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

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). Green open access

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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.

Type: Article
Title: On a BSD-type formula for L-values of Artin twists of elliptic curves
Open access status: An open access version is available from UCL Discovery
DOI: 10.1515/crelle-2020-0036
Publisher version: https://doi.org/10.1515/crelle-2020-0036
Language: English
Additional information: © 2020 Vladimir Dokchitser, Robert Evans and Hanneke Wiersema, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0).
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10112651
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