Weil type representations and automorphic forms

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)

Automorphic forms and - adic representations 4 Takeshi Saito

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...

Analytic continuation of representations and estimates of automorphic forms

0.1. Analytic vectors and their analytic continuation. Let G be a Lie group and (π,G, V ) a continuous representation of G in a topological vector space V . A vector v ∈ V is called analytic if the function ξv : g 7→ π(g)v is a real analytic function on G with values in V . This means that there exists a neighborhood U of G in its complexification GC such that ξv extends to a holomorphic functi...