Profinite rigidity explores the extent to which non-isomorphic groups can be distinguished by their finite quotients. Many interesting examples of this phenomenon arise in the context of group extensions—short exact sequences of groups with a fixed kernel and quotient. This talk will outline two main mechanisms that govern profinite rigidity in this setting and provide concrete examples of families of extensions that cannot be distinguished by their finite quotients.

The talk is based on my DPhil thesis.

We show how to explicitly compute equations for everywhere locally soluble 3-coverings of Jacobians of genus 2 curves with a rational Weierstrass point, using the notion of visibility introduced by Cremona and Mazur.  These 3-coverings are abelian surface torsors, embedded in the projective space $\mathbb{P}^8$ as degree 18 surfaces. They have points over every $p$-adic completion of $\mathbb{Q}$, but no rational points, and so are counterexamples to the Hasse principle and represent non-trivial elements of the Tate-Shafarevich group.  Joint work in progress with Tom Fisher.