On factorizations of maps between curves

TL;DR

This paper investigates the unique factorization properties of covers between algebraic curves, especially under conditions involving abelian monodromy groups, with applications to polynomials, elliptic curves, and Lattès maps.

Contribution

It establishes the uniqueness of factorization sequences for certain covers of curves when the monodromy group contains a transitive abelian subgroup, extending to various classes of maps.

Findings

Factorization length is uniquely determined under abelian monodromy conditions.

Sequences of monodromy and automorphism groups are uniquely determined for such factorizations.

Results apply to polynomials, elliptic curve isogenies, and Lattès maps.

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.