Four conjectures by Zhi-Hong Sun

Abstract

We prove some results conjectured by Zhi-Hong Sun regarding the value $\mathrm{mod}p$ of ${\epsilon }_d^{\frac{p-1}{4}}$, where ${\epsilon }_d$ is a unit of norm $-1$ in some fields $\mathbb{Q}(\sqrt{d})$, with $\left(\frac{-1}{p}\right)=\left(\frac{d}{p}\right)=1$. The answer is given in terms of how $p$ writes as $p=f(x,y)=u^2+v^2$, with $x,y,u,v\in \mathbb{Z}$, where $f$ is a certain quadratic form of determinant $-4d$.