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

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.