Mathematische Zeitschrift

, Volume 275, Issue 1–2, pp 509–527 | Cite as

Weil representations associated with finite quadratic modules

Article

Abstract

To any finite quadratic module, that is, a finite abelian group together with a non-degenerate quadratic form, it is possible to associate a representation of \(\mathrm{Mp}_{2}(\mathbb Z )\), the metaplectic cover of the modular group. This representation is usually referred to as a Weil representation and our main result is a general explicit formula for its matrix coefficients. This result completes earlier work by Scheithauer in the case when the representation factors through \(\mathrm{SL}_{2}(\mathbb Z )\). Furthermore, our formula is given in a such a way that it is easy to implement efficiently on a computer.

Keywords

Weil representation Metaplectic group Finite quadratic module 

Mathematics Subject Classification (2000)

11F27 20C25 

Notes

Acknowledgments

I would like to thank Nils Scheithauer for clarifying details of [15], Nils-Peter Skoruppa for sharing thoughts about Weil representations in general, and the manuscript [19], in particular. I would also like to thank Stephan Ehlen for his extensive assistance with proof reading the manuscript and in simplifying Theorem 4.1.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Mathematical SciencesDurham UniversityDurhamUK

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