The elliptic sieve and Brauer groups

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.

32 pages

References in corpus (4)

• The sign of an elliptic divisibility sequence
• A recurrence relation for elliptic divisibility sequences
• The Signs in Elliptic Nets
• Rational points on conic bundles over elliptic curves