K_{1} N _{F}}. Using [12, Prop 2.11], we deduce that acts by scalar multiplication on each Ω_{π}^{K_{1} N _{F}}, so there exists a morphism θ_{π} \overline{Q} such that T acts as θ_{π} T on Ω_{π}^{K_{1} N _{F}}. The multiplicity one and strong multiplicity one theorems [14] ensure that the characters θ_{π} _{π Π K_{1} N _{F}} are pairwise distinct.

For any χ G, let π f χ be the automorphic representation of GL_{2} A_{f} corresponding to the modular form f χ. We have π f χ π f χ det , where χ A_{f}^{}Q_{0} C^{}denotes the adèlization of χ, sending ϖ_{p} to χ p for every p m. Since π f Π K_{1} N , it follows that π f χ Π K_{1} N _{F} .

Lemma 15. For any prime p Nm, we have

θ_{π f χ} T p a_{p}χ p  (43)
θ_{π f χ} T p, p χ p ^{2}.  (44)

Proof. We know that θ_{π f} T p a_{p} and θ_{π f} T p, p 1. The equalities (43) and (44) follow formally from the fact that χ det is equal to χ p on the

double coset K_{1} N _{F} \begin{matrix}ϖ_{p} & 0 \\ 0 & 1 \end{matrix} K_{1} N _{F}.

Let e_{f χ} Ω^{1} X_{1} N _{F} \overline{Q} Ω_{π f χ}^{K_{1} N _{F}} be the projection induced by (42). The multiplicity one theorems imply that e_{f χ} .

Proposition 16. The element e_{χ}e_{F} is divisible by e_{f χ} in G.

Proof. Since e_{χ}, e_{F} and e_{f χ} are commuting projections, it suffices to prove that the image of e_{χ}e_{F} is contained in the image of e_{f χ} . We know that the image of φ^{}Ω^{1} E Ω^{1} X_{1} N lies in the kernel of T p a_{p} T_{X_{1} N}. Therefore the image of φ_{F}^{}lies in the kernel of ν_{F} T p a_{p}. Using Lemma 13, it follows that in G we have

T p σ_{p}e_{F} a_{p}e_{F} .  (45)

Applying e_{χ} to both sides and using the identity e_{χ}σ_{p} \overline{χ} p e_{χ} yields

T p e_{χ}e_{F} a_{p}χ p e_{χ}e_{F} .  (46)

The same argument shows that T p, p e_{χ}e_{F} χ p ^{2}e_{χ}e_{F}. The proposition now follows from Lemma 15 and the multiplicity one theorems.

6 Proof of the main results

Recall that φ X_{1} N E is a modular parametrization, and that φ_{F} is the base change of φ to F. We have a commutative diagram

\xymatrix{K_{2} X_{1} N _{F} Q \ar[r] \ar[d]_{φ_{F} _{}} & H^{1} X_{1} N _{F} C , R ^{} \ar[d]^{φ_{F} _{}} \\ K_{2} E_{F} Q \ar[r] & H^{1} E_{F} C , R ^{}}(47)

where the horizontal maps are the regulator maps on X_{1} N _{F} and E_{F}. The strategy of the proof is to use Beilinson’s theorem on X_{1} N _{F} and then to get back to E_{F} using the Hecke algebra. Let _{E F} φ_{F} _{}_{X_{1} N F} K_{2} E_{F} Q. By [15, 1.1.2(iii)], we have E F H_{Z}^{2} E_{F} , Q 2 . We want to prove that R_{E F} reg_{E F} _{E F} is a Q-structure satisfying (4). Since _{X_{1} N F} is stable by the Hecke algebra, the spaces _{E F} and R_{E F} are stable by G. For any χ G, let R_{χ} e_{χ} R_{E F} \overline{Q} and H_{χ} e_{χ} H_{E F} \overline{Q} . We want to compare R_{χ} and H_{χ}. We have

φ_{F}^{}R_{χ} e_{χ}φ_{F}^{}R_{E F} \overline{Q}
e_{χ}e_{F} reg_{X_{1} N F} _{X_{1} N F} \overline{Q} .  (48)

Similarly, we have

φ_{F}^{}H_{χ} e_{χ}e_{F} H_{X_{1} N F} \overline{Q} .  (49)

We will build on the following theorem of Schappacher and Scholl. Let λ_{χ} be the unique element of \overline{Q} R ^{}such that for every ψ \overline{Q} C, we have ψ λ_{χ} L^{}f χ^{ψ}, 0 C^{}. By [15, 1.2.4 and 1.2.6], we have

e_{f χ} reg_{X_{1} N F} _{X_{1} N F} \overline{Q} λ_{χ} e_{f χ} H_{X_{1} N F} \overline{Q} .  (50)

By Prop. 16, the equality (50) remains true when e_{f χ} is replaced by e_{χ}e_{F}, so that φ_{F}^{}R_{χ} λ_{χ} φ_{F}^{}H_{χ} by (48) and (49). Since φ_{F}^{}is injective, we get R_{χ} λ_{χ} H_{χ}. Put V H^{1} E_{F} C , R ^{}and V_{χ} e_{χ} V \overline{Q} for any χ G.

Lemma 17. The R G -module V is free of rank 1.

Proof. By Poincaré duality V Hom_{Q} Ω^{1} E_{F} , R , and Ω^{1} E_{F} Ω^{1} E F is free of rank 1 over Q G by the normal basis theorem.

We will use the following lemma from linear algebra. Recall that if B is an Aalgebra and N is a B-module, an A-structure of N is an A-submodule M N such that M _{A} B^{}N.

Lemma 18. Let M be a Q G -submodule of V. The following conditions are equivalent :

(i) M is a Q-structure of the real vector space V.

(ii) For any χ G, the space M_{χ} e_{χ} M \overline{Q} is a \overline{Q}-structure of the \overline{Q} Rmodule V_{χ}.

Moreover, if these conditions hold, then M is free of rank 1 over Q G .

Proof. The implication i ii follows from the isomorphisms M_{χ} \overline{Q} \overline{Q} R e_{χ} M \overline{Q} R V_{χ}. Let us assume (ii). By Lemma 17, the \overline{Q} R-module V_{χ} is free of rank 1, so that dim\overline{Q} M_{χ} 1. Since M \overline{Q} _{χ G} M_{χ}, we get dim_{Q} M d. Moreover M \overline{Q} R generates V \overline{Q} over \overline{Q} R, so that any Q-basis of M is actually free over R. Finally, if (i) holds, then M is isomorphic to the regular representation of G by Lemma 17, so that M is free of rank 1 over Q G .

Using Lemma 18 with the Q-structure H_{E F}, we see that H_{χ} is a \overline{Q}-structure of V_{χ}. By Lemma 8, the 1-form η_{χ} is a \overline{Q}-basis of H_{χ}.

Proof of Theorem 2. Since R_{χ} λ_{χ} H_{χ} is a \overline{Q}-structure of V_{χ}, Lemma 18 implies that R_{E F} is a Q-structure of V. Moreover, the determinant of R_{E F} \overline{Q} with respect to H_{E F} \overline{Q} is represented by δ _{χ G} λ_{χ} \overline{Q} R ^{}. Note that σ λ_{χ} λ_{χ^{σ}} for any σ Gal \overline{Q} Q , so that δ lies in the image of R^{} in \overline{Q} R ^{}. Using the natural evaluation map \overline{Q} R ^{}^{ι} C^{}, we get in fact δ _{χ G} L^{}f χ, 0 . Since the natural map R^{}Q^{}\overline{Q} R ^{}\overline{Q}^{}is injective, we conclude that det R_{E F} L^{d} E_{F} , 0 det H_{E F} by (8).

Proof of Theorem 1. We know from Theorem 2 that R_{E F} is a Q-structure of V. Since R_{E F} is stable by G, it is free of rank 1 over Q G by Lemma 18. Let γ _{E F} such that R_{E F} Q G reg_{E F} γ . Replacing γ by a suitable integer multiple, we may assume that γ has a representative γ F E ^{}F E ^{}. Let ℓ β γ . For any χ G, we have R_{χ} μ_{χ} ℓ H_{χ} by Prop. 9, where μ_{χ} ℓ is given by (26). It follows that μ_{χ} ℓ λ_{χ} \overline{Q}^{}. Since λ_{χ} and μ_{χ} ℓ belong to Q χ R, we have in fact μ_{χ} ℓ λ_{χ} Q χ ^{}. Moreover, the definitions of λ_{χ} and μ_{χ} ℓ show that

τ λ_{χ} λ_{χ^{τ}} and τ μ_{χ} ℓ μ_{χ^{τ}} ℓ  τ Gal Q χ Q .  (51)

Lemma 19. Let a_{χ} _{χ G} be a family of algebraic numbers, with a_{χ} Q χ ^{}, such that τ a_{χ} a_{χ^{τ}} for any χ and any τ Gal Q χ Q . Then there exists a unique a Q G ^{}such that for every χ G, we have χ a a_{χ}.

Proof. The canonical morphism of Q-algebras Ψ Q G _{χ G} Q χ is injective and its image is contained in the subalgebra W of families b_{χ} _{χ} satisfying τ b_{χ} b_{χ^{τ}} for any χ and τ. Writing G as a disjoint union of Galois orbits, we have dim_{Q} W #G d, so that Ψ is an isomorphism.

Using Lemma 19 with a_{χ} μ_{χ} ℓ λ_{χ}, we get a Q G ^{}such that μ_{χ} ℓ χ a λ_{χ} for any χ. Since μ_{χ} aℓ \overline{χ} a μ_{χ} ℓ , replacing ℓ with a suitable integer multiple of aℓ results in μ_{χ} ℓ _{Q^{}}λ_{χ} for any χ. Evaluating everything in C yields (1).

Proof of Corollary. Let us first recall the Dedekind-Frobenius formula for group determinants. If a G C is an arbitrary function, let A be the matrix a gh^{1} _{g,h G}. Then

det A _{χ G} _{g G}χ g a g .  (52)

Let ℓ Z E \overline{Q} ^{G_{F}} be a divisor satisfying the identities (1) of Theorem 1. Assume first F is real. Put ℓ_{i} ℓ^{σ_{i}^{1}} for 1 i d. Using (52) with a σ D_{E} ℓ^{σ} yields

det D_{E} ℓ_{i}^{σ_{j}}  _{1 i,j d} _{Q^{}}_{χ G}πL^{}E χ, 0 _{Q^{}}π^{d}L E_{F} , 2  (53)

where the last relation follows from (8) and Prop. 4. Assume now F is complex. Put ℓ_{i} ℓ^{σ_{i}^{1}} for 1 i d 2. We use (52) with the function a σ D_{E} ℓ^{σ} J_{E} ℓ^{σ} . Indexing the lines and columns of A by

σ_{1}, \overline{σ_{1}}, …, σ_{d 2}, \overline{σ_{d 2}}, we see that A consists of blocks of the form \begin{matrix}x y & x y \\ x y & x y \end{matrix} ,

where x D_{E} ℓ^{σ_{j}σ_{i}^{1}} and y J_{E} ℓ^{σ_{j}σ_{i}^{1}} . Elementary operations on the lines and columns of A thus gives

det A 2^{d} det D_{E} ℓ_{i}^{σ_{j}}  _{1 i,j d 2} det J_{E} ℓ_{i}^{σ_{j}}  _{1 i,j d 2} .  (54)

On the other hand, we have

_{σ G}χ σ a σ  ;  ;  ;  ;  ;  ; _{σ G} χ σ D_{E} ℓ^{σ} if χ is even, _{σ G} χ σ J_{E} ℓ^{σ} if χ is odd,  (55)

so that we conclude as in the first case.

Further remarks and a conjecture

The proof of Theorem 2 relies crucially on the hypothesis that F Q is abelian. Since the field of constants of a modular curve is always an abelian extension of Q, it is not possible to cover a non-abelian base change of E by a usual modular curve. In fact, in the case F Q is not abelian, we have no example of a (non CM) elliptic curve E over Q for which we can prove Zagier’s conjecture for L E_{F} , 2 . However, Theorem 1 suggests the following conjecture for Artintwisted L-values. For simplicity, we restrict to the case F is totally real.

Conjecture 20. Let E be an elliptic curve defined over Q, and let F be a finite Galois totally real extension of Q. There exists a divisor ℓ Z E \overline{Q} ^{Gal \overline{Q} F} satisfying Goncharov and Levin’s conditions such that for every Artin representation ρ Gal F Q GL_{d} C , we have

L^{d} E ρ, 0 _{Q^{}}π^{d} det _{σ G}ρ σ D_{E} ℓ^{σ} .  (56)

Conversely, for every ℓ Z E \overline{Q} ^{Gal \overline{Q} F} satisfying Goncharov and Levin’s conditions and for every ρ Gal F Q GL_{d} C , we have

π^{d} det _{σ G}ρ σ D_{E} ℓ^{σ} L^{d} E ρ, 0 Q tr ρ ,  (57)

where Q tr ρ is the field generated by the traces of ρ.

Note that the identities (56) and (57) are compatible with taking direct sums of Artin representations. In fact, Conjecture 20 is a refinement of Zagier’s conjecture for L E_{F} , 2 , in the sense that taking the product over irreducibles ρ with multiplicities dim ρ gives the conjecture for E_{F}. Note that the analytic continuation and the functional equation of L E ρ, s are only conjectural in general. It would be interesting to investigate the rational factors arising in Theorem 1. As a matter of fact, even for F Q, we don’t know how to predict the rational factor appearing in Zagier’s conjecture. The Bloch-Kato conjecture predicts the exact value of L E_{F} , 2 (at least up to a unit in the ring of integers of F), but the link between both conjectures remains to be worked out. In fact, in this setting it may be more natural to investigate the equivariant Tamagawa number conjecture of Burns and Flach [9, Part 2, Conjecture 3], which predicts the equivariant L-value L _{F} E, 2 R G ^{}up to a unit in an order of Q G . Taking norms down to Q, this predicts L E_{F} , 2 up to sign. The deep work of Gealy [10] on the Bloch-Kato conjecture for modular forms, which uses Kato’s Euler system, could be used to tackle this equivariant conjecture. Note also that if F is abelian and real, then Theorem 1 gives a link between L _{F} E, 2 and the vector-valued elliptic dilogarithm { \atop D_{E}} ℓ _{σ G} D_{E} ℓ^{σ} σ . Finally, although the divisor ℓ produced by Theorem 1 is inexplicit in general, it would be interesting to try to bound the number field generated by the support of ℓ, as well as the heights of the points involved.

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François Brunault ÉNS Lyon, UMPA 46 allée d’Italie 69007 Lyon France francois.brunault@ens-lyon.fr