Euler Systems, p-adic Deformations, and the Birch-Swinnerton-Dyer Conjecture

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This project is concerned with research in number theory. A central focus in this area of mathematics is understanding the mechanism whereby local information can be packaged to get access to the global information of interest, such as the solutions to polynomial equations. Certain analytic objects, the so-called L-functions, are expected to encode such mechanism. The celebrated conjecture of Birch and Swinnerton-Dyer (one of the millenium prize problems) revolves around this theme, as does Dirichlet's class number formula from the nineteenth century. This project aims to enhance our understanding of the Birch and Swinnerton-Dyer conjecture and of closely related problems. Progress in these directions may have an impact on areas, such as cryptography, exploiting the complexity of the arithmetic of elliptic curves.Euler systems build a bridge between certain arithmetic objects and their analytic counterparts (L-functions), hence providing very powerful tools for tackling problems on the passage from local to global. The project aims to exploit the Euler systems for tensor products and triple products of modular forms, and especially their variation in p-adic families, to obtain new results on fundamental open problems in number theory, such as Greenberg's conjecture on the generic order of vanishing of L-functions in Hida families, the p-part of the Birch and Swinnerton-Dyer formula in ranks 0 and 1, and the construction of explicit classes in Selmer groups of elliptic curves of rank 2, curves that lie just beyond our current understanding of the Birch and Swinnerton-Dyer conjecture.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.