THE CONJECTURE OF TATE AND VOLOCH ON p-ADIC PROXIMITY TO TORSION
THE CONJECTURE OF TATE AND VOLOCH ON p-ADIC PROXIMITY TO TORSION
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TATE和VOLOCH关于p-ADIC接近挠率的猜想
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发表时间：
2000
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通讯作者：
T. Scanlon
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作者：
T. Scanlon
Tate和Voloch证明了Cp上半阿贝尔簇的扭点到子簇的p-adic距离是一致有界的。利用代数模型论的方法和Sen关于Hodge-Tate型Galois表示的一个结果证明了Qp上半交换簇上扭点的猜想。作为他们关于p进单位根的线性形式定理的推广，Tate和Voloch提出：猜想：（Tate，Voloch）设G是Cp上的半交换簇。设X ∈ G是定义在Cp上的子簇.则存在一个常数N ∈ N使得对任意扭点<$G（Cp）或<$G ∈ X或λ（<$，X）≤ N .在上面的陈述中，Cp表示Qp的代数闭包的完备化，λ（·，X）是X函数的p-adic邻近。在[Sc]中，这个猜想是在假设G是在Qp上定义的，并且Qp是一个素数到p阶的扭点的情况下证明的。有人建议，同样的证明方法在没有后一种限制的情况下也会起作用。这一建议在本说明中得到贯彻。主要定理如下。主要定理：设K是Qp的有限扩张。设G是定义在K上的半阿贝尔簇。设X ∈ G是定义在Cp上的闭子簇.存在一个仅依赖于X的常数N ∈ N，使得对任意挠点<$G（Cp）或<$G ∈ X或λ（<$，X）≤ N .在上面的语句中，λ（·，X）是到X的p-adic接近函数。在[Sc]中，这个函数被表示为“d（·，X）”，并被称为“距离函数”，但是为了匹配文献中常见的符号，我们恢复为“λ”而不是“d”。证明的主要定理通过分析（由于森）的作用惯性集团的泰特模的p-可分群的G和完成结合这一论点的主要定理[Sc]。日期：1999年2月4日，1999年3月20日修订。1991年数学学科分类。主要：11 D88;次要：03 C60、11 U 09。由NSF MSPRF支持。本文的研究发生在MSRI在1998年春季场计划模型理论。
Tate and Voloch have conjectured that the p-adic distance from torsion points of semi-abelian varieties over Cp to subvarieties may be uniformly bounded. We prove this conjecture for torsion points on semi-abelian varieties over Qp using methods of algebraic model theory and a result of Sen on Galois representation of Hodge-Tate type. As a generalization of their theorem on linear forms in p-adic roots of unity, Tate and Voloch conjectured: Conjecture: (Tate, Voloch) Let G be a semi-abelian variety over Cp. Let X ⊆ G be a subvariety defined over Cp. Then there is a constant N ∈ N such that for any torsion point ζ ∈ G(Cp)tor either ζ ∈ X or λ(ζ,X) ≤ N . In the above statement, Cp denotes the completion of the algebraic closure of Qp and λ(·, X) is the p-adic proximity to X function. In [Sc] this conjecture was proved under the assumptions that G is defined over Qp and that ζ is a torsion point of order prime-to-p. It was suggested that the same method of proof would work without the latter restriction. This suggestion is carried out in this note. The main theorem is the following. Main Theorem: Let K be a finite extension of Qp. Let G be a semi-abelian variety defined over K. Let X ⊆ G be a closed subvariety defined over Cp. There is a constant N ∈ N depending only on X such that for any torsion point ζ ∈ G(Cp)tor either ζ ∈ X or λ(ζ,X) ≤ N . In the above statement, λ(·, X) is the p-adic proximity to X function. In [Sc], this function was denoted by “d(·, X)” and called a “distance function,” but to match the notation common in the literature, we revert to “λ” instead of “d.” The proof of the Main Theorem passes through an analysis (due to Sen) of the action of the inertia group on the Tate module of the p-divisible group of G and is completed by combining this argument with the main theorem of [Sc]. Date: 4 February 1999, revised 20 March 1999. 1991 Mathematics Subject Classification. Primary: 11D88; Secondary: 03C60, 11U09. Supported by an NSF MSPRF. The research for this paper took place at MSRI during the Spring 1998 Model Theory of Fields program.