About an analogue of Ihara's lemma for Shimura curves

Abstract

The object of this work is to present the status of art of an open problem: to provide an analogue for the cohomology of Shimura curves of the Ihara's lemma [Ihara73] which holds for modular curves. We will formulate our conjecture and locate it in the more general setting of the congruence subgroup problem. We will exploit the relationship between cohomology of Shimura curves and certain spaces of modular forms to establish some consequences of the conjecture about congruence modules of modular forms and about the problem of raising the level.