arXiv 1405.47532014-05-19math.AG

关于曲线间映射的因子分解

On factorizations of maps between curves

Dijana Kreso, Michael E. Zieve

备注：: 23 pages

摘要：

我们研究了不同的方式写覆盖曲线$ϕ\colon C\to D$在字段$K$作为一个组合$ϕ=ϕ_n\circϕ_{n-1}\circ\dots\circϕ_1$，其中每个$ϕ_i$是覆盖曲线在$K$的程度至少$2$，不能写为两个较低的程度覆盖的组合。证明了如果单值群$\textrm{Mon}(ϕ)$有一个可迁阿贝尔子群，则序列$(\degϕ_i)_{1\le i\le n}$在$ϕ$的置换下是唯一确定的，特别是序列长度$n$是唯一确定的.我们证明了序列$(\textrm{Mon}(ϕ_i))_{1\le i\le n}$和$(\textrm{Aut}(ϕ_i))_{1\le i\le n}$的类似结论。当$ϕ$在$D(\overline{K})$中的某点上是tamely和完全分歧的，以及当$ϕ$是一维代数群的态射（或这种态射的坐标投影）时，存在这样的传递阿贝尔子群。因此，例如，我们的结果适用于分解的多项式的程度不能整除由$\textrm{char}(K)$，添加剂多项式，椭圆曲线isogenies，和Lattès地图。

英文摘要：

We examine the different ways of writing a cover of curves $ϕ\colon C\to D$ 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 $\textrm{Mon}(ϕ)$ 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 $(\textrm{Mon}(ϕ_i))_{1\le i\le n}$ and $(\textrm{Aut}(ϕ_i))_{1\le i\le n}$. 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 $\textrm{char}(K)$, additive polynomials, elliptic curve isogenies, and Lattès maps.

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摘要：

英文摘要：

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