Diagonal hypersurfaces and elliptic curves over finite fields and hypergeometric functions

Abstract

Let $D_{\lambda }^{d,k}$ denote the family of diagonal hypersurface over a finite field ${\mathbb{F}}_q$ given by \begin{align*} D_λ^{d,k}:X_1^d+X_2^d=λdX_1^kx_2^{d-k}, \end{align*} 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 ${\mathbb{P}}^1({\mathbb{F}}_q)$. It is easy to see that $\#D_{\lambda }^{d,k}$ is equal to the number of distinct zeros of the polynomial $y^d-d\lambda y^k+1\in {\mathbb{F}}_q[y]$ in ${\mathbb{F}}_q$. In this article, we prove that $\#D_{\lambda }^{d,k}$ is also equal to the number of distinct zeros of the polynomial $y^{d-k}(1-y{)}^k-(d\lambda {)}^{-d}$ in ${\mathbb{F}}_q$. 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.

25 pages