Matteo Longo: p-adic families of Heegner points on Shimura curves and p-adic L-functions

Number Theory Seminar: Matteo Longo (University of Padova)

p-adic families of Heegner points on Shimura curves and p-adic L-functions

Heegner points (and their higher counterparts, Heegner cycles) are powerful tools in the study of the arithmetic of elliptic curves and modular forms. The question of their variation along p-adic families of modular forms (over modular curves) and the relation with BDP p-adic L-functions has been addressed through several different approaches by many authors: B. Howard in 2007, Castella in 2020, Jetchev-Loeffler-Zerbes and  Büyükboduk-Lei in 2021. I will report on some progress and partial results on analogues of these theories in the case of modular forms over quaternionic Shimura curves (defined over the field of rational numbers). The main ingredient is the systematic use of Serre-Tate expansion on Shimura curves as a replacement of Fourier expansions of elliptic modular forms, following an old approach by A. Mori. This is a work in collaboration with P. Magrone and E. Walchek.