Curves of given $p$-rank with trivial automorphism group

TL;DR

This paper constructs smooth projective curves over algebraically closed fields of characteristic p with specified p-rank and trivial automorphism groups, and hyperelliptic curves with only the hyperelliptic involution, analyzing their moduli spaces.

Contribution

It proves the existence of such curves with prescribed p-rank and automorphism properties, including trivial automorphism groups and hyperelliptic cases, using moduli space dimension computations.

Findings

Existence of curves with given p-rank and trivial automorphism group.

Existence of hyperelliptic curves with only the hyperelliptic involution.

Dimension calculations of moduli spaces with automorphisms.

Abstract

Let $k$ be an algebraically closed field of characteristic $p >0$. Suppose $g \geq 3$ and $0 \leq f \leq g$. We prove there is a smooth projective $k$-curve of genus $g$ and $p$-rank $f$ with no non-trivial automorphisms. In addition, we prove there is a smooth projective hyperelliptic $k$-curve of genus $g$ and $p$-rank $f$ whose only non-trivial automorphism is the hyperelliptic involution. The proof involves computations about the dimension of the moduli space of (hyperelliptic) $k$-curves of genus $g$ and $p$-rank $f$ with extra automorphisms.