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Introduction

  • Gérard Lion
  • Michèle Vergne
Part of the Progress in Mathematics book series (PM, volume 6)

Abstract

We have constructed in Part 1 a projective representation R of the symplectic group. As discovered by A. Weil, this representation plays a central role in the transformation properties of the classical Jacobi θ-serie \(\theta (z) = \sum\limits_{n} {{{e}^{{i\pi {{n}^{2}}z}}}}\) and higher dimensional θ-series, when interpreted as suitable coefficients of this representation R. We indicate now the nature of the relation between R and theta series:
  • Let D be the Siegel upper half-plane, i.e.

  • D = {Z. (n × n) complex symmetric matrices, such that Im Z ≫ 0}

Keywords

Modular Form Congruence Subgroup High Weight Vector Theta Series Lagrangian Subspace 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1980

Authors and Affiliations

  • Gérard Lion
    • 1
  • Michèle Vergne
    • 2
  1. 1.X U.E.R. de Sciences EconomiquesUniversité de ParisNanterreFrance
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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