Proceedings of the American Mathematical Society

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ISSN 1088-6826(e) ISSN 0002-9939(p)

The local zeta function for the non-trivial characters associated with the singular Jordan algebras

Author(s): Margaret M. Robinson
Journal: Proc. Amer. Math. Soc. 124 (1996), 2655-2660.
MSC (1991): Primary 11R52, 11F85
MathSciNet review: 1328374
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Abstract: This paper investigates the local integrals

$\begin{displaymath}Z_m(t,\chi )=\int _{H_m(O_C)} \chi ( \det (x)) | \det (x) |^s dx\end{displaymath}$

where $O_C$ represents the integers of a composition algebra over a non-archimedean local field $K$ and $\chi$ is a non-trivial character on the units in the ring of integers of $K$ extended to $K^*$ by setting $\chi (\pi )=1$ . The local zeta function for the trivial character is known for all composition algebras $C$ . In this paper, we show in the quaternion case that $Z(t, \chi )=0$ for all non-trivial characters and then compute the local zeta function in the ramified quadratic extension case for $\chi$ equal to the quadratic character. In this latter case, $Z(t, \chi )=0$ for any character of order greater than $2$ .

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