Gaussian Hypergeometric Series and Traces of Hecke Operators
by
Sharon Frechette
College of the Holy Cross
Coauthors: Ken Ono (University of Wisconsin), Matthew Papanikolas (Texas A & M)

We establish a simple inductive formula for the trace Tr_k^new(\Gamma₀(8), p) of the p-th Hecke operator on the space S_k^new(\Gamma₀(8)) of newforms of level 8 and weight k in terms of the values of ₃ F₂-hypergeometric functions over the finite field F_p. Using this formula when k=6, we prove a conjecture of Koike relating Tr₆^new(\Gamma₀(8), p) to the values ₆F₅(1)_p and ₄F₃(1)_p. Furthermore, we find new congruences between Tr_k^new(\Gamma₀(8), p) and generalized Apéry numbers.

Date received: January 27, 2004