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 ${\mathbb{F}}_p$.

14 pages