Classical hypergeometric functions play interesting roles in number theory and arithmetic geometry. For example, F. Beukers has shown that Apery's theorem on the irrationality of zeta(3) can be conceptualized in terms of values of hypergeometric functions. For another example, the periods of the Legendre family of elliptic curves can be expressed in terms of simple 2F1 functions.
In this talk I will introduce a class of hypergeometric functions defined over finite fields, originally defined by J. Green and D. Stanton, which are closely connected to both classical hypergeometric functions and other number theoretic objects. In particular, we will see that these functions can be used to count points on varieties over finite fields and that they also generate the Fourier coefficients of certain modular forms. Joint work with S. Frechette and K. Ono.