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

### Abstract: A number of authors have considered the weighted sum of various sets of curves over finite fields $k := F_q$. We denote this $\sum_{[C]} 1/|Aut_k(C)|$ where $[C]$ is the $k$-isomorphism classes of the curves, and $Aut_k(C)$ is the automorphism group of $C$ over $k$. These types of curves include elliptic curves over fields of any characteristic $p$ (Howe), hyperelliptic curves over fields of characteristic two (Van der Geer, Van der Vlugt), as well as Artin-Schreier curves of genus two and three over fields of characteristic two (Nart et al.). We extend the work of these authors by considering Artin-Schreier curves with a more general genus $g$ over fields of any characteristic $p$. We will discuss our results and methods of counting, which include looking at ramification divisors, finding associated rational models $y^p – y = u(x)$, and examining the actions of $PGL_2(k)$ on the models. We will end by discussing open cases and a conjecture.

Sarah Frei - MS final examination
Date: Thursday, July 31, 2014
Place: Weber, 201
Time: 10:00 a.m.

Title: The a-number of hyperelliptic curves

Advisor: Dr. Rachel Pries

Committee:
Dr. Jeff Achter
Dr. Sarah Sloane

Abstract:
It is known that for a smooth hyperelliptic curve to have a large $a$-number, the genus must be small relative to the characteristic of the field over which the curve is defined. It was proven by Elkin that for a genus $g$ hyperelliptic curve to have $a_C=g-1$, the genus is bounded by $g<\frac{3p}{2}$. In this paper, we show that this bound can be lowered to $g <p$ for a genus $g$ hyperelliptic curve with $a_C=g-1$. The method of proof is to enforce that the Cartier-Manin matrix have rank one and examine what restrictions that places on the affine equation defining the hyperelliptic curve. In an attempt to lower the bound further, we discuss what happens when $g=p-1$. We then use this bound to summarize what is known about the existence of such curves when $p=3,5$ and $7$.

## The AMS announces MATHEMATICS RESEARCH COMMUNITIES Snowbird Resort, Utah

The AMS invites mathematicians just beginning their research careers to become part of Mathematics Research Communiities, a program to develop and sustain long-lasting cohorts for collaborative research projects in many areas of mathematics. Women and underrepresented minorities are especially encouraged to participate. The AMS will provide a structured program to engage and guide all participants as they start their careers. The program will include:

**One-week summer conference for each topic
**Special Session at the national meeting
**Discussion networks by research topic
**Funding for additional collaborations
**Longitudinal study of early-career mathematicians

June 8-14, 2014
Cluster Algebras

June 15-21, 2014
Algebraic and Geometric Methods in Applied Discrete Mathematics

June 24-30, 2014
Mathematics of Quantum Phases of Matter and Quantum Information

June 24-30, 2014
Network Science

For details, go to: http://www.ams.org/programs/research-communities/mrc

2014 MSRI SUMMER GRADUATE SCHOOLS
This coming summer MSRI is hosting three summer graduate schools in Berkeley, CA and co-sponsoring two* additional schools at off-site locations:

Dispersive Partial Differential Equations June 16 – 27, 2014 at MSRI

Séminaire de Mathématiques Supérieures 2014: Counting Arithmetic Objects June 23 – July 4, 2014 in Montréal, Canada

IAS/PCMI 2014: Geometry and Materials June 29 – July 19, 2014 in Park City, Utah

Stochastic Partial Differential Equations July 7 – 18, 2014 at MSRI

Geometry and Analysis July 28 – August 8, 2014 at MSRI