On some determinants involving Jacobi symbols

In this paper we study some conjectures on determinants with Jacobi symbol entries posed by Z.-W. Sun. For any positive integer $n\equiv3\pmod4$, we show that $(6,1)_n=[6,1]_n=(3,2)_n=[3,2]_n=0$ and $(4,2)_n=(8,8)_n=(3,3)_n=(21,112)_n=0$ as conjectured by Sun, where $(c,d)_n=\bigg|\left(\frac{i^2+cij+dj^2}n\right)\bigg|_{1\le i,j\le n-1}$ and $[c,d]_n=\bigg|\left(\frac{i^2+cij+dj^2}n\right)\bigg|_{0\le i,j\le n-1}$ with $(\frac{\cdot}n)$ the Jacobi symbol. We also prove that $(10,9)_p=0$ for any prime $p\equiv5\pmod{12}$, and $[5,5]_p=0$ for any prime $p\equiv 13,17\pmod{20}$, which were also conjectured by Sun. Our proofs involve character sums over finite fields.

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