Séminaire d'arithmétique à Lyon

Hida theory for Pizer's quaternionic orders

by Luca Dall'Ava (Università degli Studi di Padova)

M7.411 (ENS Lyon, UMPA)




In this talk, we discuss a quaternionic Control Theorem, in the spirit of Hida and Greenberg-Stevens, considering a generalization of Eichler orders proposed by Pizer. These orders allow higher level structure at the primes where the quaternion algebra ramifies. Interestingly, the quaternionic modular forms associated with these orders live in Hecke-eigenspaces whose rank might be 2 and not necessarily 1, as in the Eichler case. The proved Control Theorem deals with this higher multiplicity situation. Time permitting, we will discuss some work-in-progress developments on recovering strong multiplicity 1, and an expected generalization of Chenevier's p-adic Jacquet-Langlands correspondence with these interesting level structures. This last part is joint work with Aleksander Horawa.