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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Author: 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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Additional Information

Margaret M. Robinson
Affiliation: Department of Mathematics, Statistics, and Computer Science, Mount Holyoke College, South Hadley, Massachusetts 01075
Email: robinson@mhc.mtholyoke.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03420-X
PII: S 0002-9939(96)03420-X
Received by editor(s): July 5, 1994
Received by editor(s) in revised form: March 27, 1995
Communicated by: William W. Adams
Article copyright: © Copyright 1996 American Mathematical Society