Course description: We will explain the use of spectral theory of automorphic forms to study diophantine problems as well as to problems concerning Lfunctions. Specifically, we review the Maass form spectral theory for GL(2). This is used to investigate averages of Lfunctions in families, which in turn are used to prove nontrivial upper bounds and nonvanishing of special values of Lfunctions. 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:

"Introduction to the spectral theory of automorphic forms," Revista Mat. Iberoamericana, 1995

"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:

Waldpurger's paper in J. Math Pures Appl., 60 (9), (1981), 375484 or perhaps more explicit versions of it such as W.Kohnen in Math Ann., 271 No 2, (1985), 237268, and S. KatokP. Sarnak in Israel Math J., 84, (1993), 193222

The paper by Duke, Friedlander, and Iwaniec in Invent. Math, 112, (1993), 18 and perhaps the second in this series, i.e., part II which appears in Inventiones a year or two later.
