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

TL;DR

This paper explores the eigenvectors of Pascal matrices, linking them to prolate spheroidal functions and elliptic curve point counts over finite fields, revealing new explicit formulas and modular relationships.

Contribution

It provides explicit formulas for eigenvector generating functions of Pascal matrices and connects these to elliptic curve point counts modulo primes, bridging linear algebra, special functions, and number theory.

Findings

Eigenvectors of Pascal matrices are analogs of prolate spheroidal functions.

Explicit formulas for eigenvector generating functions at eigenvalue 1.

Modulo p, generating functions relate to elliptic curve point counts.

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 $\mathbb F_p$.