Arithmetic of L-values

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Arithmetic of L-valuesPI: Glenn Stevens Proposal 0071065Project AbstractThe principal investigator proposes to continue his work on specialvalues of L-functions and their relation to arithmetic geometry.The proposed research would improve our understanding of bothautomorphic forms and p-adic cohomology by providing new tools forstudying the former and by providing concrete examples of the latter.Specifically, the research would extend the applicability of ExplicitReciprocity Laws to include p-adic analytic families of galoisrepresentations. This, in turn, would provide a new tool forstudying the arithmetic properties of two-variable p-adicL-functions. It would also clarify how analytic families of galoisrepresentations degenerate at semistable non-crystalline points andwould describe monodromy at such points in terms of a deformation inthe weight direction, thus enlarging the standard picture ofmonodromy in potentially useful ways. This research would alsocomplement Kato's recent work on values of L-functions and K-theory ofmodular curves by providing tools for deforming Kato's theory inp-adic analytic families. In related work, the PI hopes to developa p-adic Eichler-Shimura correspondence that would relate histheory of overconvergent modular symbols to Katz's theory ofoverconvergent modular forms and to construct analytic families ofnon-ordinary half-integral weight modular forms by generalizing ap-adic theta lifting developed in earlier work of the PI. Finally,the PI intends to generalize these ideas to automorphic forms onother reductive algebraic groups. This research offers promisingtools for the construction of p-adic analytic families ofnon-ordinary automorphic representations together with naturaldeformation spaces of Galois representations, andmultivariable p-adic L-functions. This is connected with anumber of investigations, including p-adic monodromy, Jochnowitz'sconjectures on the square root of the theta operator, and thep-adic deformation theory of Galois representations.The investigations of this proposal belong to the generalmathematical area of Arithmetic Geometry. This ultramodern researcharea combines two of the oldest branches of mathematics: numbertheory and geometry. New insights arising out of thiscombination are producing increasingly powerful tools to solvelongstanding problems like Fermat's Last Theorem, which haveresisted the strongest efforts of over three centuries ofmathematicians. In addition, though Arithmetic Geometry is sometimesregarded as the purest of pure mathematics, it has also beendeveloping insightful new techniques leading to dramatic progress insuch applied areas as error-correcting codes and cryptography.