Higher Rank Selmer Groups

Brandeis University, Waltham MA

Investigators

Abstract

Number theory, the study of properties and hidden structures of integers, is one of the oldest branch of mathematics. An important aspect of it is the study of equations, or systems of equations, for which we are looking for solutions that are integers. These equations are named diophantine equations after Diophantus, a Greek mathematician of the Roman era. Diophantine equations are still the object of intense study nowadays. Independently, in the 19th century, a new branch of mathematics, born of the marriage between the calculus invented by Newton and Leibniz in the 17th century, and complex numbers discovered by algebraists of the 16th century, was developed: complex analysis, the study of so-called holomorphic functions, whose variable is a complex number. In a revolutionnary paper of 1859, Riemann observed that this new branch could be successfully used to solve previously intractable questions in number theory. Building on his work, mathematicians understood how to associate to any diophantine equation an holomorphic function, called its L-function, and make it a fundamental tool in studying the equation. Yet empirically observed subtle relations between the number of solutions of certain diophantine equations, and the analytic property of their holomorphic L-functiona remain unproved and deeply mysterious. The project will employ new methods to study these relations. The project will focus on the conjecture of Bloch-Kato, which predicts a relation between the arithmetic properties of diophantine equations (or rather their elementary components, called motives) and their holomorphic L-function. More precisely, this conjectures predicts that the rank of the Selmer group of a motive (which is an arithmetic object attached to the motive, closely related, in some fundamental cases, to the solutions of the diophantine equation defining it) is equal to the order at 1 of the L-function of that motive. The aim of this project is to prove, in many cases, one direction in this conjectural equality, namely that the Selmer groups has rank at least the order of vanishing. The proposed method involves the study of eigenvarieties, or universal families of automorphic forms, at critical points, the method of decomposition of critical p-adic L-functions of Dasgupta and the PI, and the local study of deformation of critical Galois representations developed by the PI and Chenevier, Bergdall, Breuil and others. 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.