A conference on L-functions and modularity of algebraic varieties will be held July 15–23, 2022, immediately following ICM 2022.
Ever since Langlands, motivic L–functions have been expected to be automorphic, meaning that they can be analytically continued and satisfy functional equations. The famous Shimura–Taniyama–Weil conjecture, or the Breuil–Conrad–Diamond–Taylor theorem, holds that 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. That is a direct consequence of the existence of a weight of two cusp forms whose Mellin transform is L.
In dimension two, 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 two with respect to the Siegel paramodular groups of genus two. 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 three 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.
The conference will feature a research school on the subject on July 15–17. The school will provide direct and intensive dialog between established mathematicians and younger participants, who will have an opportunity to give short talks.