**NB:** A

PDF
version of this announcement (suitable for posting) is also available.

## Fourier Expansions of Modular Forms on Finite Upper Half Planes

### Audrey Terras

Math. Dept., U.C.S.D.

###
Thursday, October 13, 2005

L01 Carson Hall, 4 pm

Tea 3:30 pm, Math Lounge

**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.