Education is not the filling of a pail, it is the lighting of a fire. - W. B. Yeats

### The conference website:http://www.math.colostate.edu/~betten/COCOA15/cocoa15.html

Anne Ho - PhD defense
Date: Friday, July 17, 2015
Place: Weber, 201
Time: 10:00 a.m.

Title: Counting Artin-Schreier Curves Over Finite Fields

Abstact:  Several authors have considered the weighted sum of various types of curves over finite fields $k :=F_q$. This is denoted $\sum_{[C]} 1/|Aut_k(C)|$ where $[C]$ ranges over the $k$-isomorphism classes of the curves, and $Aut_k(C)$ is the automorphism group of $C$ over $k$.  We examine a related weighted sum for Artin-Schreier curves $C$ over finite fields of any characteristic $p$ and of genus $g = d(p-1)/2$ for $1 \leq d \leq 5$ as well as Artin-Schreier curves with one, two, three, and four branch points.  This generalizes the work of Cardona, Nart, and Pujolas for the $p = 2, g =2$ case and also the work of Nart and Sadornil for the the $p = 2, g = 3$ case.  In this talk, we will present our results, methods of counting, and the geometric connections to the moduli space of Artin-Schreier covers.  Then, we will discuss the four branch point c ases and the techniques used to determine the number of orbits of $n$-sets of $\mathbb{P}^1(\overline{k})$ under $PGL_2(k)$, which is a necessary step in finding the weighted sum.