Elliptic Curves, p-adic Deformations, and Iwasawa Theory

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A central problem in number theory is understanding the (integer, or rational) solutions to polynomial equations with integer coefficients. Elliptic curves are a class of such equations that has been studied since antiquity, yet for which many questions still remain open. Elliptic curves are a central object of study in this research. Since the fundamental work of Gross-Zagier and Kolyvagin, powerful tools from algebraic geometry and representation theory have guided research in this direction. The PI aims to leverage some recent developments in those areas to further our understanding of the arithmetic of elliptic curves and related problems.More specifically, the research is divided into two related parts. The projects in the first part focus on the construction of linearly independent Selmer classes for higher rank elliptic curves. Darmon and Rotger have proposed very precise conjectures in the rank 2 setting, and these projects are expected to yield new results in this direction. The second part focuses on problems in Iwasawa theory, with applications towards the Birch and Swinnerton-Dyer conjecture, a non-vanishing conjecture by Greenberg in the case of root number -1, and the conjectured local indecomposability of the Galois representations associated to ordinary non-CM forms.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.