Explore a mathematical lecture that delves into two significant results in algebraic geometry and number theory. Begin with an examination of a new case of the Birch and Swinnerton-Dyer (BSD) conjecture for elliptic curves of height 1 over global function fields of genus 1, derived from a joint work with Hamacher and Zhao. Discover how this result is obtained by specializing a more general theorem on the Tate conjecture, with a focus on the key geometric idea involving rigidity properties of variations of Hodge structures to study deformation of line bundles in positive and mixed characteristic. Then, investigate a generalization of these deformation results, recently developed with Urbanik, demonstrating that for sufficiently large arithmetic families of smooth projective varieties, there exists an open dense subscheme of the base over which all line bundles in positive characteristics can be obtained by specializing those in characteristic 0.