Vol. 4, No. 1, 2020

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Two-cover descent on plane quartics with rational bitangents

Nils Bruin and Daniel Lewis

Vol. 4 (2020), No. 1, 73–89
Abstract

We implement two-cover descent for plane quartics over with all 28 bitangents rational and show that on a significant collection of test cases, it resolves the existence of rational points. We also review a classical description of the relevant moduli space and use it to generate examples. We observe that local obstructions are quite rare for such curves and only seem to occur in practice at primes of good reduction. In particular, having good reduction at 11 implies having no rational points. We also gather numerical data on two-Selmer ranks of Jacobians of these curves, providing evidence these behave differently from those of general abelian varieties due to the frequent presence of an everywhere locally trivial torsor.

Keywords
plane quartics, rational points, local-to-global obstructions, bitangents, descent obstructions, two-covers
Mathematical Subject Classification 2010
Primary: 11G30, 14H30
Secondary: 11D41, 14H50
Milestones
Received: 28 February 2020
Accepted: 29 April 2020
Published: 29 December 2020
Authors
Nils Bruin
Department of Mathematics
Simon Fraser University
Burnaby BC
Canada
Daniel Lewis
Department of Mathematics
The University of Arizona
Tucson, AZ
United States