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.
Mathematics Subject Classification (2010): 11F03, 11F30, 11M36, 30F35
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Volume I, McGraw-Hill, 1965.
A.F. Beardon, The Geometry of Discrete Groups, Graduate Texts in Mathematics, Springer-Verlag, 1995.
Y. Colin de Verdière, Une nouvelle démonstration du prolongement méromorphe des séries d’Eisenstein, C. R. Acad. Sci. Paris Sér. I Math. 293 (1981), 361–363.
I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, 1980.
H. Huber, Über eine neue Klasse automorpher Funktionen und ein Gitterpunktproblem in der hyperbolischen Ebene. I, Comment. Math. Helv. 30 (1956), 20–62.
H. Iwaniec, Spectral Methods of Automorphic Forms, Graduate Studies in Mathematics, Vol. 53, Amer. Math. Soc., 2002.
J. Jorgenson, J. Kramer, Bounding the sup-norm for automorphic forms, Geom. Funct. Anal. 14 (2004), 1267–1277.
J. Jorgenson, J. Kramer, Canonical metrics, hyperbolic metrics and Eisenstein series for \({\mathrm{PSL}}_{2}(\mathbb{R})\), unpublished preprint.
J. Jorgenson, J. Kramer, Sup-norm bounds for automorphic forms and Eisenstein series, in Arithmetic Geometry and Automorphic Forms, J. Cogdell et al. (eds.), ALM 19, 407–444, Higher Education Press and International Press, Beijing-Boston, 2011.
J. Jorgenson, C. O’Sullivan, Convolution Dirichlet series and a Kronecker limit formula for second-order Eisenstein series, Nagoya Math. J. 179 (2005), 47–102.
S.S. Kudla, J.J. Millson, Harmonic Differentials and Closed Geodesics on a Riemann Surface, Invent. Math. 54 (1979), 193–211.
H. Neunhöffer, Über die analytische Fortsetzung von Poincaréreihen, S.-B. Heidelberger Akad. Wiss. Math.-Natur. Kl. (1973), 33–90.
A.-M. v. Pippich, The arithmetic of elliptic Eisenstein series, Ph.D. thesis, Humboldt-Universität zu Berlin (2010).
P. Sarnak, Estimates for Rankin–Selberg L-functions and quantum unique ergodicity, J. Funct. Anal. 184 (2001), 419–453.
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Additional information
To the memory of Serge Lang
Rights and permissions
Copyright information
© 2012 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Kramer, J., von Pippich, AM. (2012). Elliptic Eisenstein series for \({PSL}_{2}(\mathbb{Z})\) . In: Goldfeld, D., Jorgenson, J., Jones, P., Ramakrishnan, D., Ribet, K., Tate, J. (eds) Number Theory, Analysis and Geometry. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-1260-1_19
Download citation
DOI: https://doi.org/10.1007/978-1-4614-1260-1_19
Published:
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4614-1259-5
Online ISBN: 978-1-4614-1260-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)