On some determinants involving cyclotomic units

Abstract

For each odd prime $p$, let $\zeta_p$ denote a primitive $p$-th root of unity. In this paper, we study the determinants of some matrices with cyclotomic unit entries. For instance, we show that when $p\equiv 3\pmod4$ and $p>3$ the determinant of the matrix $\(\frac{1-\zeta_p^{j^2k^2}}{1-\zeta_p^{j^2}}\)_{1\le j,k\le (p-1)/2}$ can be written as $(-1)^{\frac{h(-p)+1}{2}}(a_p+b_pi\sqrt{p})$ with $a_p,b_p\in\frac12\Z$ and $$\begin{cases}\nu_p(a_p)=\nu_p(b_p)=\frac{p-3}{8}&\mbox{if}\ p\equiv 3\pmod8, \\\nu_p(a_p)=\nu_p(b_p)+1=\frac{p+1}{8}&\mbox{if}\ p\equiv 7\pmod8,\end{cases}$$ where $\nu_p(x)$ denotes the $p$-adic order of a $p$-adic integer $x$, and $h(-p)$ denotes the class number of the field $\Q(\sqrt{-p})$. Meanwhile, let $(\frac{\cdot}{p})$ denote the Legendre symbol. We have $$2^{\frac{p+1}{2}}a_pb_p=(-1){^\frac{p+1}{4}}p^{\frac{p-3}{4}}\det [S(p)],$$ and $$2^{\frac{p-1}{2}}(a_p^2-pb_p^2)=\frac{p-1}{2}(-p)^{\frac{p-3}{4}}\det [S(p)],$$ where $\det [S(p)]$ is the determinant of the $\frac{p-1}{2}$ by $\frac{p-1}{2}$ matrix $S(p)$ with entries $S(p)_{j,k}=(\frac{j^2+k^2}{p})$ for any $1\le j,k\le (p-1)/2$.