Martí Roset Julià

Let (G, G') be a pair of subgroups of a group that are mutual centralizers. The theory of theta correspondences provides a lift from automorphic forms for G to automorphic forms for G'. The study of this lift allows the translation of theorems from one type of automorphic form to the other. Moreover, the Fourier coefficients of the lift in G' can be understood from the original automorphic form in G. These lifts can be constructed in p-adic analytic families, leading to arithmetic properties regarding their Fourier coefficients.

The p-adic Kudla program is a novel program that aims to relate generating series of cycles on p-adic symmetric spaces to derivatives of p-adic families of modular forms. 

Inspired by this program, we have expressed a generating series of Heegner points on a Shimura curve in terms of the p-adic uniformization of the curve to relate it to a p-adic family of theta series. This establishes the modularity of this generating series, recovering a new proof of the Gross--Kohnen--Zagier theorem. Furthermore, it is feasible to explore variations of this program for p-adic symmetric spaces and p-adic cycles lacking archimedean counterparts, thus presenting new avenues for investigation.

The theory of complex multiplication produces elements in abelian extensions of imaginary quadratic fields as values of modular functions on the upper half-plane at imaginary quadratic points. Darmon and Vonk, building upon the works of Darmon and Dasgupta, proposed a generalization of this theory to construct elements in abelian extensions of real quadratic fields. There, the upper half-plane is replaced by the p-adic upper half-plane. Although this space does not have interesting invariant functions for the modular group, the 1-cocycles for such a group valued on meromorphic functions are interesting and can be evaluated at real quadratic points. 

This framework gives rise to numerous conjectures, some of which have already been proven, concerning the significance of these values. Furthermore, it paves the way for potential extensions of the theory to other base fields.