A generalized Igusa local zeta function and local densities of quadratic forms
The paper proves that a certain formal power series attached to the local densities of quadratic forms can be realized as a coefficient of the Laurent expansion of a generalization of Igusa's local zeta function. Let \(A\) and \(B\) be non-degenerate symmetric matrices of degrees \(m\) and \(n\), respectively with entries of \(p\)-adic integers \(\mathbb{Z}_ p\). We define a local density attached to the symmetric matrices \(A\), \(B\) by \[ \alpha_ p(B,A)= \lim_{e\to\infty} p^{(-mn+ (n(n+1)/2))e} \# \{\overline{X}\in M_{mn} (\mathbb{Z}_ p)/p^ e M_{mn}(\mathbb{Z}_ p);\;A[X]\equiv B\bmod q\cdot p^ e\}, \] (\(A[X]:={}^ tXAX\)). Then we define a formal power series \(P(B,A; x_ 1,\dots,x_ n)\) by \[ P(B,A; x_ 1,\dots, x_ n)+ \sum_{r_ 1,\dots,r_ n=0}^ \infty \alpha_ p(B [\text{diag} (p^{r_ 1},\dots, p^{r_ n})], A)x_ 1^{r_ 1}\cdots x_ n^{r_ n}. \] The main purpose of the present paper is to show that the series \(P(B,A; x_ 1,\dots, x_ n)\) appears as a coefficient of the Laurent expansion of a generalized Igusa's local zeta function.
generalization of Igusa's local zeta function
scientific article; zbMATH DE number 495479