Motives and D-modules

这项拨款的主要焦点将是与曲线上的线性微分方程相关的周期行列式或全局epsilon因子。这个整体的ε因子类似于作用于有限域上曲线上的轴的上同调的frobenius的行列式。在这两种情况下，主要的结果都是一个乘积公式，根据亚纯微分形式的选择，将全局因子表示为局部因子的乘积。研究者将集中研究这两种理论之间的相似之处。特别地，他将考虑幂级数域上约化群上的不规则形式连接与合适的自同构高斯和型d模之间是否存在Langlands型对应关系。许多数学上有趣的数字，例如π，都是作为周期出现的，即有理函数的合适积分。其他人，例如e，显然没有。如果允许积分，被积项不一定是有理数，但满足一个有有理数系数的线性微分方程，就会得到一个更大的周期集合（包括e）。现代动机理论是研究第一类时期的有力工具。这项拨款的重点是人们对微分方程周期的动机结构的期望。主要的问题是关于这些周期和某些数字之间的惊人类比，这些数字被称为局部的epsilon因子，与多项式方程mod p的算术研究有关。

DMS-0400451Spencer J. BlochThe principal focus of this grant will be the period determinant, or global epsilon factor, associated to a linear differential equation on a curve. This global epsilon factor is analogous to the determinant of frobenius acting on cohomology of a sheaf on a curve over a finite field. In both cases, the main result is a product formula expressing the global factor as a product of local epsilon factors depending on the choice of a meromorphic differential form. The investigator will focus on analogies between these two theories. In particular, he will consider whether there exists a Langlands style correspondence between irregular formal connections and suitable automorphic Gauss sum style D-modules on reductive groups over power series fields.Many numbers of mathematical interest, for example pi, arise as periods, i.e. suitable integrals of rational functions. Others, for example e, apparently do not. If one permits integrals where the integrand is not necessarily rational but satisfies a linear differential equation with rational coefficients, one is led to a larger collection of periods (including e). The modern theory of motives is a powerful tool for studying periods of the first sort. This grant focuses on what sort of motivic structure one might expect for periods of differential equations. The main questions concern a surprising analogy between these periods and certain numbers called local epsilon factors associated to the study of the arithmetic of polynomial equations mod p.