# Algebraic curves with many automorphisms

## TL;DR

This paper investigates the relationship between the size of automorphism groups of algebraic curves and their p-rank, establishing bounds for when the p-rank is zero, with a focus on curves of even and odd genus.

## Contribution

It provides new bounds on automorphism group sizes that guarantee zero p-rank for algebraic curves, especially improving understanding for even genus cases.

## Key findings

- For even genus g, if |Aut(X)| ≥ 900 g^2, then X has zero p-rank.
- For any genus g ≥ 2, if Aut(X) has a solvable subgroup G with |G| > 252 g^2, then X has zero p-rank.
- The proofs utilize advanced group theory, including the Hurwitz and Deuring Shafarevich formulas.

## Abstract

Let $X$ be a (projective, geometrically irreducible, nonsingular) algebraic curve of genus $g \ge 2$ defined over an algebraically closed field $K$ of odd characteristic $p$. Let $Aut(X)$ be the group of all automorphisms of $X$ which fix $K$ element-wise. It is known that if $|Aut(X)|\geq 8g^3$ then the $p$-rank (equivalently, the Hasse-Witt invariant) of $X$ is zero. This raises the problem of determining the (minimum-value) function $f(g)$ such that whenever $|Aut(X)|\geq f(g)$ then $X$ has zero $p$-rank. For {\em{even}} $g$ we prove that $f(g)\leq 900 g^2$. The {\em{odd}} genus case appears to be much more difficult although, for any genus $g\geq 2$, if $Aut(X)$ has a solvable subgroup $G$ such that $|G|>252 g^2$ then $X$ has zero $p$-rank and $G$ fixes a point of $X$. Our proofs use the Hurwitz genus formula and the Deuring Shafarevich formula together with a few deep results from finite group theory characterizing finite simple groups whose Sylow $2$-subgroups have a cyclic subgroup of index $2$. We also point out some connections with the Abhyankar conjecture and the Katz-Gabber covers.