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Poincaré families along vertical lines

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Abstract

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.

Keywords

  • Vertical Line
  • Holomorphic Function
  • Eisenstein Series
  • Cusp Form
  • Automorphic Form

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© 1994 Springer Basel AG

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Bruggeman, R.W. (1994). Poincaré families along vertical lines. In: Families of Automorphic Forms. Modern Birkhäuser Classics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0346-0336-2_11

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  • DOI: https://doi.org/10.1007/978-3-0346-0336-2_11

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0346-0335-5

  • Online ISBN: 978-3-0346-0336-2

  • eBook Packages: Springer Book Archive