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Programme on Lie Groups 2001

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Topics on Analytic Theory of Automorphic Forms

Professor Wenzhi Luo
The Ohio State University, USA

This series of 8 lectures will present a survey of some recent development in the analytic aspect of automorphic forms, with special emphasis on modular forms, Maass forms and the associated L-functions. The lectures will center on the following closely related themes:

1. Weyl's law.

2. The Siegel zero for symmetric square L-function (a la Hoffstein and Lockhart).

3. Selberg's eigenvalue conjecture.

4. Ergodicity of eigenfunctions of the Laplacian.

5. Equidistribution of Hecke eigenforms.

6. Prime geodesic theorem.

7. Determination of modular forms by twists of critical L-values.

8. Density of integer points on varieties.

References

1. J. Hoffstein and P. Lockhart, Coefficients of Maass forms and the Siegel zero, with an appendix by D. Goldfeld, J. Hoffstein and D. Lieman, An effective zero free region, Annals of Math. 140, 1994, 161-181.

2. W. Luo, On the nonvanishing of Rankin-Selberg L-functions, Duke Math. J. Vol.69, 1993, 411-425.

3. W. Luo, Z. Rudnick and P. Sarnak, On Selberg's eigenvalue conjecture, Geometric and Functional Analysis, Vol.5, 1995, 387-401.

4. W. Luo and P. Sarnak, Quantum ergodicity of eigenfunctions on PSL(2,Z)\H, IHES Publ. Math. 81, 1995, 207-237.

5. W. Luo and D. Ramakrishnan, Determination of modular forms by twists of critical L-values, Invent. math. 130, 1997, 371-398.

6. W. Luo, Rational points on complete intersections over F_p, Inter. Math. Res. Notices, 16, 1999, 901-907.

7. W. Luo, Z.Rudnick and P.Sarnak, On the generalized Ramanujan conjecture for GL(n), Proc. Symp. Pure Math. 66, part 2, Amer. Math. Soc., 1999, 301-310.

8. W. Luo, Nonvanishing of L-values and the Weyl law, to appear in Annals of Math.

Dates / Time:	May Lectures May 23 (Wed), 25 (Fri), 31 (Thur) at 10:30am - 12:00noon June Lectures June 13 (Wed), 15 (Fri), 20 (Wed), 22 (Fri), 26 (Tue) at 4:00 - 5:30pm
Place:	Room 517, Meng Wah Complex