Pascal's Matrix, Point Counting on Elliptic Curves and Prolate Spheroidal Functions
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915479930535936 |
|---|---|
| author | Casper, W. Riley |
| author_facet | Casper, W. Riley |
| contents | 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 $\mathbb F_p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_08494 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pascal's Matrix, Point Counting on Elliptic Curves and Prolate Spheroidal Functions Casper, W. Riley Spectral Theory Classical Analysis and ODEs Number Theory 33C90, 11C20, 34L10, 11T06 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 $\mathbb F_p$. |
| title | Pascal's Matrix, Point Counting on Elliptic Curves and Prolate Spheroidal Functions |
| topic | Spectral Theory Classical Analysis and ODEs Number Theory 33C90, 11C20, 34L10, 11T06 |
| url | https://arxiv.org/abs/2508.08494 |