An explicit formula for local densities of quadratic forms (Q1273719)

Let \(S\) and \(T\) be two integral nondegenerate symmetric matrices of \(\mathbb{Z}_p\) of degree \(m\) and \(n\) respectively. Assume \(m>n\geq 1\). Then the local density of representing \(T\) by \(S\) is defined as \[ \alpha_p(T,S)=\lim_{t\to\infty}(p^t)^{n(n+1)/2-mn}A_t(T,S), \] where \(A_t(T,S)=\sharp\{X\in M_{m,n}(\mathbb{Z}_p/p^t):X^tSX\equiv T\bmod p^t\}\). Suppose \(S\) is half-integral and \(p^{-1}S\) is not and let \(0\leq l\leq 1\). There is a polynomial \(\alpha(X,T,S)\) with \(\alpha(1,T,S)=\alpha_p(T,S)\). The main theorem of the paper under review states that, if \(n=1\), there is an explicitly constructed polynomial \(R_1(X,T,S)\) such that \[ \alpha(X,p^lT,p^lS)=1+p^lX^lR_1(X,T,S)+(1-p^{-1})lp^lX^l. \] A similar formula is given for \(n=2\) and \(p\neq 2\). The method used in the proof of this result is quite different from others in this subject. The author relates the local density to a Whittaker integral and then computes the integral directly.