Abstract
Abstract This is an investigation into the possible existence and consequences of a Birch–Swinnerton-Dyer-type formula for L-functions of elliptic curves twisted by Artin representations. We translate expected properties of L-functions into purely arithmetic predictions for elliptic curves, and show that these force some peculiar properties of the Tate–Shafarevich group, which do not appear to be tractable by traditional Selmer group techniques. In particular, we exhibit settings where the different p-primary components of the Tate–Shafarevich group do not behave independently of one another. We also give examples of “arithmetically identical” settings for elliptic curves twisted by Artin representations, where the associated L-values can nonetheless differ, in contrast to the classical Birch–Swinnerton-Dyer conjecture.
Highlights
The Birch–Swinnerton-Dyer conjecture classically provides a connection between the arithmetic of elliptic curves and their L-functions
In this article we focus on factorisation of L-functions: when E=Q is an elliptic curve and F=Q a finite extension, L.E=F; s/ factorises as a product of L-functions of twists of E by Artin representations L.E; ; s/
We would like to give a BSD-type formula for the leading term at s D 1 for L.E; ; s/, but, as we shall explain, there is a significant barrier to this
Summary
The Birch–Swinnerton-Dyer conjecture classically provides a connection between the arithmetic of elliptic curves and their L-functions. We will show that Conjecture 4 can be used to establish purely theoretical results, such as the following case of the Birch–Swinnerton-Dyer conjecture for twists of elliptic curves by dihedral Artin representations (below D2pq denotes the dihedral group of order 2pq). For the “L-function side” of the sought Birch–Swinnerton-Dyer formula for twists we use the following modification of the leading term of L.E; ; s/ at s D 1 This is very carefully chosen so as to mesh well with the Birch–Swinnerton-Dyer conjecture over number fields, the functional equation and Deligne’s period conjecture for Artin twists of elliptic curves (see Section 2.4) at the same time. The following notation is used throughout the paper: E: an elliptic curve defined over Q. cv.E=F /: the local Tamagawa number of E=Fv. GF : the absolute Galois group Gal.Q=F / of a number field F Â Q: Frobp: (arithmetic) Frobenius element at a prime p. Every elliptic curve over Q of conductor N admits a modular parametrisation X1.N / ! E with Manin constant 1
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