Hilbert modular surfaces, Hirzebruch-Zagier cycles and a p-adic Gross-Zagier formula for Hilbert modular forms

Speaker: Iván Blanco Chacón (UCD)

Date:  Monday, 28th September 2015

Time:  4pm – 5pm

Location:  Agriculture. 1.01 (Seminar Room).

Abstract:

We explain our recent joint work with Ignacio Sols, in which we solve a question posed by Henri Darmon in 2014. For a rational elliptic curve of conductor N_1, we consider the attached cuspform provided by the modularity theorem. The modular curve is generically embedded in the Hilbert modular surface of level one. We consider a level N_2 > 3 so that the corresponding Hilbert modular surface is a fine moduli space of HBAS. Choosing a suitable prime p, by the use of recent work of Hida, we construct a 2-variable p-adic L-function whose special values interpolates the p-adic Abel jacobi map at a Hirzebruch-Zagier cycle. Our intention is to construct non-geometrical Euler-Kato systems in the elliptic curve, non-reachable by modular or Shimura parametrisations.

Series: Algebra and Number Theory