Special value formulas of L

Special value formulas of L-functions
Wei Zhang
Time: 4 lectures, about 8 hours.
We will mainly focus on the cental value formula of Rankin L-series
established by Waldspurger in [2]. If the time permits, we can discuss a
similar formula of Ichino for triple product L-series.
1. Introduction to special value formula of L-function. [Review basic
concepts for automorphic forms on GL(2).]
2. Weil representation, theta series and Siegel-Weil formula
3. Waldspurger’s formula for Rankin L-series (I). [Global periods.]
4. Waldspurger’s formula for Rankin L-series (II). [Local period, inte-
gral of matrix coefficients.]
Prerequisite: Adeles, Hecke characters. And very basic ideas on repre-
sentation theory of GL2(Qp). A nice reference is the first 6 chapters of the
book by Gelbart ([1])
References
[1] Gelbart, Stephen S. Automorphic forms on adele groups. Annals of
Mathematics Studies, No. 83. Princeton University Press, Princeton,
N.J.; University of Tokyo Press, Tokyo, 1975. x+267 pp.
[2] Waldspurger, J.-L. Sur les valeurs de certaines fonctions Lautomorphes
en leur centre de symtrie. (French) [Values of certain automorphic L-
functions at their center of symmetry] Compositio Math. 54 (1985), no.
2, 173–242.
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