Numerical evidence for the equivariant Birch and Swinnerton-Dyer conjecture (Part II)

Abstract

Abstract. We continue the study of the Equivariant Tamagawa Number Conjecture for the base change of an elliptic curve begun by the author in 2009. We recall that the methods developed there, apart from very special cases, cannot be applied to verify the l-part of the ETNC if l divides the order of the group. In this note we focus on extensions of l-power degree (l an odd prime) and describe methods for computing numerical evidence for ETNC l . For cyclic l-power extensions we also express the validity of ETNC l in terms of explicit congruences.

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

“…If p does not divide the order of G, then, without any hypothesis on the reduction of A at p, it is straightforward to use the techniques developed in [15, §1.7] to give an explicit interpretation of eTNC p (although, of course, obtaining a full proof in this case still remains a very difficult problem). However, if p divides the order of G, then even obtaining an explicit interpretation of eTNC p has hitherto seemed to be a very difficult problem -see, for example, the considerable efforts made by Bley [3,4] in this direction.…”

“…More precisely, we show that for certain elliptic curves A the validity of eTNC p with p = 3 follows from that of the relevant cases of the Birch and Swinnerton-Dyer conjecture for a family of S 3 -extensions of number fields (see Corollary 6.2) and provide examples (with p = 5 and p = 7) in which our approach allows eTNC p to be verified by numerical computations (see §6. 3).…”

La conjecture équivariante de Tamagawa

“…Pour plus de détails et un cas spécial où tous les calculs sont accomplis le lecteur peut consulter [3,Sec. 4].…”

“…Numériquement on vérifie la conjecture de rationalité parce que u est proche de Soit ε 0 ∈ Ext 2 Z l [G] (H 1 l , H 0 l ) la classe correspondant à la suite exacte correcte, c'est-à-dire, la suite exacte qui est définie par RΓ c,l (cf. [3,Sec. 4]).…”

“…Calculs dans K 0 (Z[G], Q). -On reprend le diagramme fondamental (3). Avec quelques simplifications, les conjectures qu'on veut étudier dans la suite prennent la forme d'une équation…”