On Mordell–Weil groups and congruences between derivatives of twisted Hasse–Weil L-functions

Abstract

Let A be an abelian variety defined over a number field k and let F be a finite Galois extension of k. Let p be a prime number. Then under certain not-too-stringent conditions on A and F we compute explicitly the algebraic part of the p-component of the equivariant Tamagawa number of the pair h 1 (A /F )(1), Z[Gal(F/k)] . By comparing the result of this computation with the theorem of Gross and Zagier we are able to give the first verification of the p-component of the equivariant Tamagawa number conjecture for an abelian variety in the technically most demanding case in which the relevant Mordell-Weil group has strictly positive rank and the relevant field extension is both non-abelian and of degree divisible by p. More generally, our approach leads us to the formulation of certain precise families of conjectural p-adic congruences between the values at s = 1 of derivatives of the Hasse-Weil L-functions associated to twists of A, normalised by a product of explicit equivariant regulators and periods, and to explicit predictions concerning the Galois structure of Tate-Shafarevich groups. In several interesting cases we provide theoretical and numerical evidence in support of these more general predictions.

Congruences for critical values of higher derivatives of twisted Hasse–WeilL-functions

“…This limitation of our previous methods is consistent with those occurring in all existing verifications of p-components of the eTNC for any elliptic curves. Indeed, in the settings of the theoretical verifications obtained by the first author in [3], by Burns, Wuthrich and the second author in [9], or of the recent extensions of these results by Burns and the second author in [8]; as well as of the numerical verifications carried out both in [9] and in our previous article [4]; a full verification of this conjecture was only ever achieved in situations which forced the Z p [G]-module A(F ) p to be projective. On the other hand, even for n = 1 (meaning that the extension F/k has degree p), the result [8,Thm.…”

“…In [12], a close analysis of the finite support cohomology of Bloch and Kato for the base change of the p-adic Tate module of A t is carried out under certain technical hypotheses on A and F . A consequence of this analysis is an explicit reinterpretation of the eTNC p in terms of a natural 'equivariant regulator' (see [12,Th. 5.1]).…”

“…However, it is well-known that in many cases of interest the modular element θ A,G vanishes, thus rendering any such properties trivial, and it would therefore be desirable to carry out an analogous study for elements interpolating leading terms rather than values at s = 1 of the relevant Hasse-Weil L-functions, normalised by appropriate regulators. Although the aim to study such elements already underlies the results of [12], one of the main advantages of confining ourselves to the special case in which the given extension of number fields F/k is cyclic of prime-power degree is that we are led to defining completely explicit 'twisted regulators' from our computation of the aforementioned equivariant regulator of [12]. Furthermore, we arrive at very explicit statements without having to restrict ourselves to situations in which the relevant Mordell-Weil groups are projective when considered as Galois modules.…”

“…In Section 2 we present our main results and in Section 4 we supply the proofs. In order to prepare for the proofs we recall in Section 3 the relevant material from [12]. In the final Section 5 we present some numerical computations.…”

On non-abelian higher special elements of $p$-adic representations

“…In particular, for any primitive central idempotent e of A one has that e(Φ) is invertible over Ae only if e = ee 0 , so we deduce that nr A (Φ) = e 0 nr A (Φ). Combining this equality with (10), (11) and (12) we finally find that…”