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Annales mathématiques du Québec

, Volume 39, Issue 1, pp 61–89 | Cite as

A p-adic approach to the Weil representation of discriminant forms arising from even lattices

  • Shaul Zemel
Article

Abstract

Suppose that M is an even lattice with dual \(M^{*}\) and level N. Then the group \(Mp_{2}(\mathbb {Z})\), which is the unique non-trivial double cover of \(SL_{2}(\mathbb {Z})\), admits a representation \(\rho _{M}\), called the Weil representation, on the space \(\mathbb {C}[M^{*}/M]\). The main aim of this paper is to show how the formulae for the \(\rho _{M}\)-action of a general element of \(Mp_{2}(\mathbb {Z})\) can be obtained by a direct evaluation which does not depend on “external objects” such as theta functions. We decompose the Weil representation \(\rho _{M}\) into p-parts, in which each p-part can be seen as subspace of the Schwartz functions on the p-adic vector space \(M_{\mathbb {Q}_{p}}\). Then we consider the Weil representation of \(Mp_{2}(\mathbb {Q}_{p})\) on the space of Schwartz functions on \(M_{\mathbb {Q}_{p}}\), and see that restricting to \(Mp_{2}(\mathbb {Z})\) just gives the p-part of \(\rho _{M}\) again. The operators attained by the Weil representation are not always those appearing in the formulae from 1964, but are rather their multiples by certain roots of unity. For this, one has to find which pair of elements, lying over a matrix in \(SL_{2}(\mathbb {Q}_{p})\), belong to the metaplectic double cover. Some other properties are also investigated.

Keywords

Weil Representations Discriminant Forms Lattices Representations of p-adic Groups 

Résumé

Soit M un treillis pair de dual \(M^{*}\) et de niveau N. Alors le groupe \(Mp_{2}(\mathbb {Z})\), qui est l’unique revêtement non-trivial double de \(SL_{2}(\mathbb {Z})\), admet une représentation \(\rho _{M}\), dite la représentation de Weil, sur l’espace \(\mathbb {C}[M^{*}/M]\). Le but premier de cet article est de montrer que les formules pour l’action \(\rho _{M}\) d’un élément quelconque de \(Mp_{2}(\mathbb {Z})\) peuvent être obtenues via une évaluation directe qui ne dépend pas “ d’objets externes”   tels les fonctions thêta. Nous décomposons la représentation \(\rho _{M}\) de Weil en p-parties, chacune de ces p-parties pouvant être vue comme un sous-espace des fonctions de Schwartz sur l’espace vectoriel p-adique \(M_{\mathbb {Q}_{p}}\). Nous considérons alors la représentation de Weil de \(Mp_{2}(\mathbb {Q}_{p})\) sur l’espace des fonctions de Schwartz sur \(M_{\mathbb {Q}_{p}}\), et constatons que nous restreindre à \(Mp_{2}(\mathbb {Z})\) redonne précisément la p-partie de \(\rho _{M}\). Les opérateurs touchés par la représentation de Weil ne sont pas toujours ceux qui apparaissent dans les formules de 1964, mais sont plutôt leurs multiples par certaines racines de l’unité. Concrètement, il faut trouver quelle paire d’éléments, provenant d’une matrice de \(SL_{2}(\mathbb {Q}_{p})\), appartiennent au revêtement double métaplectique. Nous investiguons aussi d’autres propriétés.

Mathematics Subject Classification

11F27 22D10 

Notes

Acknowledgments

The initial stage of this research has been carried out as part of my Ph.D. thesis work at the Hebrew University of Jerusalem, Israel. The final stage of this work was supported by the Minerva Fellowship (Max-Planck-Gesellschaft). I am deeply indebted to E. Lapid for his proposal to look for a p-adic proof to the factoring of the Weil representation through (a double cover of) \(SL_{2}(\mathbb {Z}/N\mathbb {Z})\), which initiated my work on this paper (and the corresponding part in my Ph.D. thesis). I would also like to thank my Ph.D. advisor R. Livné and H. M. Farkas for their help. I also thank J. Bruinier, N. Scheithauer and F. Strömberg for fruitful discussions while writing this paper and for referring me to [11]. Special thanks also to T. Yang, for referring me to [18]. I am also grateful to the two referees, whose remarks have greatly contributed to the improvement of the presentation of the results of this paper.

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Copyright information

© Fondation Carl-Herz and Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Fachbereich Mathematik, AG 5Technische Universität DarmstadtDarmstadtGermany

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