Dokchitser, V; Evans, R; Wiersema, H; (2020) On a BSD-type formula for L-values of Artin twists of elliptic curves. Journal für die reine und angewandte Mathematik 10.1515/crelle-2020-0036. (In press).

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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.

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