On the de-Rham cohomology of hyperelliptic curves

TL;DR

This paper provides an explicit basis for the first de-Rham cohomology of hyperelliptic curves, explores the non-splitting of the Hodge-de-Rham sequence in characteristic p>2, and analyzes the hyperelliptic involution's action.

Contribution

It introduces an explicit basis for the de-Rham cohomology of hyperelliptic curves and investigates the Hodge-de-Rham sequence and involution action in various characteristics.

Findings

Explicit basis for de-Rham cohomology of hyperelliptic curves

Examples where the Hodge-de-Rham sequence does not split in characteristic p>2

Hyperelliptic involution acts as -1 on cohomology when p>2, identity when p=2

Abstract

For any hyperelliptic curve X, we give an explicit basis of the first de-Rham cohomology of X in terms of \v{C}ech cohomology. We use this to produce a family of curves in characteristic p>2 for which the Hodge-de-Rham short exact sequence does not split equivariantly; this generalises a result of Hortsch. Further, we use our basis to show that the hyperelliptic involution acts on the first de-Rham cohomology by multiplication by -1, i.e., acts as the identity when p=2.