Metaplectic covers of G L n and the Gauss-Schering lemma

Richard Hill

Journal de théorie des nombres de Bordeaux (2001)

  • Volume: 13, Issue: 1, page 189-199
  • ISSN: 1246-7405

Abstract

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The Gauss-Schering Lemma is a classical formula for the Legendre symbol commonly used in elementary proofs of the quadratic reciprocity law. In this paper we show how the Gauss Schering Lemma may be generalized to give a formula for a 2 -cocycle corresponding to a higher metaplectic extension of GL n / k for any global field k . In the case that k has positive characteristic, our formula gives a complete construction of the metaplectic group and consequently an independent proof of the power reciprocity law for k .

How to cite

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Hill, Richard. "Metaplectic covers of $GL_n$ and the Gauss-Schering lemma." Journal de théorie des nombres de Bordeaux 13.1 (2001): 189-199. <http://eudml.org/doc/248702>.

@article{Hill2001,
abstract = {The Gauss-Schering Lemma is a classical formula for the Legendre symbol commonly used in elementary proofs of the quadratic reciprocity law. In this paper we show how the Gauss Schering Lemma may be generalized to give a formula for a $2$-cocycle corresponding to a higher metaplectic extension of GL$_n/k$ for any global field $k$. In the case that $k$ has positive characteristic, our formula gives a complete construction of the metaplectic group and consequently an independent proof of the power reciprocity law for $k$.},
author = {Hill, Richard},
journal = {Journal de théorie des nombres de Bordeaux},
language = {eng},
number = {1},
pages = {189-199},
publisher = {Université Bordeaux I},
title = {Metaplectic covers of $GL_n$ and the Gauss-Schering lemma},
url = {http://eudml.org/doc/248702},
volume = {13},
year = {2001},
}

TY - JOUR
AU - Hill, Richard
TI - Metaplectic covers of $GL_n$ and the Gauss-Schering lemma
JO - Journal de théorie des nombres de Bordeaux
PY - 2001
PB - Université Bordeaux I
VL - 13
IS - 1
SP - 189
EP - 199
AB - The Gauss-Schering Lemma is a classical formula for the Legendre symbol commonly used in elementary proofs of the quadratic reciprocity law. In this paper we show how the Gauss Schering Lemma may be generalized to give a formula for a $2$-cocycle corresponding to a higher metaplectic extension of GL$_n/k$ for any global field $k$. In the case that $k$ has positive characteristic, our formula gives a complete construction of the metaplectic group and consequently an independent proof of the power reciprocity law for $k$.
LA - eng
UR - http://eudml.org/doc/248702
ER -

References

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  5. [5] D. Kazhdan, S.J. Patterson, 'Metaplectic forms. Publ. Math. I.H.E.S.59 (1984), 35-142 + Corrigendum Ibid62 (1985), 419. Zbl0559.10026MR743816
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  7. [7] T. Kubota, Topological covering of SL(2) over a local field. J. Math. Soc. Japan19 (1967), 114-121. Zbl0166.29603MR204422
  8. [8] T. Kubota, On Automorphic Functions and the Reciprocity Law in a Number field. Lecture notes vol. 2, Kyoto Univ., 1969. Zbl0231.10017MR255490
  9. [9] T. Kubota, Geometry of numbers and class field theory. Japan J. Math.13 (1987), 235-275. Zbl0639.12004MR921585
  10. [10] T. Kubota, Foundations of Class Field Theory through properties of space figures, (in Japanese). Sugaka44 (1992), 1-12. Zbl0760.11034
  11. [11] H. Matsumoto, Sur les sous-groupes arithmétiques des groupes semi-simples déployés. Ann. scient. Ec. Norm. Sup. (4) 2 (1969), 1-62. Zbl0261.20025MR240214
  12. [12] J. Milnor, Introduction to Algebraic K-Theory. Annals of Mathematics Studies72 (1971). Zbl0237.18005MR349811
  13. [13] E. Schering, Verallgemeinerung des Gaußischen Criterium für den quadratischen Restcharakter einer Zahl in Bezug auf eine andere. Werke Band I, 285-286. JFM08.0093.04

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