Arithmetic Aspects of Special Values of L-Functions

Understanding the structure of the set of rational points on an elliptic curve (essentially a cubic equation) has been an aim in number theory for over a century. It has connections to open problems (such as the congruent number problem) buried in antiquity. The Birch and Swinnerton-Dyer (BSD) conjecture predicts a mystifying relation among the rational points and the associated L-function. The intriguing L-functions appear in various areas of mathematics and also in physics. Even though analytic in appearance, the L-functions reveal striking affinity to arithmetic. The PI seeks to study certain arithmetic aspects of special values of L-functions. The first part of the project aims to establish a p-adic criterion for an elliptic curve to have analytic rank (the vanishing order of the associated L-function at its center) one. It will build on the recent progress towards the BSD conjecture (due to Skinner, Zhang and others) and aims to remove some of the key hypotheses. The other part of the project aims to study mod p non-vanishing of special values of L-functions in vertical and horizontal families. It will build on the progress during the last two decades (due to Hida, Michel--Venkatesh 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.