Extreme Values of Quadratic Twists of Modular L-Functions

Abstract

In this talk, I will discuss the extreme (minimum and maximum) values of quadratic twists of modular L-functions associated with a normalized modular form of weight k, evaluated at a horizontal mesoscopic shift from the critical point. I will also discuss some potential arithmetic applications, including the growth of the heights of Heegner points and the order of the Tate-Shafarevich group.