The elliptic sieve and Brauer groups

Subham Bhakta; Daniel Loughran; Simon Rydin Myerson; Masahiro Nakahara

doi:10.1112/plms.12520

The elliptic sieve and Brauer groups

Subham Bhakta, Daniel Loughran, Simon Rydin Myerson, Masahiro Nakahara

Department of Mathematical Sciences

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Abstract

A theorem of Serre states that almost all plane conics over (Formula presented.) have no rational point. We prove an analogue of this for families of conics parametrised by elliptic curves using elliptic divisibility sequences and a version of the Selberg sieve for elliptic curves. We also give more general results for specialisations of Brauer groups, which yields applications to norm form equations.

Original language	English
Pages (from-to)	1884-1922
Number of pages	39
Journal	Proceedings of the London Mathematical Society
Volume	126
Issue number	6
Early online date	11 May 2023
DOIs	https://doi.org/10.1112/plms.12520
Publication status	Published - 30 Jun 2023

Funding

We thank Gergely Harcos and Efthymios Sofos for helpful comments and advice on some of the proofs. We also thank the anonymous referee for a careful reading of the paper which led to an improvement of Theorem 1.5 . D. Loughran and M. Nakahara were sponsored by EPSRC grant EP/R021422/2. S. Bhakta and S. L. Rydin Myerson were supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant ID 648329). S. L. Rydin Myerson was supported by DFG project number 255083470, and by a Leverhulme Early Career Fellowship. We thank the ZORP online seminar and its organisers. Initially there were two separate teams working on this problem using similar methods. But during a gathertown meeting on ZORP on 11 December 2020, we became aware of each other during discussions with Efthymios Sofos. Each team had a slightly different viewpoint and results, so we decided to combine to create, in our view, an ultimately superior paper.

Funders	Funder number
Engineering and Physical Sciences Research Council	EP/R021422/2
Leverhulme Trust
European Research Council
Deutsche Forschungsgemeinschaft	255083470
Horizon 2020	648329

Access to Document

10.1112/plms.12520Licence: CC BY

https://arxiv.org/abs/2109.03746

Cite this

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The elliptic sieve and Brauer groups

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Quantitative arithmetic geometry

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The elliptic sieve and Brauer groups

Abstract

Funding

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Quantitative arithmetic geometry

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