Number Theory Seminar

I will talk about two results. The first is a new case of the BSD conjecture, contained in a joint work with Hamacher and Zhao. Namely, we prove the conjecture for elliptic curves of height 1 over a global function field of genus 1 under a mild assumption. This is obtained by specializing a more general theorem on the Tate conjecture. The key geometric idea is an application of rigidity properties of the variations of Hodge structures to study deformation of line bundles in positive and mixed characteristic. Then I will talk about a generalization of such deformation results recently obtained with Urbanik. Namely, we show that for a sufficiently big arithmetic family of smooth projective varieties, there is an open dense subscheme of the base over which all line bundles in positive characteristics can be obtained by specializing those in characteristic 0.