9701017 Hida After several years of basic research on the Arithmetic of modular L-values,p-adic Hecke algebras, Galois representations and their Selmer groups, number theorists can now start building up the theory in a little more sophisticated way. After the proof of the Shimura-Taniyama conjecture by A. Wiles and R. Taylor, which include the proof of Mazur's conjecture on Hecke algebras and deformation rings of modular Galois representations, there are now effective tools to deal with non-abelian Selmer groups; in particular, those of adjoint modular Galois representations. This award will support research into five projects connected to Selmer groups. From the view point of Iwasawa's theory, number theorists need to construct p-adic analytic L-functions on the spectrum of the Hecke algebra to supply tools to describe such Selmer groups(Project II). Since adjoint L-values are often non-critical, one needs to find out how to compute the algebraic part of the L-values in an automorphic way feasible enough to connect them directly to Selmer groups even in non-critical case (Project I).Since the work of Taylor and Wiles is now generalized to the Hilbert modular case by K. Fujiwara, one can try to attack the two variable main conjecture for the adjoint Selmer groups (Project III). Two key tools in studying this problem are: (i) analysis of base change via Hecke algebras and deformation rings, and (ii) theory of p-adic nearly ordinary Hecke algebras for symplectic groups constructed by J. Tilouine and E. Urban. Thus it is natural to study how functorial operations on Galois representations are reflected by deformation rings and Hecke algebras (Project IV) and to generalize the theory to more general groups, perhaps unitary groups (Project V). 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 w ithstood generations. Among its many consequences are new error correcting codes. Such codes are essential for both modern computers (hard disks) and compact disks.
