On the minimal degree of morphisms between algebraic curves

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.