p-Adic流形的Zeta函数，物理学家的Dwork理论
The Zeta-Function of a p-Adic Manifold, Dwork Theory for Physicists
摘要：
在这篇文章中，我们审查的观察，由于最初的Dwork，即zeta函数的算术品种，原来定义的领域与p个元素，是一个超行列式。我们回顾这一观察的背景下，一个参数的家庭五次三倍，并研究zeta函数作为一个函数的参数λ。由于消去，无限矩阵的超行列式简化为有限矩阵的（普通）行列式U（n），对应于Frobenius映射在某些上同调群上的作用。U（λ）的参数依赖性由关系U（λ）=E^{-1}（λ ^p）U（0）E（λ）给出，其中E（λ）是由流形的周期形成的朗斯基矩阵。周期由收敛于$|ϕ|_p < 1$的级数定义。感兴趣的值是那些满足以下条件的值：对于非零的，我们有|\vph|_p=1。我们解释如何的过程中的p-adic解析延续适用于这种情况。矩阵U（n）分解成秩为4和秩为2的子矩阵，我们能够从这个角度解释以前通过数值计算得出的一些观察结果。
英文摘要：
In this article we review the observation, due originally to Dwork, that the zeta-function of an arithmetic variety, defined originally over the field with p elements, is a superdeterminant. We review this observation in the context of a one parameter family of quintic threefolds, and study the zeta-function as a function of the parameter ϕ. Owing to cancellations, the superdeterminant of an infinite matrix reduces to the (ordinary) determinant of a finite matrix, U(ϕ), corresponding to the action of the Frobenius map on certain cohomology groups. The parameter-dependence of U(ϕ) is given by a relation U(ϕ)=E^{-1}(ϕ^p)U(0)E(ϕ) with E(ϕ) a Wronskian matrix formed from the periods of the manifold. The periods are defined by series that converge for $|ϕ|_p < 1$. The values of ϕthat are of interest are those for which ϕ^p = ϕso, for nonzero ϕ, we have |\vph|_p=1. We explain how the process of p-adic analytic continuation applies to this case. The matrix U(ϕ) breaks up into submatrices of rank 4 and rank 2 and we are able from this perspective to explain some of the observations that have been made previously by numerical calculation.
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