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 journalArticlepeer-review

10 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 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.

Original languageAmerican English
Pages (from-to)61-89
Number of pages29
JournalAnnales Mathematiques du Quebec
Volume39
Issue number1
DOIs
StatePublished - 1 Jun 2015
Externally publishedYes

Bibliographical note

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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