In this project the investigator and his collaborator intend to develop an analogue in the p-adic setting of the Riemann-Hilbert correspondence between perverse sheaves and D-modules on algebraic varieties over the complex numbers. The Hilbert correspondence generalizes De Rham theory, and establishes a deep connection between the topology of a given complex algebraic variety (as encoded in the category of perverse sheaves on the variety) and the behavior of systems of differential operators defined on the variety (which are encoded as D-modules on the variety; that is, as sheaves of modules over the sheaf of rings of differential operators on the variety). The proposed p-adic analogue would perform a similar function for varieties over the p-adic numbers. It would yield an equivalence between the (currently conjectural) category of ``crystalline perverse sheaves'' on such a variety, and the (again conjectural) category of weakly admissible filtered D-modules equipped with a Frobenius operator. The category of crystalline perverse sheaves is believed to carry both geometric and also arithmetic information about the variety to which it is attached, and would for example be a natural ingredient in a p-adic analogue of Beilinson's theory of regulators. This gives some hint of the important role that such a category of sheaves can be expected to play in the local analysis at p of systems of Diophantine equations.
The problem of solving equations is one of the most basic in mathematics, going back at least to the mathematicians of ancient Greece, such as Diophantus. He studied the problem of solving equations in whole numbers; such equations are now known as Diophantine equations. Since the work of Descartes and Fermat, it has been understood that geometry provides a powerful tool for analyzing systems of equations, even if one is at first more interested in the equations from an arithmetic point of view. For this reason, the development of powerful geometric tools is important for progress in the theory of Diophantine equations. In this project, the investigator and his collaborator intend to develop such tools, by extending known techniques in the usual so-called archimedean geometry to the context of non-archimedean, or p-adic, geometry. This geometry, which has a strong arithmetic flavor, provides a crucial geometric setting for the analysis of Diophantine equations, and these techniques are expected to yield several new developments in that analysis. Such developments are important not only because they enrich what continues to be one of the center pieces of the mathematical tradition, but because the theory of Diophantine equations has deep interconnections with the theory of discrete processes, and especially with the theory of codes, so that progress in theory of Diophantine equations can be expected to yield progress in these fields.
