On factorizations of maps between curves

Abstract

We examine the different ways of writing a cover of curves $\phi\colon C\to D$ over a field $K$ as a composition $\phi=\phi_n\circ\phi_{n-1}\circ\dots\circ\phi_1$, where each $\phi_i$ is a cover of curves over $K$ of degree at least $2$ which cannot be written as the composition of two lower-degree covers. We show that if the monodromy group $\textrm{Mon}(\phi)$ has a transitive abelian subgroup then the sequence $(\deg\phi_i)_{1\le i\le n}$ is uniquely determined up to permutation by $\phi$, so in particular the length $n$ is uniquely determined. We prove analogous conclusions for the sequences $(\textrm{Mon}(\phi_i))_{1\le i\le n}$ and $(\textrm{Aut}(\phi_i))_{1\le i\le n}$. Such a transitive abelian subgroup exists in particular when $\phi$ is tamely and totally ramified over some point in $D(\overline{K})$, and also when $\phi$ is a morphism of one-dimensional algebraic groups (or a coordinate projection of such a morphism). Thus, for example, our results apply to decompositions of polynomials of degree not divisible by $\textrm{char}(K)$, additive polynomials, elliptic curve isogenies, and Latt\`es maps.