Motives and D-modules

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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.