Motives and D-modules

该补助金的主要重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因子。这个整体的上同调因子类似于弗罗贝纽斯作用于有限域上曲线上的层的上同调的行列式。在这两种情况下，主要的结果是一个产品的公式表示的整体因素作为一个产品的本地的非线性因素取决于选择的亚纯微分形式。研究者将着重于这两种理论之间的类比。特别是，他将考虑是否存在一个朗兰兹风格之间的对应关系不规则的正式联系和适当的自守高斯和风格的D-模块约化群在幂级数领域。许多数字的数学利益，例如π，出现的时期，即适当的积分的合理职能。其他人，例如E，显然没有。如果一个允许积分的被积函数不一定是理性的，但满足一个线性微分方程的合理系数，导致一个更大的收集周期（包括e）。现代动机理论是研究第一类时期的有力工具。这个补助金的重点是什么样的动机结构，人们可能会期望的时期微分方程。主要的问题是，这些周期与某些称为局部周期因子的数字之间有着惊人的相似性，这些因子与多项式方程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.