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) automorphic representations. Tensor product decomposition and the newforms for cuspidal automorphic representations. Central character and the character of classical newform. Lecture 1 (Sep. 15, 2008) 1. Cyclotomic Theory and Adeles 1.1. Cyclotomic theory. Let us start by admiring a beautiful theorem. Theorem 1.1. For an odd prime p: (i) ∃x, y ∈ Z, p = x2 + y2 ⇐⇒ p ≡ 1 (mod 4). (ii) ∃x, y ∈ Z, p = x2 + 2y2 ⇐⇒ p ≡ 1, 3 (mod 8). (iii) ∃x, y ∈ Z, p = x2 − 2y2 ⇐⇒ p ≡ 1, 7 (mod 8). Compare this theorem in elementary number theory with the Galois theory of corresponding quadratic fields, contained in Q(ζ8):