THE NUMBER OF F q-POINTS ON DIAGONAL HYPERSURFACES WITH MONOMIAL DEFORMATION

We consider the family of diagonal hypersurfaces with monomial deformation D d ,A, h : x 1 d + x 2 d +<middle dot><middle dot><middle dot>+ x n d - d A x h 1 1 x h 2 2 . . . x h n n = 0 , where d = h 1 + h 2 + <middle dot> <middle dot> <middle dot> + h n with gcd ( h 1 , h 2 , . . . , h n ) = 1. We first provide a formula for the number of F q-points on D d ,A, h in terms of Gauss and Jacobi sums. This generalizes a result of Koblitz, which holds in the special case d | q - 1. We then express the number of F q-points on D d ,A, h in terms of a p-adic hypergeometric function previously defined by the author. The parameters in this hypergeometric function mirror exactly those described by Koblitz when drawing an analogy between his result and classical hypergeometric functions. This generalizes a result by Sulakashna and Barman, which holds in the case gcd ( d , q - 1 ) = 1. In the special case h 1 = h 2 = <middle dot> <middle dot> <middle dot> = h n = 1 and d = n , i.e., the Dwork hypersurface, we also generalize a previous result of the author which holds when q is prime.