On the minimal degree of morphisms between algebraic curves

TL;DR

This paper establishes an upper bound on the minimal degree of non-constant morphisms between algebraic curves over number fields, utilizing isogeny estimates between abelian varieties to advance understanding of morphism degrees.

Contribution

It introduces a new upper bound on the minimal degree of morphisms between algebraic curves over number fields, based on isogeny estimates, which was not previously known.

Findings

Provides an explicit upper bound on morphism degrees

Utilizes isogeny estimates between abelian varieties

Advances the understanding of morphism degrees in algebraic geometry

Abstract

Given smooth, projective, geometrically integral algebraic curves $X$ and $Y$ defined over a number field $K$, assuming that there is a non-constant $K$-morphism $\varphi \colon X \to Y$, we give an upper bound on the minimum of the degrees of such morphisms. The proof is based on isogeny estimates between abelian varieties.