Finite Abelian Groups and Character Sums

These are some notes I have written up for myself. I have compiled what I feel to be some of the most important facts concerning characters on finite abelian groups, especially from a number theoretic point of view. I decided to distribute these because I couldn’t find a single reference that covered everything that I thought was relevant to the material I will discuss on Wednesday. Of course, there is significantly more in these notes than I plan to cover in the talk. The talk will most likely cover the definition and very basic properties of characters on finite abelian groups (most importantly the orthogonality relations), the definition of the Fourier transform along with the inversion formula, the Gauss and Weyl sum estimates and their relation to each other, and if I have time, the Polya-Vinogradov estimate. 1. Finite Abelian Groups 1.1. Examples. The three most important examples of finite abelian groups for us will be 1. The group of residues modulo some integer q, denoted Z/qZ with additive notation. 2. The group of q roots of unity, {ωn}q−1 n=0 with multiplicative notation. Throughout, ω = e is a primitive q root of 1. This group is isomorphic to Z/qZ in the obvious way. 3. The group U(q) = Z/qZ∗ = {n ∈ Z/qZ : (n, q) = 1}, the multiplicative group of units modulo q. 1.2. Fundamental Theorems on Finite Abelian Groups. We will implicitly use the following theorems from elementary group theory. Lagrange’s Theorem. If H is a subgroup of G then |H| divides |G|. Chinese Remainder Theorem. If (N,M) = 1 then Z/NMZ ∼= Z/NZ⊕ Z/MZ. Fundamental Theorem of Finite Abelian Groups. If G is a finite abelian group then