Quadratic Diophantine Equations, the Class Number, and the Mass Formula

with a given q ∈ F×. In particular, in the classical case with F = Q and V = Q, we usually assume that φ is Z-valued on Z and q ∈ Z. The purpose of the present article is to present some new ideas on various arithmetical questions on such an equation. We start with some of our basic symbols and terminology. For a set X we denote by #X or #{X} the number (≤ ∞) of elements of X. For an associative ring R with identity element, we denote by R× the group of invertible elements of R and by Mn(R) the ring of all square matrices of size n with entries in R. We then put GLn(R) = Mn(R) and denote by 1n the identity element of Mn(R). For two square matrices A and B of size m and n we denote by diag[A, B] the square matrix of size m+ n with A and B in diagonal blocks and zeros in the remaining blocks. Now, given (V, φ) as above, we always assume that φ is nondegenerate. We also put n = dim(V ) and define, as usual, the orthogonal group O(V ) and the special orthogonal group SO(V ) by