Poincaré families along vertical lines

  • Roelof W. Bruggeman
Part of the Modern Birkhäuser Classics book series (MBC)


Theorem 10.2.1 gives the meromorphic continuation of Poincaré series in (φ, s). One may ask where the resulting Poincaré families are holomorphic. Propositions 10.2.12 and 10.2.14 give some results in this direction. The final Chapters 11 and 12 of Part I of this book discuss the singularities of Poincaré families at points (φ, s)with φV r and Re s ≥ 0. At which of these points are the Poincaré families not holomorphic? How bad are the singularities? It turns out that often there is a relation with the presence of eigenfunctions of the selfadjoint extension of the Casimir operator. The restriction to φV r is essential for these results: in general it is difficult even to define the extension A0 · exp (φ), l 0) for other φ. The restriction to points with Re s0 is less essential; the functional equation can be used to obtain assertions for other s.


Vertical Line Holomorphic Function Eisenstein Series Cusp Form Automorphic Form 
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Copyright information

© Springer Basel AG 1994

Authors and Affiliations

  • Roelof W. Bruggeman
    • 1
  1. 1.Mathematisch InstituutUniversiteit UtrechtUtrechtThe Netherland

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