Title: Isogeny decompositions, the parity conjecture and the Tate—Shafarevich group.

Abstract: This talk concerns the arithmetic of curves that admit a finite group of automorphisms. We begin by presenting a method for decomposing Jacobians of curves, up to isogeny, using basic Galois theory and representation theory of finite groups. As an application, we outline a new proof of the parity conjecture for elliptic curves that makes use of the arithmetic of genus 3 curves with automorphisms. Finally, we show that every square-free natural number appears as the non-square part of the order of some $\Sha(A/\mathbb{Q})$, where $A/\mathbb{Q}$ is an abelian variety arising from an explicit isogeny decomposition. This is joint work with Vladimir Dokchitser, Holly Green and Adam Morgan.