On factorizations of maps between curves

Abstract

We examine the different ways of writing a cover of curves $\varphi :C\to D$ over a field $K$ as a composition $\varphi ={\varphi }_n\circ {\varphi }_{n-1}\circ \cdots \circ {\varphi }_1$, where each ${\varphi }_i$ 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 $\text{Mon}(\varphi )$ has a transitive abelian subgroup then the sequence $(\mathrm{deg}{\varphi }_i{)}_{1\le i\le n}$ is uniquely determined up to permutation by $\varphi$, so in particular the length $n$ is uniquely determined. We prove analogous conclusions for the sequences $(\text{Mon}({\varphi }_i){)}_{1\le i\le n}$ and $(\text{Aut}({\varphi }_i){)}_{1\le i\le n}$. Such a transitive abelian subgroup exists in particular when $\varphi$ is tamely and totally ramified over some point in $D(\overline{K})$, and also when $\varphi$ 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 $\text{char}(K)$, additive polynomials, elliptic curve isogenies, and Lattès maps.

23 pages

References in corpus (2)

• On Ritt's polynomial decomposition theorems
• Analogues of the Jordan-Holder theorem for transitive G-sets

Cited by in corpus (1)

• Functional composition of polynomials: indecomposability, Diophantine equations and lacunary polynomials