Taelman discovered an analog of BSD conjecture for Drinfeld modules and
proved it for the coefficient ring F_q[t]. His methods do not generalize
easily to more complicated rings. A different approach to Taelman's BSD
conjecture is provided by the theory of shtuka cohomology. This approach
allows one to treat all the coefficient rings in a uniform way. It leads
to a rather conceptual proof of the conjecture which resonates well with
equivariant Tamagawa number conjecture for motives.
My aim is to explain the shtuka-theoretic proof of Taelman's analog
of the BSD conjecture. The course can be naturally divided into three
1. Statement of the conjecture, overview of the proof and introduction to shtuka cohomology.
2. Shtuka models of Drinfeld modules and their cohomology.
3. Regulator theory and trace formula for elliptic shtukas.