Congruences of algebraic automorphic forms and supercuspidal representations

Overconvergent Algebraic Automorphic Forms

I present a general theory of overconvergent p-adic automorphic forms and eigenvarieties for connected reductive algebraic groups G whose real points are compact modulo centre, extending earlier constructions for forms of GLn due to Buzzard, Chenevier and Yamagami. This leads to some new phenomena, including the appearance of intermediate spaces of “semiclassical” automorphic forms; this gives ...

On Higher Congruences between Automorphic Forms

We prove a commutative algebra result which has consequences for congruences between automorphic forms modulo prime powers. If C denotes the congruence module for a fixed automorphic Hecke eigenform π0 we prove an exact relation between the p-adic valuation of the order of C and the sum of the exponents of p-power congruences between the Hecke eigenvalues of π0 and other automorphic forms. We a...

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)