We propose a conjectural construction of determinants global points on modular elliptic curves over arbitrary number fields, generalizing both the p-adic Heegner via Čerednik–Drinfeld uniformization and definition classical Stark–Heegner points. In alignment with Nekovář Scholl's plectic conjectures, we expect non-triviality these to control Mordell–Weil group higher rank curves. provide some i...

In this paper we prove several Lehmer type inequalities for Drinfeld modules which will enable us to prove certain Mordell-Weil type structure theorems for Drinfeld modules.

This is an extended version of the talk I gave at the summer school in Göttingen in July 2007. We discuss the Mordell-Weil Sieve and some applications.

We compute all K3 surfaces with Picard rank 20 over Q. Our proof uses modularity, the Artin-Tate conjecture and class group theory. With different techniques, the result has been established by Elkies to show that Mordell-Weil rank 18 over Q is impossible for an elliptic K3 surface. We also apply our methods to general singular K3 surfaces, i.e. with geometric Picard rank 20, but not necessaril...