Arizona Winter School 2001
Sarnak's Course and Project Descriptions

Course description:

We will explain the use of spectral theory of automorphic forms to study diophantine problems as well as to problems concerning L-functions. Specifically, we review the Maass form spectral theory for GL(2). This is used to investigate averages of L-functions in families, which in turn are used to prove nontrivial upper bounds and nonvanishing of special values of L-functions. The diophantine applications include the solution of Hilbert's eleventh problem as well as other problems of equidistribution in arithmetic.
Some useful preparatory reading:
  • The two books of H.Iwaniec:
    1. "Introduction to the spectral theory of automorphic forms," Revista Mat. Iberoamericana, 1995
    2. "Topics in classical automorphic forms," AMS, 1999
  • My book "Some applications of modular forms," Cambridge Univiversity Press, 1990

Projects for students to digest and present:

  1. Waldpurger's paper in J. Math Pures Appl., 60 (9), (1981), 375-484 or perhaps more explicit versions of it such as W.Kohnen in Math Ann., 271 No 2, (1985), 237-268, and S. Katok-P. Sarnak in Israel Math J., 84, (1993), 193-222
  2. The paper by Duke, Friedlander, and Iwaniec in Invent. Math, 112, (1993), 1-8 and perhaps the second in this series, i.e., part II which appears in Inventiones a year or two later.