A recursion formula for local densities
Let \(S\) and \(T\) be non-degenerate matrices of degree \(s\) and \(t\) \((s\geq t\geq 1)\) with entries in the ring \(\mathbb{Z}_p\) of \(p\)-adic integers. Then the local density \(\alpha_p(T,S)\) is defined as follows: Set \[ A_e(T,S)= \{X\in M(s,t;\mathbb{Z}_p/_{p^e}\mathbb{Z}_p);{^t XSX}\equiv T\text{ mod }p^e\} \] and \(\alpha_p(T,S)= \lim_{e\to\infty}p^{-est+et(t- 1)/2}\# A_e(T,S)\). The paper under review contains some useful recursion formulas for local densities \(\alpha_p(T,S)\) in the case of a unimodular matrix \(X\). These formulas determine a denominator of the formal power series \[ R((B_1,\dots, B_s),A;x_1,\dots,x_s)= \sum_{e_1\geq\cdots\geq e_s\geq 0}\alpha_p (p^{e_1}B_1\perp\cdots\perp p^{e_s}B_s,A) x^{e_1}_1\cdots x^{e_s}_s, \] which is known to the rational. Here \(B_1,\dots,B_s\), \(A\) are non-degenerate symmetric matrices and \(A\) is unimodular.
scientific article; zbMATH DE number 1025608
scientific article