Factorization Tests and Algorithms Arising from Counting Modular Forms and Automorphic Representations

Hyperbolicity, automorphic forms and Siegel modular varieties

We study the hyperbolicity of compactifications of quotients of bounded symmetric domains by arithmetic groups. We prove that, up to an étale cover, they are Kobayashi hyperbolic modulo the boundary. Applying our techniques to Siegel modular varieties, we improve some former results of Nadel on the non-existence of certain level structures on abelian varieties over complex function fields.

Week 1: Automorphic Forms and Representations

Brief introduction to cyclotomic theory over Q using adeles. Discussion of the definitions of modular forms and automorphic forms. Introducing the adelic automorphic forms via strong approximation theorem. Discussion of the connected components of Shimura varieties (modular curves). Smooth/admissible representations of locally finite groups. Definition and admissibility of (cuspidal) automorphi...

Automorphic forms and - adic representations 4

In Carayol’s note [4], a geometric construction of the Galois representations associated to Hilbert modular forms and the compatibility with the local Langlands correspondence are discussed. In loc. cit., the compatibility is established in the case = p where the Galois representation is an -adic representation and p is the prime divided by the prime p of the totally real field where the restri...

Galois Representations and Automorphic Forms (MasterMath)

Galois representations and modular forms

This note is based on a series of lectures given at the summer school held on July 17-29, 2006 at IHES. The purpose of the lectures is to explain the basic ideas in the geometric construction of the Galois representations associated to elliptic modular forms of weight at least 2.