A ruled residue theorem for function fields of hyperelliptic curves

TL;DR

This paper investigates the extensions of valuations from a base field to function fields of hyperelliptic curves, establishing finiteness results for certain transcendental residue extensions under specific characteristic conditions.

Contribution

It proves a finiteness theorem for residually transcendental valuation extensions to hyperelliptic curve function fields, extending valuation theory in algebraic geometry.

Findings

Finitely many valuation extensions with transcendental but not ruled residue fields

Results depend on residue characteristic being zero or greater than the hyperelliptic degree

Provides new insights into valuation extensions in hyperelliptic contexts

Abstract

We study residually transcendental extensions of a valuation $v$ on a field $E$ to function fields of hyperelliptic curves over $E$. We show that $v$ has at most finitely many extensions to the function field of a hyperelliptic curve over $E$, for which the residue field extension is transcendental but not ruled, assuming that the residue characteristic of $v$ is either zero or greater than the degree of the hyperelliptic curve.