A remark on the Breuer's conjecture related to the Maillet's matrix (Q946984)

From the introduction: We are concerned with a question asked by F. Momose whether the \(\varphi(n)/2\) real numbers \(\cot(u\pi/n)\) (\(0<u<n/2),\ (u,n)=1\)) are independent over \(\mathbb Q\) or not. Here \(\varphi(n)\) denotes Euler's phi-function. In his book [Characters and automorphism groups of compact Riemann surfaces. Cambridge: Cambridge University Press (2000; Zbl 0952.30001)], \textit{T. Breuer} settled the equivalent problem in the case where \(n\) is a prime power with \(n>2\), and conjectured the \(\mathbb Q\)-independence for any \(n\) with \(n>2\). In this note, reducing to the nonvanishing of \(L(1,\chi)\), we prove the following, which implies \(\mathbb Q\)-independence of \(\cot(u\pi/n)\)'s. Proposition. Let \(n\) be an integer with \(n>2\). Then the rank of the matrix \(\left(\frac 12 -\left\langle\frac{au^*_b}{n}\right\rangle\right)\) is equal to \(\varphi(n)/2\), where \(a,b\) range over the set \(\{1,\dots,n-1\}\) and \(\{1,\dots,\varphi(n)/2\}\), respectively. The Maillet determinant , that is the determinant of the matrix of the Proposition above in the case where \(a\) ranges over the set \(\{u_1,\dots,u_{\varphi(n)/2}\}\) has been studied in various ways. It is equal to the first factor of the class number, up to a non-zero factor in the case where \(n\) is a prime power. In this note we deduce this Proposition by proving the following Theorem. Let \(n\) be an integer with \(n>2\). Then the following holds: \[ \Delta(\zeta_n^{u_1}/(1-\zeta_n^{u_1}),\dots,\zeta_n^{u_{\varphi(n)/2}},1,\zeta_n,\zeta_n^2,\dots,\zeta_n^{\varphi(n)/2-1}=\pm \frac 2{Qw}(n\sqrt{-1}-1)^{\varphi(n)/2}\cdot h_n^{-}\cdot d\cdot \prod_{\chi \text{ odd}}L_\chi\,. \]