Proceedings of the American Mathematical Society

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

where

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 $2$

1.: L.E. Dickson, Linear Groups with an exposition of the Galois field theory", B.G. Teubner, Leibzig, 1901.
2.: H.-D. Ebbinghaus et al., Numbers, Springer-Verlag, New York (1991). MR 91h:00005
3.: C.F. Gauss, Hundert Theoreme uber die neuen Transscendenten, Werke III, (1876), 461-469.
4.: J. Igusa, On the arithmetic of Pfaffians, Nagoya Math. J. 47 (1972), 169-198. MR 51:12798
5.: J. Igusa, Geometry of absolutely admissible representations, Number Theory, Algebraic Geometry and Commutative Algebra, Kinokuniya, Tokyo (1973), 373-452. MR 51:3319
6.: J. Igusa, Complex powers and asymptotic expansions. I, J. Reine Angew. Math. 268/269 (1974), 110-130; II, ibid 278/279 (1975), 307-321. MR 53:8018; MR 50:254
7.: J. Igusa, Forms of Higher Degree, Tata Inst. Lect. Notes 59, Springer-Verlag (1978). MR 80m:10020
8.: J. Igusa, Some results on p-adic complex powers, Amer. J. Math. 106 (1984), 1013-1032. MR 86f:11046
9.: J. Igusa, On functional equations of complex powers, Invent. Math. 85 (1986), 1-29. MR 87j:11134
10.: N. Jacobson, Structure and representation of Jordan algebras, Amer. Math. Soc. Colloq. Publ. 39 (1969). MR 40:4330
11.: M. Robinson, The Igusa local zeta function associated with the singular cases of the determinant and the Pfaffian, J. Number Theory 57 (1996), 385--408.

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