A certain formal power series of several variables attached to local densities of quadratic forms. I
Let \(p\) be an odd prime number. Let \(S\in M_ s (\mathbb{Z}_ p)\), \(T\in M_ t (\mathbb{Z}_ p)\) \((s\geq t\geq 1)\) be symmetric non-singular matrices with entries in the \(p\)-adic integers \(\mathbb{Z}_ p\). Set \[ A_ e (T,S)= \{\overline {X}\in M_{st} (\mathbb{Z}_ p)/ p^ e M_{st} (\mathbb{Z}_ p);\;{}^ t XSX\equiv T\bmod p^ e\}. \] Then a local density \(\alpha_ p (T,S)\) is defined by \[ \alpha_ p (T,S)= \lim_{e\to \infty} p^{(-2 st+t (t+1))\cdot e/2} A_ e (T,S). \] With it we attach the following power series \[ P(T,S; x_ 1,\dots, x_ t)= \sum_{r_ 1,\dots, r_ t=0}^ \infty \alpha_ p \bigl( \langle p^{r_ 1}, \dots, p^{r_ t} \rangle\cdot T\cdot \langle p^{r_ 1}, \dots, p^{r_ t} \rangle, S\bigr) \cdot x_ 1^{r_ 1} \cdots x_ t^{r_ t}, \] where \(\langle a_ 1, \dots, a_ t\rangle\in M_ t (\mathbb{Z}_ p)\) is the diagonal matrix with diagonal entries \(a_ 1, \dots, a_ t\). The main result of the paper under review states that for a diagonal matrix \(T\) the power series \(P(T,S; x_ 1,\dots, x_ t)\) is a rational function in \(x_ 1, \dots, x_ t\) and its denominator is determined. It depends only on \(s\), \(t\) and \(p\). The proof of the main theorem uses a certain induction formula for local densities with congruence conditions.
scientific article; zbMATH DE number 751310