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$D_{\lambda }^{d,k}:X_1^d+X_2^d=\lambda d{X}_1^k{x}_2^{d-k},$ where $d\ge 2$, $1\le k\le d-1$, and $\gcd (d,k)=1$. Let $\#D_{\lambda }^{d,k}$ denote the number of points on $D_{\lambda }^{d,k}$ in $P^1\left(F_q\right)$. It is easy to see that $\#D_{\lambda }^{d,k}$ is equal to the number of distinct zeros of the polynomial $y^d-d\lambda y^k+1\in F_q\left[y\right]$ in $F_q$. In this article, we prove that $\#D_{\lambda }^{d,k}$ is also equal to the number of distinct zeros of the polynomial $y^{d-k}{(1-y)}^k-{\left(d\lambda \right)}^{-d}$ in $F_q$. We express the number of distinct zeros of the polynomial $y^{d-k}{(1-y)}^k-{\left(d\lambda \right)}^{-d}$ 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.