Involutive modular transformations on the Siegel upper half plane and an application to representations of quadratic forms
It is appropriate to extract some sentences from the authors' summary: We establish a one-to-one correspondence between the set of conjugacy classes of elliptic transformations in Sp(n,\({\mathbb{Z}})\) which satisfy \(X^ 2+I=0\) \((resp.\quad X^ 2+X+I=0)\) and the set of Hermitian forms of rank n over \({\mathbb{Z}}[\sqrt{-1}]\) \((resp.\quad {\mathbb{Z}}[(-1+\sqrt{- 3})/2])\) of determinant \(\pm 1\). As an application, we generalize, to positive symmetric integral matrices S of rank n, the classical fact that any divisor of \(m^ 2+1\) (resp. \(m^ 2+m+1)\) can be represented by the quadratic form \(F(X,Y)=X^ 2+Y^ 2\) (resp. \(X^ 2+XY+Y^ 2)\) with relatively prime integers X,Y.
conjugacy classes of elliptic transformations
scientific article; zbMATH DE number 3937250
scientific article