Metaplectic covers of 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 -cocycle corresponding to a higher metaplectic extension of GL for any global field . In the case that has positive characteristic, our formula gives a complete construction of the metaplectic group and consequently an independent proof of the power reciprocity law for .

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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  10. [10] T. Kubota, Foundations of Class Field Theory through properties of space figures, (in Japanese). Sugaka44 (1992), 1-12. Zbl0760.11034
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