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Seminario di Teoria dei Numeri: “Shimura curves: moduli interpretation and rational points”

May 9, 2013 h. 10:30 - Room 1BC45 - Carlos de Vera

ARGOMENTI: Seminari

Seminario di Teoria dei Numeri
“Shimura curves: moduli interpretation and rational points”

Carlos de Vera (UPC Barcelona)
May 9, 2013 h. 10:30
Room 1BC45

Abstact:
Shimura curves have great arithmetic significance, for they are moduli spaces of abelian surfaces with quaternionic multiplication (or fake elliptic curves). The study of diophantine properties of these curves is therefore of fundamental importance in number theory.
After recalling the moduli interpretation of Shimura curves, we will explain two strategies for studying the existence of global points on both Shimura curves and their Atkin-Lehner quotients. The first of them requires introducing the notion of Galois representations over fields of moduli, inspired on the work of Ellenberg and Skinner on the modularity of Q-curves. And the second one is a descent strategy for which we need to exploit the Cerednik-Drinfel'd theory of p-adic uniformization of Shimura curves.
Part of the talk will be based on a work in collaboration with V. Rotger.

Rif. Int. A. Bertapelle