Collaborative Research: P-adic Variation of Supersingular Iwasawa Invariants

University Of Massachusetts Amherst, Amherst MA

Investigators

Abstract

Abstract for collaborative award DMS-0439264 and DMS-0440708 of Pollack and Weston: Iwasawa theory is concerned with the relation between the Galois theoretic and p-adic analytic aspects of arithmetic objects. The investigators propose to study the Iwasawa theory of modular forms with supersingular reduction. A primary focus of this work is the behavior of Iwasawa invariants in p-adic analytic families of modular forms such as the eigencurve of Coleman and Mazur. In fact, the definitions of these invariants, well known in the ordinary case, are not yet known in general in the supersingular case. In order to define and study the algebraic Iwasawa invariants the PI's intend to use the p-adic Hodge theory of Fontaine to study the growth of Selmer groups of modular forms over cyclotomic fields. The definitions of the analytic invariants should be related to special values of modular L-functions; exhibiting the desired behavior in families will involve a study of congruences of special values of modular L-functions, a recurring theme in much recent work. An eventual goal of this project is to show that the main conjecture of Iwasawa theory can be checked for an entire family by checking it for a single form in the family. The investigators also intend to study the generalization of the Riemann-Hurwitz type formula of Kida, Hachimori and Matsuno, which describe the change in p-adic Iwasawa invariants over p-extensions of number fields, to modular forms of higher weight. Number theory, often considered the oldest mathematical discipline, has in recent times developed remarkable applications to cryptography. Many of these applications involve arithmetic geometric objects known as elliptic curves. Modular forms, the primary object of study in this proposal, are generalizations of elliptic curves which play a fundamental role in modern number theory. The questions investigated in this proposal deal with invariants which are closely related to those of interest in cryptography and may perhaps yield some insight into them.