Collaborative Research: P-adic Variation of Supersingular Iwasawa Invariants

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Abstract for collaborative award DMS-0439264 and DMS-0440708 of Pollack and Weston:Iwasawa theory is concerned with the relation between the Galois theoreticand p-adic analytic aspects of arithmetic objects. The investigatorspropose to study the Iwasawa theory of modular forms with supersingularreduction. A primary focus of this work is the behavior of Iwasawainvariants in p-adic analytic families of modular forms such as theeigencurve of Coleman and Mazur. In fact, the definitions of theseinvariants, well known in the ordinary case, are not yet known in generalin the supersingular case. In order to define and study the algebraicIwasawa invariants the PI's intend to use the p-adic Hodge theory ofFontaine to study the growth of Selmer groups of modular forms overcyclotomic fields. The definitions of the analytic invariants should berelated to special values of modular L-functions; exhibiting the desiredbehavior in families will involve a study of congruences of special valuesof modular L-functions, a recurring theme in much recent work. Aneventual goal of this project is to show that the main conjecture ofIwasawa theory can be checked for an entire family by checking it for asingle form in the family. The investigators also intend to study thegeneralization of the Riemann-Hurwitz type formula of Kida, Hachimori andMatsuno, which describe the change in p-adic Iwasawa invariants overp-extensions of number fields, to modular forms of higher weight.Number theory, often considered the oldest mathematical discipline, has inrecent times developed remarkable applications to cryptography. Many ofthese applications involve arithmetic geometric objects known as ellipticcurves. Modular forms, the primary object of study in this proposal, aregeneralizations of elliptic curves which play a fundamental role in modernnumber theory. The questions investigated in this proposal deal withinvariants which are closely related to those of interest in cryptographyand may perhaps yield some insight into them.