L-functions and modularity of algebraic varieties

It has been expected since Langlands that motivic L-functions are automorphic and can therefore be analytically continued and satisfy functional equations. By the famous Shimura-Taniyama-Weil conjecture, or the Breuil-Conrad-Diamond-Taylor theorem, the complete L-function of an elliptic curve over Q extends to an analytic function in the entire complex plane and satisfies a functional equation; this is a direct consequence of the existence of a weight 2 cusp form whose Mellin transform is L.

        In dimension 2, an analogue of the Shimura-Taniyama-Weil conjecture was proposed by A. Brumer and K. Kramer in 2010, linking isogeny classes of abelian surfaces defined over  Q  and certain Hecke eigenforms of weight 2 with respect to the Siegel paramodular groups of genus 2.

        One of the core subjects at the conference will be the case that lends itself to consideration next after elliptic curves and abelian surfaces: that of weight 3 four-dimensional Galois representations of Calabi-Yau type, conjecturally connected to certain weight three paramodular cusp forms.

        In a broader context, we will consider new applications of automorphic forms and Borcherds products in arithmetic, geometry and physics.