Yun and Zhang compute the Taylor series expansion of an automorphic L-function over a function field, in terms of intersection pairings of certain algebraic cycles on the so-called moduli stack of shtukas. This generalizes the Waldspurger and Gross-Zagier formulas, which concern the first two coefficients.

The goal of the seminar is to develop the background necessary to state their formula, and then indicate the structure of the proof. If time allows, we may also discuss applications to the Birch and Swinnerton-Dyer conjecture for elliptic curves over function fields.

Tentative plan:

(a) Automorphic forms over function fields and their L-functions.

(b) Waldspurger's formula

(c) Relative trace formula

(d) Intersection theory on smooth algebraic varieties (and smooth stacks)

(e) Moduli spaces of Drinfeld's stukas

(f) Formula of Yun and Zhang

(g) Outline of proof

P.S. Those, who want to get an impression about the subject may consult

http://web.stanford.edu/~tonyfeng/HigherGrossZagier.pdf,

but we are going to assume less background and to go in a much slower pace.

## Date:

Sunday, 31 March, 2019 - 14:00 to 16:00

Repeats every week every Sunday until Sun Jun 23 2019 except Sun Apr 21 2019