Rationality of a certain formal power series attached to local densities of quadratic forms
Let \(p\) be a prime number. Let \(A\) and \(B\) be non-degenerate symmetric matrices of degree \(m\) and \(n\) with \(m\geq n\geq 1\), respectively, with entries in the ring \(\mathbb{Z}_ p\) of \(p\)-adic integers. We define a local density \(\alpha_ p (B, A)\) by \[ \alpha_ p (B,A)= \lim_{e\to \infty} p^{(-mn+ \langle n\rangle )e} \#\{ \overline {X}\in M_{mn} (\mathbb{Z}_ p)/ p^ e M_{mn} (\mathbb{Z}_ p);\;{}^ t XAX\equiv B\bmod p^ e\}, \] where \(\langle n\rangle= n(n+ 1)/2\) and \(M_{mn} (\mathbb{Z}_ p)\) denotes the set of \((m,n)\)-matrices with entries in \(\mathbb{Z}_ p\). For the matrices \(A\) and \(B\) stated above, and an \(l\)-tuple \(\vec n:= (n_ 1,\dots, n_ l)\) of positive integers \(n_ 1,\dots, n_ l\) such that \(n_ 1+ \cdots+ n_ l=n\), we define a formal power series \[ P(B,A, \vec n; x_ 1,\dots, x_ l)= \sum_{r_ 1,\dots, r_ l=0}^ \infty \alpha_ p \bigl(B [\text{diag} (p^{r_ 1} E_{n_ 1},\dots, p^{r_ l} E_{n_ l}) ], A\bigr) x_ 1^{r_ 1}\dots x_ l^{r_ l}, \] where for an \((m,n)\)-matrix \(C\) and an \((m,m)\)-matrix \(D\) we write \(D[ C]:= t{}^ t CDC\), and for square matrices \(A_ 1,\dots, A_ r\). In this paper, the author proves the rationality of the function for arbitrary symmetric matrices in his original way.
scientific article; zbMATH DE number 599653
scientific article