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Elliptic Eisenstein series for \({PSL}_{2}(\mathbb{Z})\)

  • Jürg KramerEmail author
  • Anna-Maria von Pippich
Chapter

Abstract

Let \(\Gamma\subset \mathrm{{ PSL}}_{2}(\mathbb{R})\)be a Fuchsian subgroup of the first kind acting by fractional linear transformations on the upper half-plane \(\mathbb{H}\), and let \(\Gamma \setminus \mathbb{H}\)be the associated finite volume hyperbolic Riemann surface. Associated to any cusp of \(\Gamma \setminus \mathbb{H}\), there is the classically studied non-holomorphic (parabolic) Eisenstein series. In [11], Kudla and Millson studied non-holomorphic (hyperbolic) Eisenstein series associated to any closed geodesic on \(\Gamma \setminus \mathbb{H}\). Finally, in [9], Jorgenson and the first named author introduced so-called elliptic Eisenstein series associated to any elliptic fixed point of \(\Gamma \setminus \mathbb{H}\). In this article, we study elliptic Eisenstein series for the full modular group \(\mathrm{{PSL}}_{2}(\mathbb{Z})\). We explicitly compute the Fourier expansion of the elliptic Eisenstein series and derive from this its meromorphic continuation.

Key words

Eisenstein series automorphic functions Fourier coefficients meromorphic continuation 

Notes

Acknowledgements

We would like to express our thanks to J. Jorgenson for his valuable advice in the course of the write-up of this article. Furthermore, we would like to thank J. Funke, O. Imamoglu, and U. Kühn for helpful discussions. Both authors acknowledge support from the DFG Graduate School Berlin Mathematical Schooland the DFG Research Training Group Arithmetic and Geometry.

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany

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