Mini Course: Relative trace formula and applications

Prerequisite: We shall assume knowledge in graduate-level algebraic number theory (including the adelic language and Tate’s thesis), as well as basics of modular (or automorphic) forms and Lie (or algebraic) groups. We shall try to make the class accessible to more people and recall some necessary facts.

Course Description: The relative trace formula was invented by Jacquet to reprove Waldspurger’s famous theorem relating toric periods to central values of automorphic L-functions for GL(2). We shall give an introduction to Jacquet’s approach and discuss certain applications and generalisations, notably the Jacquet-Rallis trace formula. Our tentative plan is as follows.

1. Motivations and preliminaries

2. The Petersson-Bruggeman-Kuznetsov formula, after Knightly-Li

3. Geometric side of Jacquet’s relative trace formulae

4. Spectral side of Jacquet’s relative trace formulae

5. Introduction to the Gan-Gross-Prasad conjecture for unitary groups

6. From the Arthur-Selberg trace formula to the Jacquet-Rallis trace formula, after Zydor

7. Ingredients of the proof of the unitary GGP conjecture, after W. Zhang et al.

Reference:

 1. D. Bump, Automorphic forms and representations.

 2. J. R. Getz and H. Hahn, An introduction to automorphic representations---with a view toward trace formulae.

 3. A. Knightly and C. Li, A relative trace formula proof of the Petersson trace formula.

 4. J.-L. Waldspurger, Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie.

 5. H. Jacquet, Sur un résultat de Waldspurger.

 6. J. Guo, On the positivity of the central critical values of automorphic L-functions for GL(2).

 7. W. T. Gan, B. H. Gross and D. Prasad, Symplectic local root numbers, central critical L values, and restriction problems in the representation theory of classical groups,

 8. Raphaël Beuzart-Plessis, Relative trace formulae and the Gan–Gross–Prasad conjectures.

 9. J. Arthur, An introduction to the trace formula.

 10. H. Jacquet and S. Rallis, On the Gross-Prasad conjecture for unitary groups.

 11. M. Zydor, Les formules des traces relatives de Jacquet-Rallis grossières.

 12. P.-H. Chaudouard, On relative trace formulae: the case of Jacquet-Rallis.

 13. W. Zhang, Fourier transform and the global Gan-Gross-Prasad conjecture for unitary groups.

 14. W. Zhang, Automorphic period and the central value of Rankin-Selberg L-function.

Schedule & Venue: