Local densities of $p$-adic quaternion hermitian forms

Abstract

We consider local densities for $p$-adic quaternion hermitian forms (hermitian forms over a division quaternion algebra over a ${\mathfrak p}$-adic field $k$). The author has studied such forms in connection with spherical functions on the space of quaternion hermitian forms in the previous paper. Obtaining good explicit formulas of local densities is an interesting but difficult problem in general, and without explicit formulas we may study some general theory on local densities. In this paper, we give two results on the relations among local densities. First, regarding the local density $\mu(B, A)$ (of representations of $B$ by $A$) as functions of $B$, we study their linear independence when $A$ varies by scaling hyperbolic planes. Then, following Kitaoka for symmetric forms, we introduce a formal power series $P(B, A; X) = \sum_{r \geq 0}\, \mu(\pi^rB, A)X^r$, where $\pi$ is a prime element of $k$. We give an explicit polynomial $f(X)$ for which $f(X)\cdot P(B, A; X)$ becomes a polynomial of $X$. Both results are based on Gauss sum expressions of local densities and Fourier transforms of functions on the space of quaternion hermitian forms. Finally, as an appendix, we list the previous results on symmetric forms and hermitian forms for comparison.