Motives and D-modules

DMS-0400451 Spencer J.Bloch这笔赠款的主要关注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epsilon因子类似于Frobenius的行列式，它作用于有限域上曲线上的一束上同调。在这两种情况下，主要结果都是一个乘积公式，该公式将全局因子表示为局部epsilon因子的乘积，具体取决于亚纯微分形式的选择。研究人员将专注于这两种理论之间的类比。特别地，他将考虑在幂级数域上的约化群上的非正规形式联络与适当的自同构的Gauss和型D-模之间是否存在朗兰兹式对应。许多数学感兴趣的数字，例如pi，以周期形式出现，即有理函数的适当积分。其他人，例如e，显然不是这样。如果允许积分，其中被积函数不一定是有理的，但满足有理系数的线性微分方程式，则会导致更大的周期集合(包括e)。现代动机理论是研究第一类时期的有力工具。这项资助的重点是人们可能期待的微分方程式周期的动机结构。主要问题涉及这些周期和某些数字之间令人惊讶的相似之处，称为局部epsilon因子，与研究多项式方程mod p的算术有关。

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.