Genus zero tensor products

TL;DR

This paper investigates conditions under which the tensor product of two finite extensions of the rational function field results in a genus zero field, revealing a bound on ramification for such cases.

Contribution

It establishes a new criterion linking genus zero tensor products to ramification constraints in finite extensions of rational function fields.

Findings

If $L ensor_K M$ has genus 0, then at least one extension is ramified over at most four valuations.

The result connects the genus of tensor products with ramification properties of the extensions.

Provides a new perspective on the structure of tensor products in function field extensions.

Abstract

Let $L$ and $M$ be finite extensions of $K = \mathbb{C}(t)$. If $L\otimes _K M$ is a field of genus $0$, then at least one of $L$ and $M$ is ramified over at most four valuations of $K$.