Rational points on elliptic curves over function fields

Abstract: In this talk I would like to discuss older and more recent techniques of finding rational points on elliptic curves over functions fields. We will focus mostly on the case of the rational function field and describe how the structure of the Picard lattice of the associated elliptic surface can be used

to make the computations effective. I will describe some classical results due to Shioda, Kumar and van Luijk. I will emphasize the computational and practical aspect of the problem. In the final part, we will  discuss certain applications of the general theory, including computations on rational elliptic surfaces, K3 surfaces, construction of certain motives and relation to del Pezzo surfaces.