Gauss Sums and p-adic Division Algebras
- PDF / 7,327,844 Bytes
- 199 Pages / 461 x 684 pts Page_size
- 28 Downloads / 271 Views
987
Colin J. Bushnell Albrecht Frohlich
Gauss Sums and p-adic Division Algebras
Springer-Verlag Berlin Heidelberg New York Tokyo 1983
Authors
Colin J. Bushnell Department of Mathematics King's College, Strand London WC2R 2LS, England
Albrecht Frohlich Imperial College of Science and Technology Department of Mathematics, Huxley Building Queen's Gate, London SW7 2BZ, England
AMS Subject Classifications (1980): 12B27, 12B37,22E50 ISBN 3-540-12290-7 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-12290-7 Springer-Verlag New York Heidelberg Berlin Tokyo This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1983 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
§l
Arithmetic of local division algebras
§2
Introduction to Gauss sums
§3
Functional equation
§4
One-dimensional representations
§S
The basic correspondence
§6
The basic inductive step
§7
The general induction process
§8
Representations of certain group extensions
§9
Trace calculations
§lO
Induction constants for Galois Gauss sums
§ll
Synthesis of results
§12 Modified correspondences
I N T ROD U C T ION
Let F be a finite field extension of the p-adic rational field let D be a central F-division algebra of finite dimension n 2 • have two principal aims. congruence
and
These notes
The first of these is to develop the theory of a
T(n) attached to an irreducible admissible representX
ation n of the multiplicative group D of D.
For the second, we assume
* The final version of this set of notes was while the authors were visiting the University of Illinois at in the Fall of 1981 as participants in a Special Year of Algebra aIld Algebraic Number Theory organised by I. Reiner. The first was on sabbatical leave from King's College London and was partially supported by the NSF. The second was the G.A. Miller Visiting Professor. Both would like to thank the University of Illinois for its hospitality during this period.
IV
that the index n of the division algebra D is not divisible by the residual characteristic p of F.
In this situation, Corwin and Howe [3] have
constructed a bijective correspondence between these representations
and
continuous irreducible representations a, of degree dividing n, of the Weil group W of F. F
Koch and Zink then took up the subject again, and gave a
more complete account in [10].
Our second aim is then to derive a precise
comparison between the constant equation attached to
in the Godement-Jacquet functional
and the Langlands local constant W(a) of the
corresponding representation a of
WF• We shall see that the root number
Data Loading...
