On Factorizations of Maps between Curves

We examine the different ways of writing a cover of curves φ : C → D over a field K as a composition φ = φn ◦ φn−1 ◦ · · · ◦ φ1, where each φi 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 Mon(φ) has a transitive abelian subgroup then the sequence (deg φi)1≤i≤n is uniquely determined up to permutation by φ, so in particular the length n is uniquely determined. We prove analogous conclusions for the sequences (Mon(φi))1≤i≤n and (Aut(φi))1≤i≤n. Such a transitive abelian subgroup exists in particular when φ is tamely and totally ramified over some point in D(K), and also when φ 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 char(K), additive polynomials, elliptic curve isogenies, and Lattès