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Abstract: I will report on some joint work with Anthony Shaheen. First I will discuss the finite upper half plane H_q attached to a finite field F_q with q elements and compare it to the usual Poincare upper half plane. Here q = p^n, and I assume that the prime p is odd and that n > 1. A reference is Terras, Fourier Analysis on Finite Groups and Applications, Cambridge, 1999. The group G = GL(2; F_q) (2x2 non-singular matrices with entries in F_q) acts on H_q by fractional linear transformation. A finite analogue of a modular form is a complex-valued function on H_q with an invariance property under the action of GL(2; F_p) - the subgroup of G which is our analogue of the modular group. The talk will then discuss Fourier expansions of finite analogues of Maass-type Eisenstein series for SL(2; F_p) and GL(2; F_p). For Maass wave forms on the ususal Poincare upper half plane, these Fourier expansions involved K-Bessel functions. In the finite case they involve Kloosterman sums.
This talk will be accessible to undergraduates.
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