(Joint work with T. Keller (Groningen) and Y. Qin (Berkeley)) We consider versions for smooth varieties X over finitely generated fields K in positive characteristic p of several conjectures that can be traced back to Tate, and study their interdependence. In particular, let A=K be an abelian variety. Assuming resolutions of singularities in positive characteristic, I will explain how to relate the BSD-rank conjecture for A to the finiteness of the p-primary part of the Tate-Shafarevich group of A using rigid cohomology. Furthermore, I will discuss what is needed for a generalisation.
De Hélène Esnault
De David Loeffler
De Yunqing Tang
De Alexander Goncharov
De Spencer Bloch
De Akhil Mathew
De Veronika Ertl
De Ana Caraiani
De Jie Lin
