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On Rationality of Modular Representations

On Rationality of Modular Representations C. T. C. WALL This note falls naturally into two parts. In the first, we determine the field of definition of a modular representation (real or virtual) in terms of its Brauer character. Then we show how to obtain the complete list of fields of definition of modular irreducibles from the character table. We adopt throughout the notation of Serre [5], but need some further con- ventions. For p a rational prime, q = p write F for the field with q elements, K for q q st the field of (q- l) roots of unity over the field K of p-adic numbers: thus we can identify the ring A of integers in K with that of Witt vectors over F . K is an q q q qS unramified extension of K , and there are natural isomorphisms of Galois groups Fq*IFq*-A /A -+ K /K . It will be convenient to fix embeddings K -+C: we qS q qS q q can indeed identify C with the algebraic closure of K . We use Quillen's [4; see also second edition of 5] modification of the Brauer character, defining its value on any x to be that on http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

On Rationality of Modular Representations

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/5.2.199
Publisher site
See Article on Publisher Site

Abstract

C. T. C. WALL This note falls naturally into two parts. In the first, we determine the field of definition of a modular representation (real or virtual) in terms of its Brauer character. Then we show how to obtain the complete list of fields of definition of modular irreducibles from the character table. We adopt throughout the notation of Serre [5], but need some further con- ventions. For p a rational prime, q = p write F for the field with q elements, K for q q st the field of (q- l) roots of unity over the field K of p-adic numbers: thus we can identify the ring A of integers in K with that of Witt vectors over F . K is an q q q qS unramified extension of K , and there are natural isomorphisms of Galois groups Fq*IFq*-A /A -+ K /K . It will be convenient to fix embeddings K -+C: we qS q qS q q can indeed identify C with the algebraic closure of K . We use Quillen's [4; see also second edition of 5] modification of the Brauer character, defining its value on any x to be that on

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Jul 1, 1973

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