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Root numbers and parity phenomena

Cowland Kellock, Lilybelle; Dokchitser, Vladimir; (2023) Root numbers and parity phenomena. Bulletin of the London Mathematical Society 10.1112/blms.12931. (In press). Green open access

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Abstract

The parity conjecture has a long and distinguished history. It gives a way of predicting the existence of points of infinite order on elliptic curves without having to construct them, and is responsible for a wide range of unexplained arithmetic phenomena. It is one of the main consequences of the Birch and Swinnerton-Dyer conjecture and lets one calculate the parity of the rank of an elliptic curve using root numbers. In this handbook, we explain how to use local root numbers of elliptic curves to realise some of these phenomena, with an emphasis on explicit calculations. The text is aimed at a ‘user’ and, as such, we will not be concerned with the proofs of known cases of the parity conjecture, but instead, we will demonstrate the use of the theory by means of examples.

Type: Article
Title: Root numbers and parity phenomena
Open access status: An open access version is available from UCL Discovery
DOI: 10.1112/blms.12931
Publisher version: https://doi.org/10.1112/blms.12931
Language: English
Additional information: © 2023 The Author(s). The Bulletin of the London Mathematical Society is copyright © London Mathematical Society. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
Keywords: 11g05 (primary), 11g10, 11g40 (secondary)
UCL classification: UCL
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/10178438
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