SLIDES: tiny.cc/greentent
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 an elliptic curve 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 families of norm form equations parametrised by elliptic curves.
This is joint work with Subham Bhakta, Daniel Loughran and Masahiro Nakahara.