Iwasawa Theory and Arithmetic Geometry

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Agboola 9700937 This award will fund research on two topics in arithmetic algebraic geometry. The first is concerned with Iwasawa theory and L-functions. The PI will study new approaches to the Birch and Swinnerton-Dyer conjecture and the theory of p-adic L-functions. These approaches have their origins in the theory of arithmetic Galois module structure, and they lead to new ways of studying the Iwasawa theory of abelian varieties and p-adic representations. The second topic deals with equivariant algebraic geometry and the K-theory of higher dimensional varieties over finite fields. The PI will study reduced Grothendieck groups of equivariant vector bundles on varieties. He will develop the geometry of locally free class groups in positive characteristic, and he will explore connections with higher-dimensional class field theory and crystalline cohomology. The research described in this proposal lies in the field of arithmetic algebraic geometry. This is a subject that blends two of the oldest branches of mathematics---number theory and geometry---and which has blossomed to the point where it has solved problems that have stood for centuries. It finds applications in fields as diverse as physics, robotics, data processing and information theory.