[Paper Review] On factorizations of maps between curves

Dijana Kreso, Michael E. Zieve|arXiv (Cornell University)|May 19, 2014

Algebraic Geometry and Number Theory18 references5 citations

TL;DR

This paper establishes that for covers of curves with monodromy groups containing a transitive abelian (or more generally, transitive Dedekind or quasi-Hamiltonian) subgroup, the degrees, monodromy groups, and automorphism groups of the indecomposable factors in any complete decomposition are uniquely determined up to permutation. This generalizes Ritt's classical result on polynomial factorizations and applies to polynomials, isogenies, Lattès maps, and other arithmetic-geometric covers.

ABSTRACT

We examine the different ways of writing a cover of curves $ϕ\colon C o D$ over a field $K$ as a composition $ϕ=ϕ_n\circϕ_{n-1}\circ\dots\circϕ_1$, where each $ϕ_i$ 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 $ extrm{Mon}(ϕ)$ has a transitive abelian subgroup then the sequence $(\degϕ_i)_{1\le i\le n}$ is uniquely determined up to permutation by $ϕ$, so in particular the length $n$ is uniquely determined. We prove analogous conclusions for the sequences $( extrm{Mon}(ϕ_i))_{1\le i\le n}$ and $( extrm{Aut}(ϕ_i))_{1\le i\le n}$. Such a transitive abelian subgroup exists in particular when $ϕ$ is tamely and totally ramified over some point in $D(\overline{K})$, 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 $ extrm{char}(K)$, additive polynomials, elliptic curve isogenies, and Lattès maps.

Motivation & Objective

• To determine when the factorization of a cover of curves into indecomposable covers is uniquely determined up to permutation.
• To extend Ritt's classical result on polynomial factorizations to broader classes of covers of curves.
• To characterize the structure of automorphism groups in compositions of covers under monodromy group conditions.
• To prove that the sequence of automorphism groups of indecomposable factors divides the automorphism group of the full cover under Dedekind monodromy conditions.
• To identify natural classes of covers (e.g., polynomials, isogenies, Lattès maps) to which these uniqueness results apply.

Proposed method

• Use monodromy group theory to analyze the structure of curve covers, focusing on transitive abelian or Dedekind subgroups.
• Apply group-theoretic tools: quasi-Hamiltonian and Dedekind groups, normalizers, and subgroup lattices in monodromy groups.
• Establish a homomorphism from Aut(ψ) to Aut(θ) for compositions ψ = θ∘ρ, with kernel Aut(ρ), under Dedekind monodromy conditions.
• Use Galois-theoretic correspondences between field extensions and covers to translate group-theoretic properties into geometric ones.
• Leverage fiber product decompositions and automorphism group actions on graphs of maps to prove uniqueness of factorization components.
• Prove that |Aut(φ)| divides ∏|Aut(φ_i)| when Mon(φ) has a transitive Dedekind subgroup, using inductive group chain arguments.

Experimental results

Research questions

• RQ1 Under what conditions on the monodromy group of a cover φ: C → D is the sequence of degrees of its indecomposable factors uniquely determined up to permutation?
• RQ2 When is the sequence of monodromy groups of the indecomposable factors in a complete decomposition of φ uniquely determined up to isomorphism and permutation?
• RQ3 Does the automorphism group of a composed cover φ = φ_n ∘ … ∘ φ_1 divide the product of the automorphism groups of the factors, and under what monodromy conditions does this hold?
• RQ4 Which natural classes of covers (e.g., polynomials, isogenies, Lattès maps) satisfy the uniqueness conditions on factorization sequences?
• RQ5 Why does the conjecture that |Aut(f)| divides ∏|Aut(f_i)| for rational functions f hold in some cases but fail in others, and what group-theoretic condition explains this?

Key findings

• If the monodromy group of a cover φ has a transitive abelian subgroup, then the sequence of degrees (deg φ_i) in any complete decomposition is uniquely determined up to permutation.
• The sequences of monodromy groups (Mon(φ_i)) and automorphism groups (Aut(φ_i)) are also uniquely determined up to isomorphism and permutation when Mon(φ) has a transitive Dedekind subgroup.
• For covers with transitive Dedekind monodromy, there is a canonical homomorphism Aut(ψ) → Aut(θ) for ψ = θ∘ρ with kernel Aut(ρ), which implies structural control over automorphism groups in compositions.
• The automorphism group of a composed cover φ satisfies |Aut(φ)| divides ∏_{i=1}^n |Aut(φ_i)| when Mon(φ) has a transitive Dedekind subgroup.
• The result applies to all covers that are tamely and totally ramified over a point in D(K̄), including polynomials of degree not divisible by char(K), additive polynomials, elliptic curve isogenies, and Lattès maps.
• The conjecture that |Aut(f)| divides ∏|Aut(f_i)| for rational functions f is false in general, as shown by counterexamples like f(X) = X^3 + X^{-3} over ℂ, where the divisibility fails.