On the solutions of quadratic Diophantine equations II

Abstract

A quantity concerning the solutions of a quadratic Diophantine equation in $n$ variables coincides with a mass of a special orthogonal group of a quadratic form in dimension $n-1$, via the mass formula due to Shimura. We show an explicit formula for the quantity, assuming the maximality of a lattice in the $(n-1)$-dimensional quadratic space. The quantity is determined by the computation of a group index and of the mass of the genus of maximal lattices in that quadratic space. As applications of the result, we give the number of primitive solutions for the sum of $n$ squares with 6 or 8 and also the quantity in question for the sum of 10 squares.