摘要翻译:
设F:x-->X是定义在数域K上的一个群的态射,设V是X的一个K-子群,且O_F(P)={F^n(P):n=0,1,2,...}是X(K)中点P的轨道。我们描述了V与O_F(P)交的局部-全局原理。这一原理可以看作是Brauer-Manin阻塞的动力学模拟。本文证明了对于P^2上的幂映射,V(K)的有理点在两种情况下是Brauer-Manin通畅的:(1)V是环面的平移。(2)V是一条直线,P有一个前周期坐标。证明中的一个关键工具是关于序列中本原因子的经典Bang-Zsigmondy定理。我们还证明了与阿贝尔变体内生相关的动力系统的类似局部-全局结果。
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英文标题:
《On a Dynamical Brauer-Manin Obstruction》
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作者:
Liang-Chung Hsia and Joseph H. Silverman
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最新提交年份:
2008
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分类信息:
一级分类:Mathematics 数学
二级分类:Number Theory 数论
分类描述:Prime numbers, diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory
素数,丢番图方程,解析数论,代数数论,算术几何,伽罗瓦理论
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一级分类:Mathematics 数学
二级分类:Algebraic Geometry 代数几何
分类描述:Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇,叠,束,格式,模空间,复几何,量子上同调
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英文摘要:
Let F : X --> X be a morphism of a variety defined over a number field K, let V be a K-subvariety of X, and let O_F(P)= {F^n(P) :n=0,1,2,...} be the orbit of a point P in X(K). We describe a local-global principle for the intersection of V and O_F(P). This principle may be viewed as a dynamical analog of the Brauer-Manin obstruction. We show that the rational points of V(K) are Brauer--Manin unobstructed for power maps on P^2 in two cases: (1) V is a translate of a torus. (2) V is a line and P has a preperiodic coordinate. A key tool in the proofs is the classical Bang-Zsigmondy theorem on primitive divisors in sequences. We also prove analogous local-global results for dynamical systems associated to endomoprhisms of abelian varieties.
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PDF链接:
https://arxiv.org/pdf/0801.3045