The elliptic sieve and Brauer groups

A theorem of Serre states that almost all plane conics over Q${{\mathbb {Q}}}$ 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 g...

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Bibliographic Details
Published in:Proceedings of the London Mathematical Society 2023-06, Vol.126 (6), p.1884-1922
Main Authors: Bhakta, Subham, Loughran, Daniel, Rydin Myerson, Simon L., Nakahara, Masahiro
Format: Article
Language:English
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Summary:A theorem of Serre states that almost all plane conics over Q${{\mathbb {Q}}}$ 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.
ISSN:0024-6115
1460-244X
DOI:10.1112/plms.12520