利用约简映射检测线性相关

关于Mordell-Weil群中的算术

On arithmetic in Mordell-Weil groups

本文利用约化映射研究了阿贝尔簇的Mordell-Weil群中点的线性依赖性。特别是，我们试图通过有限数量的减少，以确定检测线性相关的条件在Mordell-Weil群。

变形数据、Belyi映射和局部提升问题

Deformation data, Belyi maps, and the local lifting problem

我们证明了某些微分形式的正特征的存在性和不存在性结果，称为{ \em好变形数据}。其中一些结果是通过Belyi映射的模$p$约简得到的。作为应用，我们解决了Sylow $p$ -子群为$p$阶的群的局部提升问题.

关于K-理论和阿贝尔簇中的约化映射和支撑问题

On reduction maps and support problem in K-theory and abelian varieties

在本文中，我们考虑约简映射$r_{v} : K_{2n+1}(F)/C_{F} \to K_{2n+1}(κ_{v})_{l}$，其中$F$是一个数域和$C_{F}$表示$K_{2n+1}(F)$的子群生成的$l$ -部分（对于所有素数$l$）的核的Dwyer-Friedlander地图和地图$r_{v} : A(F)\to A_{v}(κ_{v})_{l}$，其中$A(F)$是一个交换品种的数域。本文证明了Schinzel关于数域的$K$ -群的支撑问题的一个推广：设$P_{1}, ..., P_{s}, Q_{1}, ..., Q_{s}\in K_{2n+1}(F)/C_{F}$是数域的无穷阶点.假设对几乎每一个素数$l$，下列条件成立：对每一组正整数$m_{1}, ..., m_{s}$和对几乎每一个素数$v$ $$m_{1} r_{v}(P_{1})+... + m_{s} r_{v}(P_{s})=0 \mathrm{implies} m_{1} r_{v}(Q_{1})+... + m_{s}r_{v}(Q_{s})= 0. $$，则存在$α_{i}$，$β_{i}\in \mathbb{Z} \setminus \{0 \}$使得对每一个$i \in \{1, ... s\}$，在$B(F)$中有$α_{i} P_{i}+β_{i} Q_{i}=0$。我们还得到了数…

On residue maps for affine curves

We establish several compatibility results between residue maps in étale and Galois cohomology that arise naturally in the analysis of smooth affine algebraic curves having good reduction over discretely valued fields. These results are needed, and in fact have already been used, for the study of finiteness properties of the unramified cohomology of function fields of affine curves over number fields.

Cycles for rational maps over global function fields with one prime of bad reduction

Let $K$ be a global function field of characteristic $p$ and degree $D$ over $\mathbb F_{p}(t)$. We consider dynamical systems over the projective line $\mathbb P^1(K)$ defined by rational maps with at most one prime of bad reduction. The main result is an optimal bound for cycle lengths that only depends on $p$ and $D$. A bound for the cardinality of finite orbits is given as well. Our method is based on a careful analysis (for every prime of good reduction) of the $\mathfrak p$-adic distances between points belonging to the same finite orbit, in part motivated by previous work by Canci and P…

Bound for preperiodic points for maps with good reduction

Let $K$ be a number field and let $ϕ$ in $K(z)$ be a rational function of degree $d\geq 2$. Let $S$ be the places of bad reduction for $ϕ$ (including the archimedan places). Let $Per(ϕ,K)$, $PrePer(ϕ, K)$, and $Tail(ϕ,K)$ be the set of $K$-rational periodic, preperiodic, and purely preperiodic points of $ϕ$, respectively. The present paper presents two main results. The first result gives a bound for $|PrePer(ϕ,K)|$ in terms of the number of places of bad reduction $|S|$ and the degree $d$ of the rational function $ϕ$. This bound significantly improves a previous bound given by J. Canci and L.…

Reduction of dynatomic curves

The dynatomic modular curves parametrize polynomial maps together with a point of period $n$. It is known that the dynatomic curves $Y_1(n)$ are smooth and irreducible in characteristic 0 for families of polynomial maps of the form $f_c(z) = z^m +c$ where $m\geq 2$. In the present paper, we build on the work of Morton to partially characterize the primes $p$ for which the reduction modulo $p$ of $Y_1(n)$ remains smooth and/or irreducible. As an application, we give new examples of good reduction of $Y_1(n)$ for several primes dividing the ramification discriminant when $n=7,8,11$. The proofs i…

The local theory of unipotent Kummer maps and refined Selmer schemes

We study the Galois action on paths in the $\mathbb{Q}_\ell$-pro-unipotent étale fundamental groupoid of a hyperbolic curve $X$ over a $p$-adic field with $\ell\neq p$. We prove an Oda--Tamagawa-type criterion for the existence of a Galois-invariant path in terms of the reduction of $X$, as well as an anabelian reconstruction result determining the stable reduction type of $X$ in terms of its fundamental groupoid. We give an explicit combinatorial description of the non-abelian Kummer map of $X$ in arbitrary depth, and deduce consequences for the non-abelian Chabauty method for affine hyperbol…

Galois symbol maps for abelian varieties over a $p$-adic field

We study the Galois symbol map associated to the multiplicative group and an abelian variety which has good ordinary reduction over a $p$-adic field. As a byproduct, one can calculate the "class group" in the view of the class field theory for curves over a $p$-adic field.