Embeddings of quotient division algebras of rings of differential operators

TL;DR

This paper investigates the algebraic embeddings between quotient division rings of differential operator rings on algebraic curves, establishing genus constraints that answer a previously open question.

Contribution

It proves that a $k$-algebra embedding of $D(X)$ into $D(Y)$ implies the genus of $X$ is at most that of $Y$, resolving an open problem.

Findings

Embedding implies genus inequality of curves

Answer to the open question about algebra embeddings

Constraints on algebraic structures of differential operators

Abstract

Let $k$ be an algebraically closed field of characteristic zero, let $X$ and $Y$ be smooth irreducible algebraic curves over $k$, and let $D(X)$ and $D(Y)$ denote respectively the quotient division rings of the ring of differential operators of $X$ and $Y$. We show that if there is a $k$-algebra embedding of $D(X)$ into $D(Y)$ then the genus of $X$ must be less than or equal to the genus of $Y$, answering a question of the first-named author and Smoktunowicz.