Towards an analogue of Ihara's lemma for Shimura curves

TL;DR

This paper discusses the challenge of establishing an analogue of Ihara's lemma for Shimura curves, exploring current progress, conjectures, and implications for modular forms and congruence modules.

Contribution

It presents a partial result towards the Ihara's lemma analogue for Shimura curves and formulates related conjectures involving quaternion algebras and the congruence subgroup problem.

Findings

A direct result towards the Ihara's lemma for Shimura curves.

Formulation of a conjecture related to the congruence subgroup problem.

Implications for congruence modules and level raising of modular forms.

Abstract

The object of this work is to present the status of art of an open problem: to provide an analogue for Shimura curves of the Ihara's lemma \cite{Ihara73} which holds for modular curves. We will describe our direct result towards the "Problem of Ihara" and we will present some possible approaches to it, giving a formulation of our conjecture in terms of congruence subgroup problem for quaternion algebras. Since some modular forms can be reinterpreted as elements of the cohomology of Shimura curves, we will describe a consequence of the "Problem of Ihara" about congruence modules of modular forms and a consequence of it about the problem of raising the level of modular forms.