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

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### NEWS in the Department of Mathematics

Funding/Grant REP/Awards/Announcements/Outreach

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**Colorado State University and the Department of Mathematics hosts a summer conference**

**CoCoA 2015 - **Combinatorics and Computer Algebra

**July 19-25 2015**

**Faculty contact: Anton Betten - betten@math.colostate.edu**

**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

Advisor: Dr. Rachel Pries

Committee:

Dr. Tim Penttila

Dr. Jeff Achter

Dr. Myung Hee Lee

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.