Some determinants involving binary forms

Abstract: In this paper, we study arithmetic properties of certain determinants involving powers of $i^2+cij+dj^2$, where $c$ and $d$ are integers. For example, for any odd integer $n>1$ with $(\frac dn)=-1$ we prove that $\det [ (\frac{i^2+cij+dj^2}{n})]_{0\le i,j\le n-1}$ is divisible by $\varphi(n)^2$, where $(\frac{\cdot}{n})$ is the Jacobi symbol and $\varphi$ is Euler's totient function. This confirms a previous conjecture of the second author.