Motives and D-modules

DMS-0400451Spencer J. Bloch 这项资助的主要重点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。该全局 epsilon 因子类似于作用于有限域上曲线上层的上同调的弗罗贝尼乌斯行列式。在这两种情况下，主要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 epsilon 因子的乘积。研究者将重点关注这两种理论之间的类比。特别是，他将考虑不规则形式联系与幂级数域上的约简群上合适的自守高斯和式 D 模之间是否存在朗兰兹式对应。许多数学兴趣的数字，例如 pi，都是作为周期出现的，即有理函数的合适积分。其他的，例如 e，显然没有。如果允许积分，其中被积函数不一定是有理数，但满足具有有理系数的线性微分方程，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对微分方程周期可能期望什么样的动机结构。主要问题涉及这些周期和某些称为局部 epsilon 因子的数字之间的惊人类比，这些数字与多项式方程 mod p 的算术研究相关。

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.