Exponential sums, elliptic curves and rational points on varieties

<strong>Seminar of the School of Physical Sciences ----------------------------------------------</strong> Title: <strong>Exponential sums, elliptic curves and rational points on varieties</strong> Speaker: <strong>Stephan Baier</strong> (Tata Institute of Fundamental Research, Mumbai and Harish-Chandra Research Institute, Allahabad) Date:<strong> August 11, 2015</strong> <strong>Abstract: </strong>In this talk, I will give an overview of my research in the recent years. Roughly, it breaks down into the following categories: 1) L-functions for elliptic curves, 2) Rational points on varieties, 3) Diophantine approximation with constraints. Of central importance in my work is the use of exponential sums. More precisely, I will talk about my results on the Sato-Tate and Lang-Trotter conjectures on elliptic curves and the zero distribution of their L-functions (joint with L. Zhao), the distribution of rational points on Del Pezzo surfaces (joint with T. Browning) and rational approximation on manifolds with restricted numerator and denominator (joint with A. Ghosh).