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arXiv · 1905.04282

On a BSD-type formula for L-values of Artin twists of elliptic curves

Vladimir Dokchitser·Robert Evans·Hanneke Wiersema
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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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Vladimir Dokchitser, Robert Evans, Hanneke Wiersema. 2019-05-10. On a BSD-type formula for L-values of Artin twists of elliptic curves. https://arxiv.org/abs/1905.04282

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