A Poisson summation formula for SL(2, Z)
by
Johan Andersson
Stockholm University

In the talk we will develop a new type of formula that will allow us to expand a sum over SL(2, Z) for test functions f on SL(2, R)

å A ∈ SL(2, Z)  f(A),

into spectral objects such as Fourier coefficients for Maass wave and holomorphic cusp forms, and divisor functions. When f is SO(2) invariant this essentially reduces to the spectral expansion of an automorphic function. Our proof is as follows: First use properties of Ramanujan and Kloostermann sums to express the sum as a sum of Kloostermann sums and then apply the Kuznetsov summation formula. The formula can also be obtained when summing over integer matrices with fixed determinant D. The formula can be used as a natural starting point for the Kuznetsov-Motohashi theory for the spectral theory of the Riemann zeta-function.

Date received: March 15, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caew-11.