Vladimir Dokchitser

Decomposing L-functions of curves

A curve that admits a finite group of automorphisms can be decomposed into "motivic pieces", which correspond to the irreducible representations of the group. I will discuss their L-functions and an analogue of the Birch-Swinerton-Dyer conjecture, and mention some new methods we have for studying them. As an application, I will sketch a new proof of the parity conjecture for elliptic curves that makes use of the arithmetic of genus 3 curves with automorphisms. There will be explicit examples given throughout, in order to keep things down-to-earth. This is joint work with Holly Green, Alexandros Konstantinou and Adam Morgan.