Abstract: On the Hasse-Weil conjecture

Hasse-Weil conjectured that Zeta functions of varieties over number fields admit meromorphic continuation and satisfy a functional equation. We will explain new results in the direction of this conjecture for genus 2 curves over totally real fields. The difficulty is that genus 2 curves have non-regular hodge numbers and the Taylor-Wiles method that was successful in proving the conjecture for  genus 1 curves  (for example) breaks down in several places.  This is joint work with G. Boxer, F. Calegari and T. Gee.