Nonsplit conics in the reduction of an arithmetic curve

TL;DR

This paper establishes bounds on the number of certain valuations extending a given valuation in algebraic function fields, with applications to sums of squares over real function fields.

Contribution

It introduces bounds on nonruled residually transcendental extensions of valuations in algebraic function fields, linking them to the genus of the field.

Findings

Number of such valuations is at most genus + 1.

Provides bounds for valuations with algebraic function field residue fields.

Applications to sums of squares in real function fields.

Abstract

For an algebraic function field $F/K$ and a discrete valuation $v$ of $K$ with perfect residue field $k$, we bound the number of discrete valuations on $F$ extending $v$ whose residue fields are algebraic function fields of genus zero over $k$ but not ruled. Assuming that $K$ is relatively algebraically closed in $F$, we find that the number of nonruled residually transcendental extensions of $v$ to $F$ is bounded by $\mathfrak{g}+1$ where $\mathfrak{g}$ is the genus of $F/K$. An application to sums of squares in function fields of curves over $\mathbb{R}(\!(t)\!)$ is presented.