Exponential sums and finite field $A$-hypergeometric functions

Abstract: We define finite field $A$-hypergeometric functions and show that they are Fourier expansions of families of exponential sums on the torus. For an appropriate choice of $A$, our finite field $A$-hypergeometric function can be specialized to the finite field ${}kF{k-1}$-hypergeometric function defined by McCarthy.