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

Robinson, Margaret

doi:10.1090/S0002-9939-96-03420-X

The local zeta function for the non-trivial characters associated with the singular Jordan algebras
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by Margaret M. Robinson

Proc. Amer. Math. Soc. 124 (1996), 2655-2660

DOI: https://doi.org/10.1090/S0002-9939-96-03420-X

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Abstract:

This paper investigates the local integrals \[ Z_m(t,\chi )=\int _{H_m(O_C)} \chi ( \det (x)) | \det (x) |^s dx\] 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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