Embeddings of quotient division algebras of rings 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.