Diagonal hypersurfaces and elliptic curves over finite fields and hypergeometric functions

Let D-lambda(d,k) denote the family of diagonal hypersurface over a finite field F(q )given by D-lambda(d,k) : X-1(d) + X-2(d) = lambda dX(1)(k) x(2)(d-k) , where d >= 2, 1 < k <= d - 1, and gcd(d, k) = 1. Let # D-lambda(d,k) denote the number of points on D-lambda(d,k) in P-1(F-q). It is easy to see that #D-lambda(d,k) is equal to the number of distinct zeros of the polynomial yd - d lambda y(k) + 1 is an element of F-q[y] in Fq. In this article, we prove that #Dd,k lambda is also equal to the number of distinct zeros of the polynomial y(d-k)(1 - y)(k -) (d lambda)(-d) in F-q. We express the number of distinct zeros of the polynomial y(d-k)(1- y)(k) - (d lambda)(-d) in terms of a p-adic hypergeometric function. Next, we derive summation identities for the p-adic hypergeometric functions appearing in the expressions for #D-lambda(d,k). Finally, as an application of the summation identities, we prove identities for the trace of Frobenius endomorphism on certain families of elliptic curves. (c) 2024 Elsevier Inc. All rights reserved.