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 φ ∈ V
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 φ ∈ V
is essential for these results: in general it is difficult even to define the extension A(χ0 · exp (φ), l
0) for other φ. The restriction to points with Re s ≥ 0 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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