Elliptic curves over $\mathbb{F}_p$ and determinants of Legendre matrices

TL;DR

This paper investigates determinants of matrices with Legendre symbol entries related to elliptic curves over finite fields, confirming conjectures and exploring properties of such determinants over primes.

Contribution

It confirms a conjecture by Sun regarding the existence of infinitely many primes with zero determinants in specific Legendre symbol matrices and explores their connection to elliptic curves.

Findings

Proves the existence of infinitely many primes with zero determinants in Legendre symbol matrices.

Confirms Sun's conjecture about determinants related to elliptic curves over finite fields.

Abstract

Determinants with Legendre symbol entries have close relations with character sums and elliptic curves over finite fields. In recent years, Sun, Krachun and his cooperators studied this topic. In this paper, we confirm some conjectures posed by Sun and investigate some related topics. For instance, given any integers $c,d$ with $d\ne0$ and $c^2-4d\ne0$, we show that there are infinitely many odd primes $p$ such that $$\det\bigg[\left(\frac{i^2+cij+dj^2}{p}\right)\bigg]_{0\le i,j\le p-1}=0,$$ where $(\frac{\cdot}{p})$ is the Legendre symbol. This confirms a conjecture of Sun.