Arithmetic Geometry and Number Theory RTG Seminar: Taylor-Wiles-Kisin patching and mod l multiplicities in Shimura curves
Seminar | September 10 | 3:30-5 p.m. | 748 Evans Hall
Jeff Manning, UCLA
In the early 1990s Ribet observed that the classical mod l multiplicity one results for modular curves, which are a consequence of the q-expansion principle, fail to generalize to Shimura curves. Specifically he found examples of Galois representations which occur with multiplicity 2 in the mod l cohomology of a Shimura curve with discriminant pq and level 1.
I will describe a new approach to proving multiplicity statements for Shimura curves, using the Taylor-Wiles-Kisin patching method (which was shown by Diamond to give an alternate proof of multiplicity one in certain cases), as well as specific computations of local Galois deformation rings done by Shotton. This allows us to re-interpret and generalize Ribet's result. I will prove a mod l "multiplicity $2^k$" statement in the minimal level case, where k is a number depending only on local Galois theoretic data.
Time permitting I will also describe joint work (in progress) with Jack Shotton, in which we use these techniques to prove new cases of Ihara's Lemma for Shimura curves, which are not covered by the work of Diamond and Taylor.