Four conjectures by Zhi-Hong Sun

Abstract

We prove some results conjectured by Zhi-Hong Sun regarding the value $\mod p$ of $\varepsilon_d^{\frac{p-1}4}$, where $\varepsilon_d$ is a unit of norm $-1$ in some fields $\mathbb Q(\sqrt d)$, with $\left(\frac{-1}p\right) =\left(\frac dp\right) =1$. 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 $-4d$.