Quotients of hyperelliptic curves and etale cohomology

TL;DR

This paper investigates hyperelliptic curves with automorphism group actions, deriving explicit formulas for quotient curves and their etale cohomology, with implications for understanding Galois representations over local fields.

Contribution

It provides a closed-form expression for the quotient of hyperelliptic curves under automorphism groups and describes the etale cohomology as a G-representation, advancing arithmetic understanding.

Findings

Explicit formula for quotient curve C/G

Description of etale cohomology as G-representation

Applications to local Galois representations

Abstract

We study hyperelliptic curves C with an action of an affine group of automorphisms G. We establish a closed form expression for the quotient curve C/G and for the first etale cohomology group of C as a representation of G. The motivation comes from the arithmetic of hyperelliptic curves over local fields, specifically their local Galois representations and the associated invariants.