Towards 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 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2008
 DOI:
 10.48550/arXiv.0802.0596
 arXiv:
 arXiv:0802.0596
 Bibcode:
 2008arXiv0802.0596C
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Algebraic Geometry