International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 63918, 17 pages
doi:10.1155/IJMMS/2006/63918

Fourier expansions of complex-valued Eisenstein series on finite upper half planes

Anthony Shaheen1 and Audrey Terras2

1Department of Mathematics, California State University, Los Angeles 90032-8204, CA, USA
2Department of Mathematics, University of California, San Diego, La Jolla 92093-0112, CA, USA

Received 11 May 2006; Revised 20 July 2006; Accepted 25 July 2006

Copyright © 2006 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider complex-valued modular forms on finite upper half planes Hq and obtain Fourier expansions of Eisenstein series invariant under the groups Γ=SL(2,Fp) and GL(2,Fp). The expansions are analogous to those of Maass wave forms on the ordinary Poincaré upper half plane —the K-Bessel functions being replaced by Kloosterman sums.