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

Shaul Zemel

doi:10.1007/s40316-015-0034-6

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

Shaul Zemel^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

Abstract

Suppose that M is an even lattice with dual M^∗ and level N. Then the group Mp₂(Z) , which is the unique non-trivial double cover of SL₂(Z) , admits a representation ρ_M, called the Weil representation, on the space C[ M^∗/ M]. The main aim of this paper is to show how the formulae for the ρ_M-action of a general element of Mp₂(Z) can be obtained by a direct evaluation which does not depend on “external objects” such as theta functions. We decompose the Weil representation ρ_M into p-parts, in which each p-part can be seen as subspace of the Schwartz functions on the p-adic vector space MQp. Then we consider the Weil representation of Mp₂(Q_p) on the space of Schwartz functions on MQp, and see that restricting to Mp₂(Z) just gives the p-part of ρ_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 SL₂(Q_p) , belong to the metaplectic double cover. Some other properties are also investigated.

Original language	American English
Pages (from-to)	61-89
Number of pages	29
Journal	Annales Mathematiques du Quebec
Volume	39
Issue number	1
DOIs	https://doi.org/10.1007/s40316-015-0034-6
State	Published - 1 Jun 2015
Externally published	Yes

Bibliographical note

Funding Information:
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) , 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 []. Special thanks also to T. Yang, for referring me to []. 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.

Publisher Copyright:
© 2015, Fondation Carl-Herz and Springer International Publishing Switzerland.

Keywords

Discriminant Forms
Lattices
Representations of p-adic Groups
Weil Representations

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10.1007/s40316-015-0034-6

Cite this

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

abstract = "Suppose that M is an even lattice with dual M∗ and level N. Then the group Mp2(Z) , which is the unique non-trivial double cover of SL2(Z) , admits a representation ρM, called the Weil representation, on the space C[ M∗/ M]. The main aim of this paper is to show how the formulae for the ρM-action of a general element of Mp2(Z) can be obtained by a direct evaluation which does not depend on “external objects” such as theta functions. We decompose the Weil representation ρM into p-parts, in which each p-part can be seen as subspace of the Schwartz functions on the p-adic vector space MQp. Then we consider the Weil representation of Mp2(Qp) on the space of Schwartz functions on MQp, and see that restricting to Mp2(Z) just gives the p-part of ρM 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 SL2(Qp) , belong to the metaplectic double cover. Some other properties are also investigated.",

keywords = "Discriminant Forms, Lattices, Representations of p-adic Groups, Weil Representations",

author = "Shaul Zemel",

note = "Funding Information: 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) , 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. Livn{\'e} and H. M. Farkas for their help. I also thank J. Bruinier, N. Scheithauer and F. Str{\"o}mberg for fruitful discussions while writing this paper and for referring me to []. Special thanks also to T. Yang, for referring me to []. 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. Publisher Copyright: {\textcopyright} 2015, Fondation Carl-Herz and Springer International Publishing Switzerland.",

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doi = "10.1007/s40316-015-0034-6",

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N2 - Suppose that M is an even lattice with dual M∗ and level N. Then the group Mp2(Z) , which is the unique non-trivial double cover of SL2(Z) , admits a representation ρM, called the Weil representation, on the space C[ M∗/ M]. The main aim of this paper is to show how the formulae for the ρM-action of a general element of Mp2(Z) can be obtained by a direct evaluation which does not depend on “external objects” such as theta functions. We decompose the Weil representation ρM into p-parts, in which each p-part can be seen as subspace of the Schwartz functions on the p-adic vector space MQp. Then we consider the Weil representation of Mp2(Qp) on the space of Schwartz functions on MQp, and see that restricting to Mp2(Z) just gives the p-part of ρM 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 SL2(Qp) , belong to the metaplectic double cover. Some other properties are also investigated.

AB - Suppose that M is an even lattice with dual M∗ and level N. Then the group Mp2(Z) , which is the unique non-trivial double cover of SL2(Z) , admits a representation ρM, called the Weil representation, on the space C[ M∗/ M]. The main aim of this paper is to show how the formulae for the ρM-action of a general element of Mp2(Z) can be obtained by a direct evaluation which does not depend on “external objects” such as theta functions. We decompose the Weil representation ρM into p-parts, in which each p-part can be seen as subspace of the Schwartz functions on the p-adic vector space MQp. Then we consider the Weil representation of Mp2(Qp) on the space of Schwartz functions on MQp, and see that restricting to Mp2(Z) just gives the p-part of ρM 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 SL2(Qp) , belong to the metaplectic double cover. Some other properties are also investigated.

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

Abstract

Bibliographical note

Keywords

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