9701782 Stevens The principal investigator intends to continue his work on special values of L-functions and their relation to arithmetic geometry. The proposed research would improve our understanding of both automorphic forms and p-adic cohomology by providing new tools for studying the former and by providing concrete examples of the latter. The PI's recent proof of a conjecture of Mazur, Tate, Teitelbaum, and Coleman raises new questions about analytic families of automorphic forms, their p-adic L-functions, and families of galois representations and overconvergent F-crystals. The proposed research would clarify how such families degenerate at special non-crystalline points and would describe monodromy at such points in terms of a deformation in the weight direction, thus enlarging the standard picture of monodromy in potentially useful ways. This research would also complement Kato's recent work on values of L-functions and K_2 of modular curves by providing tools for deforming Kato's theory in p-adic analytic families. In related work, the PI hopes to develop a p-adic Eichler-Shimura correspondence that would relate his theory of overconvergent modular symbols to Katz's theory of overconvergent modular forms and to construct analytic families of non-ordinary half-integral weight modular forms by generalizing a p-adic theta lifting developed in earlier work of the PI. Finally, the PI intends to generalize these ideas to automorphic forms on other reductive algebraic groups. This research offers promising tools for the construction of p-adic analytic families of non-ordinary automorphic representations together with natural deformation spaces of Galois representations, and multivariable p-adic L-functions. This is connected with a number of new investigations, including p- adic monodromy, Jochnowitz's conjectures on the square root of theta operator on half-integral weight forms, and the p-adic deformation theory of Galois representations. This project falls into the general area of arithmetic geometry -a subject that blends two of the oldest areas of mathematics: number theory and geometry. This combination has proved extraordinarily fruitful - having recently solved problems that withstood generations. Among its many consequences are new error correcting codes. Such codes are essential for both modern computers (hard disks) and compact disks.
