Counting Artin-Schreier curves over finite fields

Abstract

Several authors have considered the weighted sum of various types of curves of a certain genus g over a finite field k := Fq of characteristic p where p is a prime and q = pm for some positive integer m. These include elliptic curves (Howe), hyperelliptic curves (Brock and Granville), supersingular curves when p = 2 and g = 2 (Van der Geer and Van der Vlught), and hyperelliptic curves of low genus when p = 2 (Cardona, Nart, Pujolàs, Sadornil). We denote this weighted sum ∑[C] 1/|Autk(C)|' where the sum is over k-isomorphism classes of the curves and Autk(C) is the automorphism group of C over k. Many of these curves mentioned above are Artin-Schreier curves, so we focus on these in this dissertation. We consider Artin-Schreier curves C of genus g = d(p - 1)/2 for 1 ≤ d ≤ 5 over finite fields k of any characteristic p. We also determine a weighted sum for an arbitrary genus g in one-, two-, three-, and four-branch point cases. In our cases, we must consider a related weighted sum ∑/[C] 1/|CentAutk(C)‹t›|' where CentAutk(C) ‹t› is the centralizer of ‹t› in Autk(C). We discuss our methods of counting, our results, applications, as well as geometric connections to the moduli space of Artin-Schreier covers.