Four conjectures by Zhi-Hong Sun

TL;DR

This paper proves conjectures by Zhi-Hong Sun about the value of certain units modulo primes, relating these values to representations of primes as sums of squares via quadratic forms.

Contribution

It provides proofs for Sun's conjectures connecting units in quadratic fields to prime representations as sums of squares.

Findings

Confirmed conjectures on units modulo primes

Expressed results in terms of quadratic form representations

Linked algebraic units to prime decomposition patterns

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 $-4d$.