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

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

MathSciNet review:
1328374

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Abstract | References | Similar Articles | Additional Information

Abstract: This paper investigates the local integrals

where represents the integers of a composition algebra over a non-archimedean local field and is a non-trivial character on the units in the ring of integers of extended to by setting . The local zeta function for the trivial character is known for all composition algebras . In this paper, we show in the quaternion case that for all non-trivial characters and then compute the local zeta function in the ramified quadratic extension case for equal to the quadratic character. In this latter case, for any character of order greater than .

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