Number Theory Seminar
In this talk we will discuss further applications of $\ell$-adic cohomology to analytic number theory.
In particular we will see that a certain non-trivial bound for bilinear sums of hyper-Kloosterman sums (below the Polya-Vinogradov range) constitute the final ingredient towards the resolution of a longstanding problem concerning moments for twists of $GL_2$ $L$-functions. In turn, these results shed some light on some recent conjectures of Mazur-Rubin regarding the distribution of modular symbols.
These are joint works with V. Blomer, E. Fouvry, E. Kowalski, D. Milicevic and W. Sawin