Conjectures on L-functions for Varieties over Function Fields and Their Relations

Explore conjectures on L-functions for smooth varieties over finitely generated fields in positive characteristic p in this mathematical lecture. Delve into joint work with T. Keller and Y. Qin, examining versions of conjectures traceable to Tate and their interconnections. Focus on abelian varieties A=K and learn how the BSD-rank conjecture relates to the finiteness of the p-primary part of the Tate-Shafarevich group using rigid cohomology, assuming resolutions of singularities in positive characteristic. Discuss requirements for generalizing these concepts and gain insights into advanced topics in algebraic geometry and number theory from Veronika Ertl of Universität Regensburg.