Pascal's Matrix, Point Counting on Elliptic Curves and Prolate Spheroidal Functions

Abstract

The eigenvectors of the $(N+1)\times (N+1)$ symmetric Pascal matrix $T_N$ are analogs of prolate spheroidal wave functions in the discrete setting. The generating functions of the eigenvectors of $T_N$ are prolate spheroidal functions in the sense that they are simultaneously eigenfunctions of a third-order differential operator and an integral operator over the critical line $\{z\in\mathbb{C}: \text{Re}(z) = 1/2\}$. For even, positive integers $N$, we obtain an explicit formula for the generating function of an eigenvector of the symmetric pascal matrix with eigenvalue $1$. In the special case when $N=p-1$ for an odd prime $p$, we show that the generating function is equivalent modulo $p$ to $(\# E_z(\mathbb F_p)-1)^2$, where $\# E_z(\mathbb F_p)$ is the number of points on the Legendre elliptic curve $y^2 = x(x-1)(x-z)$ over the finite field $\mathbb F_p$.