On Zagier’s Conjecture for Base Changes of Elliptic Curves

Let E be an elliptic curve over Q, and let F be a fi- nite abelian extension of Q. Using Beilinson's theorem on a suitable modular curve, we prove a weak version of Zagier's conjecture for L(EF,2), where EF is the base change of E to F.


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 Lfunctions, theoretical and numerical results by many authors (see [20]) suggested an extension of these conjectures to elliptic curves.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 ⊂ 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∈ Ĝ L(E ⊗ χ, s), where Ĝ is the group of 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. Fix an isomorphism E(C) ≅ C (Z + τ Z) (τ ∈ C, I(τ ) > 0) which is compatible with complex conjugation.Let D E (resp.J E ) be the Bloch elliptic dilogarithm (resp.its "imaginary" cousin) on E(C) (see §2-3 for the definitions).Fix an embedding ι ∶ Q ↪ C, so that E(Q) embeds naturally in E(C).Note that D E and J E induce linear maps on Z[E(Q)].Let Z[E(Q)] G F be the group of divisors on E(Q) which are invariant under G F ∶= Gal(Q F ).It carries a natural action of G.The main theorem of this article can be stated as follows.
Let ∈ Z[E(Q)] G F be a divisor satisfying the identities (1) of Theorem 1.For any i, define i = σ −1 i .If F is real, then we have Remarks.
(1) Wildeshaus's formulation of the conjecture [17, Conjecture, Part 2, p. 366] uses Kronecker doubles series instead of D E and J E .The link between these objects is classical (see the proof of Prop 6).We have chosen here 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 i , the determinant appearing in (2) is a group determinant, indexed by G.In fact, the eigenvalues of the matrix D E ( σ j i ) are precisely the sums ∑ σ∈G χ(σ)D E ( 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 produced by Theorem 1 satisfies Goncharov and Levin's conditions [11, (2)-( 4)].Following [20], let 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 1 , . . ., d ∈ A E F (resp. 1 , . . ., d 2 ∈ A E F ), the right-hand side of (2) (resp.( 3)) is a rational multiple of L(E F , 2) (maybe equal to zero).As in the case where the base field is Q, this strong conjecture is beyond the reach of current technology.
Theorem 2. There exists a subspace 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 In this case, we may choose to be 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(ζm) , 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 wellknown 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 .In §4, we explain the adelic setting for modular curves.In §5, we prove the divisibility we need in the Hecke algebra.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 above, which in turn motivated all the results presented here.

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 Ĝ 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 ∈ Ĝ, 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 ∤ N m.The L-function of E F has the following description.Proposition 3. The following identity holds : (5) L(E F , s) = χ∈ Ĝ L(f ⊗ χ, s).Proof.Let ρ = (ρ ) be the compatible system of 2-dimensional -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 (6) L(ρ G F , s) = L(Ind (Here we chose embeddings Q ↪ Q .)Finally, since an irreducibleadic representation of G Q is determined by the traces of all but finitely many Frobenius elements, the compatible system associated to Proof.Since each L(f ⊗ χ, s) has a simple zero at s = 0, we get ( 8) 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 ⊗χ = 1.Letting w = ∏ χ∈ Ĝ w f ⊗χ , we have (10) Taking the product over χ ∈ Ĝ yields the result.

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.
where D is the Bloch-Wigner dilogarithm function [18].Let K 2 (M(X)) be the Milnor K 2 -group associated to M(X).The regulator map on X is the unique linear map such that for any f, g ∈ M(X) × and any holomorphic 1-form ω on X, we have ( 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 M(X) ∶= ∏ ), 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].
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 (14) 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 ∈ M(E τ ) × , we have ] and [6, (6.2)],where K 2,1,τ is the linear extension of the following Eisenstein-Kronecker series on E τ : The result now follows from the formula − 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].

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 M(X(C)), so we get a natural map K 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 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 ) is the image of the map K 2 (X )⊗Q → K 2 (X) ⊗ Q for any proper regular model X 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 : . 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 (20) η + = dz + dz and η − = dz − dz τ − τ .
Lemma 7. Let f, g ∈ C(E) × and = β(f, g).We have Taking the wedge product with dz and integrating over E(C) yields ( 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 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 ∈ Ĝ, consider the idempotent It acts on For any ψ ∈ Σ, let η ± (ψ) be the 1form η ± sitting in the ψ-component of (23).Note that the embedding Proof.For any ψ ∈ Σ, we have c * η ± (ψ) = ±η ± (ψ).It follows that Since χ(−1)χ(σ) = χ(σ), we get the result. 1 The lattice Z+τ Z 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)).
The map β induces a linear map which we still denote by β.The following proposition computes explicitly the regulator map associated to E F .
Since e χ (r) and η χ belong to the same G-eigenspace, it suffices to compare their ι-components.By definition, we have 1) .Moreover But D E (P ) = D E (P ) and J E (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).

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 M K over Q.For example X(N ) = M K(N ) and X 1 (N ) = M K 1 (N ) , where The Riemann surface M K (C) can be identified with the compactifi- over Q, which is given on the complex points by (τ, h) ↦ (τ, hg).For any compact open subgroups The Hecke algebra H K is the space of functions K GL 2 (A f ) K → Q with finite support, equipped with the convolution product [4].It acts on H R be the perfect pairing induced by Poincaré duality.For any T ∈ H K , we have ⟨T η, ω⟩ = ⟨η, T ′ ω⟩, where T ′ ∈ H K is defined by T ′ (g) = T (g −1 ), so that the action of Schappacher and Scholl [15, 1.1.2]proved that whose determinant with respect to the natural Q-structure H M K is given by the leading term of L(h 1 (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 H K = ⊗ ′ p H Kp .For any prime p, let T (p) ∈ H K (resp.T (p, p) ∈ H K ) be the characteristic function of K p 0 0 1 K (resp.K p 0 0 p ), where 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) M K .
For any integer M ≥ 1, we let H (M ) K ⊂ H K be the subalgebra generated by the H Kp for p ∤ M .We use the notation T (M ) K for the corresponding subalgebra of T K .
Proof.For any prime p ∤ M , we have K p = GL 2 (Z p ) and by Satake the map Q[T, S, S −1 ] → H Kp given by T ↦ T (p) and S ↦ T (p, p) is an isomorphism.In particular H Kp is contained in the center of H K , whence the result.
Let pr ∶ A × f → Ẑ× be the projection associated to the decomposition Note that there is an exact sequence The sequence (36) induces a right action of G on M K F , and thus a left action of G on Ω 1 (M K F ).Moreover, the curve M K F can be identified with M K ⊗ F as a curve over Q, and we have a bijection The action of G on M K F (C) corresponds via (37) to the action by translation on the first factor of G × M K (C).Now let us consider the case In order to ease notations, let T = T Proof.Note that K(N m) ⊂ K 1 (N ) F , so T is commutative and commutes with G by Lemma 10.Since G is abelian, the result follows.
Lemma 13.For any prime p ∤ N m, we have Note that det K = Ẑ× .Consider the following correspondence where α F is the natural projection and β F is induced by g −1 .Using the identification M K F ≅ M K ⊗ F and the description (37) of the complex points, we obtain ) ○ (σ p ) * and thus (38).The proof of (39) is similar.

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 We have e 2 F = e F and e F ∈ T G. Proof.The first equality follows from (ϕ F ) * (ϕ F ) * = deg ϕ F .
We have e F = ν F (e) where e 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 (N m) X 1 (N ) .The result now follows from Lemma 13.The space Ω = lim →K Ω 1 (M K ) ⊗ 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 where each Ω K π is a simple T K -module.In particular T K is a semisimple algebra.By Lemma 10, the algebra T is contained in the center of T K 1 (N ) F .Using [12, Prop 2.11], we deduce that T acts by scalar multiplication on each Ω . 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 ∈ Ĝ, let π(f ⊗ χ) be the automorphic representation of GL 2 (A f ) corresponding to the modular form f ⊗χ.We have π Lemma 15.For any prime p ∤ N m, we have 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 π(f ⊗χ) be the projection induced by (42).The multiplicity one theorems imply that e f ⊗χ ∈ T .
Proposition 16.The element e χ e F is divisible by e f ⊗χ in T 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 Applying e χ to both sides and using the identity e χ σ p = χ(p)e χ yields (46) T (p)e χ e F = a p χ(p)e χ e F .
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.

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 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 ).We want to prove that R E F ∶= reg E F (P E F ) is a Q-structure satisfying (4).Since P X 1 (N ) F is stable by the Hecke algebra, the spaces P E F and R E F are stable by G.
For any χ ∈ Ĝ, let R χ = e χ (R E F ⊗ Q) and H χ = e χ (H E F ⊗ Q).We want to compare R χ and H χ .We have Similarly, we have (49) ϕ * F H χ = e χ e F (H X 1 (N ) F ⊗ Q).We will build on the following theorem of Schappacher and Scholl.Let λ χ be the unique element of (Q⊗R) × such that for every 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 by the normal basis theorem.
We will use the following lemma from linear algebra.Recall that if B is an A-algebra and N is a B-module, an A-structure of N is an 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 ∈ Ĝ, the space 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].
Then there exists a unique a ∈ Q[G] × such that for every χ ∈ Ĝ, we have χ(a) = a χ .
Proof.The canonical morphism of Q-algebras 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 Ĝ as a disjoint union of Galois orbits, we have dim Q W = # Ĝ = d, so that Ψ is an isomorphism.

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 where the last relation follows from (8) and Prop.4.

Assume now
We use (52) with the function a(σ) = D E ( σ ) + J E ( σ ).Indexing the lines and columns of A by σ 1 , σ 1 , . . ., σ d 2 , σ d 2 , we see that A consists of blocks of the form x + y x − y x − y x + y , where x = D E ( σ j σ −1 i ) and y = J E ( σ j σ −1 i ).
Elementary operations on the lines and columns of A thus gives On the other hand, we have if χ is odd, 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 Artin-twisted 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 ∈ Z[E(Q)] Gal(Q F ) satisfying Goncharov and Levin's conditions such that for every Artin representation ρ ∶ Gal(F Q) → GL d (C), we have Conversely, for every ∈ Z[E(Q)] Gal(Q F ) satisfying Goncharov and Levin's conditions and for every ρ ∶ Gal(F Q) → GL d (C), we have 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 functional equation of L(E ⊗ ρ, s) is 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.In this direction, note that if F is abelian and real, then Theorem 1 gives a link between L( F E, 2) and the vector-valued elliptic dilogarithm ⃗ D E ( ) ∶= ∑ σ∈G D E ( σ )[σ].The deep work of Gealy [10] on the Bloch-Kato conjecture for modular forms, which uses Kato's Euler system, should certainly be relevant to tackle this equivariant conjecture.
Finally, although the divisor produced by Theorem 1 is inexplicit in general, it would be interesting to try to bound the number field generated by the support of , as well as the heights of the points involved.
⊗F generated by T and G.Lemma 12.The algebra T G is commutative.