On certain elliptic conjugacy classes of the Siegel modular group
The author announces some results on the elliptic conjugacy classes in \(\Gamma_ n:=Sp(n,{\mathbb{Z}})\), the Siegel modular group of degree n. Let \(\Phi =\Phi_ m(X)\) be the mth cyclotomic polynomial and denote by \({\mathfrak H}(\Phi)\), for a subgroup \({\mathfrak H}\) of Sp(n,\({\mathbb{C}})\), the set of elements of \({\mathfrak H}\) whose minimal polynomials are \(\Phi\). Then in Theorem 1, the author states a bijective correspondence between the set of conjugacy classes in \(\Gamma_ n(\Phi)\) and the set of isometric classes of \(D(K/{\mathbb{Q}})^{-1}\)-modular (skew) Hermitian forms over the ring \({\mathfrak O}_ K\) of integers of the cyclotomic field \(K:={\mathbb{Q}}(X)/(\Phi)\). This can be viewed as an integral version of the principle of Milnor and Springer-Steinberg. Applying it to the case of regular elements \((\phi (m)=2n)\) the author gets (Theorem 2): The number of conjugacy classes in \(\Gamma_ n(\Phi)\) is equal to \(h^-(K)\), the relative class number of K, times \(2^{n+t- 1}\) which is the number of integral conjugacy classes in Sp(n,\({\mathbb{Q}})(\Phi)\) of Sp(n,\({\mathbb{R}})(\Phi)\) \((t=1\) if K has prime power conductor, and \(t=0\) otherwise). This refines a result of \textit{H. Midorikawa} [Tokyo J. Math. 6, 25-38 (1983; Zbl 0522.10017)]. In Theorem 3, the characterization of the integral conjugacy classes among the \(2^ n\) classes in Sp(n,\({\mathbb{R}})(\Phi)\), by their eigenvalues as elements of U(n), the unitary group of degree n is given.
scientific article; zbMATH DE number 3912473