The elliptic sieve and Brauer groups

TL;DR

This paper proves a new analogue of Serre's theorem for families of conics parametrized by elliptic curves, using elliptic divisibility sequences and sieve methods, with applications to Brauer groups and norm form equations.

Contribution

It introduces a novel approach combining elliptic divisibility sequences and sieve techniques to study rational points on conic families and Brauer group specializations.

Findings

Almost all such conic families have no rational points.

Established bounds for specializations of Brauer groups.

Applications to solving norm form equations.

Abstract

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