The elliptic sieve and Brauer groups

2021/09/08 by Bhakta, Subham, Loughran, Daniel, Myerson, Simon L. Rydin +1 · 3 citations
#11G05 #11N36 #14F22 #14G05 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2109.03746

Abstract

A theorem of Serre states that almost all plane conics over 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.

Cited by

• Explicit valuation of elliptic nets for elliptic curves with complex multiplication
• Brauer groups of conic bundles over elliptic curves
• Character sums of Division polynomials twisted by multiplicative functions

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