# The a-number of hyperelliptic curves

## TL;DR

This paper improves the known upper bound on the genus of hyperelliptic curves with maximal a-number in characteristic p, showing it can be lowered from 1.5p to p, and explores existence conditions for small primes.

## Contribution

It establishes a sharper bound g < p for hyperelliptic curves with a-number g-1, refining previous results and analyzing implications for small primes p=3,5,7.

## Key findings

- Bound g < p for a-number g-1 hyperelliptic curves.
- Cartier-Manin matrix rank constraints influence curve equations.
- Existence conditions for small primes p=3,5,7 summarized.

## 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, $p>0$, over which the curve is defined. It was proven by Elkin that for a genus $g$ hyperelliptic curve $C$ 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 $g <p$. The method of proof is to force the Cartier-Manin matrix to have rank one and examine what restrictions that places on the affine equation defining the hyperelliptic curve. We then use this bound to summarize what is known about the existence of such curves when $p=3,5$ and $7$.