On power series attached to local densities
Let \(p\) be an odd prime. Let \(S \in M(s; \mathbb{Z}_p)\), \(T\in M(t; \mathbb{Z}_p)\) be nondegenerate symmetric matrices with entries in the \(p\)-adic integers \(\mathbb{Z}_p\). Set \[ A_e(T,S) = \bigl\{\overline X \in M(s,t; \mathbb{Z}_p)/p^eM(s,t; \mathbb{Z}_p) \mid{^tXSX}-T \in p^e M(s,t; \mathbb{Z}_p) \bigr\}. \] Then the local density \(\alpha_p(T,S)\) is defined by \[ \alpha_p (T,S) = \lim_{e\to \infty} p^{(-2st + t(t+1)) e/2}\cdot\# A_e(T,S). \] With it we attach the following power series \[ P(T,S;x_1,\dots, x_t): = \sum^\infty_{r_1, \dots, r_t=0} \alpha_p \bigl(\langle p^{r_1}, \dots, p^{r_t}\rangle \cdot T \cdot \langle p^{r_1}, \dots, p^{r_t}\rangle, S\bigr) \cdot x^{r_1}_1 \cdots x_t^{r_t}. \] In [J. Number Theory 51, 169-209 (1995; Zbl 0822.11026) and Proc. Japan Acad., Ser. A 70, 208-211 (1994; Zbl 0822.11027)] the author studied these power series. He proved that \(P(T,S;x_1,\dots,x_t)\) is a rational function of \(x_1, \dots, x_t\). He further gave an explicit form for its denominator in the case of a diagonal matrix \(T\). Now he gives more precise results if in addition \(S\) is a unimodular matrix of degree \(\geq 2 \cdot \deg T\). The proof requires a careful analysis of the proof of the main theorem in the first article by the author mentioned here.
scientific article; zbMATH DE number 897685