A remark on Kitaoka's power series attached to local densities (Q2783757)

Let \(k\) be a \(p\)-adic number field of odd residual characteristics, \(O\) the ring of integers in \(k\), \(\pi\) a prime element of \(k\) and \(q= \#(O|p)\) with \(p= \pi O\). Let \(X_m(O)\) be the set of nondegenerate symmetric matrices of entries in \(O\). For \(A\in X_m(O)\) and \(B\in X_n(O)\) \((m\geq n)\), let \(P(B,A,X)\) be Kitaoka's power series. The formal power series \(P(B,A,X)\) is known to be a rational function of \(X\) for every \(A\) and \(B\). Several authors have investigated its denominator. Y. Hironaka (1988) has shown that NEWLINE\[NEWLINEP(B,A,X)\cdot(1-X)\cdot \prod_{j=0}^{n-1} (1-q^{(n+j-m-1)(n-j)} X^2)NEWLINE\]NEWLINE is a polynomial in \(X\). The result is best possible when \(A\) is the unit matrix of odd size and \(n=2\). In this paper a finer result has been obtained using the same method when \(A\) is of even size. NEWLINENEWLINENEWLINEThe main result is that if \(m\) is even then NEWLINE\[NEWLINEP(B,A,X)\quad\cdot \prod_{j=0}^n (1- (\varepsilon_A q^{1/2^{(n+j-m+1)}})^{(n-j)} X)NEWLINE\]NEWLINE is a polynomial in \(X\) where \(\varepsilon_A= \Delta^{m+2e(A)} (\frac{\varepsilon_1, \dots, \varepsilon_m}{p})\), NEWLINE\[NEWLINE\Delta= \begin{cases} (-1)^{f+1} &\text{ if } p\equiv 1\pmod 4,\\ -(-i)^f &\text{ if } p\equiv 3\pmod 4,\end{cases} \qquad q=p^f,NEWLINE\]NEWLINE \(e(A)= \#\{i\mid 1\leq i\leq m\), \(2|a_i\}\), \(a_i\geq 0\), \(\varepsilon_i\in O^\times\) such that \(A\) is equivalent to \(\text{diag}( \varepsilon_1 \lambda^{a_1},\dots, \varepsilon_m \lambda^{a_m})\).