Artihmetic Geometry: Iwasawa Theory, the Bloch-Kato Conjecture, and the Birch and Swinnerton-Dyer Conjecture

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Number theory is a subject in mathematics that has been developing very rapidly in recent years. This research project studies Iwasawa theory, a branch of number theory. It relates integral and rational solutions of polynomial equations to certain analytic objects known as L-functions. The project exploits novel approaches to this study, which combine new tools from the Langlands program and algebraic geometry. The new approaches have been used to prove that a majority of elliptic curves satisfy the Birch and Swinnerton-Dyer conjecture, whose general truth is still a challenging open question in number theory. This research project aims to expand the set of elliptic curves known to satisfy the conjecture.More concretely, this project studies the relationships between special values of L-functions and certain arithmetic objects, namely the Selmer groups of Galois representations. For a prime number p the main problems under study are the Iwasawa main conjectures and p-adic Bloch-Kato conjectures. The project develops a new approach towards non-ordinary Iwasawa theory: first to study Greenberg type Iwasawa main conjectures that are accessible to proof due to their ordinary nature, and then to relate them to non-ordinary Iwasawa theory using explicit reciprocity laws of special cycles. The ultimate goal is to prove, for all GL(2) modular forms (of any weight and possibly with ramification at p): the p-part of the Birch and Swinnerton-Dyer formula in the case when analytic rank is 1 or 0; the Iwasawa main conjecture; and that the vanishing of central critical L-value implies the corresponding Selmer group has rank at least 1. The project also aims to study these problems for higher rank motives, especially those associated to cusp forms on unitary groups.