Publikation:

Kronecker limit formulas for parabolic, hyperbolic and elliptic Eisenstein series via Borcherds products

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2021

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Schwagenscheidt, Markus
Völz, Fabian

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Published

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Journal of Number Theory. Elsevier. 2021, 225, S. 18-58. ISSN 0022-314X. eISSN 1096-1658. Verfügbar unter: doi: 10.1016/j.jnt.2021.01.010

Zusammenfassung

The classical Kronecker limit formula describes the constant term in the Laurent expansion at the first order pole of the non-holomorphic Eisenstein series associated to the cusp at infinity of the modular group. Recently, the meromorphic continuation and Kronecker limit type formulas were investigated for non-holomorphic Eisenstein series associated to hyperbolic and elliptic elements of a Fuchsian group of the first kind by Jorgenson, Kramer and the first named author. In the present work, we realize averaged versions of all three types of Eisenstein series for as regularized theta lifts of a single type of Poincaré series, due to Selberg. Using this realization and properties of the Poincaré series we derive the meromorphic continuation and Kronecker limit formulas for the above Eisenstein series. The corresponding Kronecker limit functions are then given by the logarithm of the absolute value of the Borcherds product associated to a special value of the underlying Poincaré series.

Zusammenfassung in einer weiteren Sprache

Fachgebiet (DDC)
510 Mathematik

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CM-values, Harmonic Maass forms, Meromorphic modular forms, Regularized Petersson inner products, Theta lifts, Weakly holomorphic modular forms

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ISO 690VON PIPPICH, Anna-Maria, Markus SCHWAGENSCHEIDT, Fabian VÖLZ, 2021. Kronecker limit formulas for parabolic, hyperbolic and elliptic Eisenstein series via Borcherds products. In: Journal of Number Theory. Elsevier. 2021, 225, S. 18-58. ISSN 0022-314X. eISSN 1096-1658. Verfügbar unter: doi: 10.1016/j.jnt.2021.01.010
BibTex
@article{vonPippich2021Krone-58556,
  year={2021},
  doi={10.1016/j.jnt.2021.01.010},
  title={Kronecker limit formulas for parabolic, hyperbolic and elliptic Eisenstein series via Borcherds products},
  volume={225},
  issn={0022-314X},
  journal={Journal of Number Theory},
  pages={18--58},
  author={von Pippich, Anna-Maria and Schwagenscheidt, Markus and Völz, Fabian}
}
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