Some determinants involving quadratic residues modulo primes

Abstract: In this paper we evaluate several determinants involving quadratic residues modulo primes. For example, for any prime $p>3$ with $p\equiv3\pmod4$ and $a,b\in\mathbb Z$ with $p\nmid ab$, we prove that $$\det\left[ 1+\tan\pi\frac{aj^2+bk^2}p \right]_{1\le j,k \le \frac{p-1}2} = \begin{cases}-2^{(p-1)/2}p^{(p-3)/4}&\text{if}\ (\frac{ab}p)=1, \p^{(p-3)/4}&\text{if}\ (\frac{ab}p)=-1, \end{cases}$$ where $(\frac{\cdot}p)$ denotes the Legendre symbol. We also pose some conjectures for further research.