JOURNAL OF NUMBER THEORY | 卷:225 |
Kronecker limit formulas for parabolic, hyperbolic and elliptic Eisenstein series via Borcherds products | |
Article | |
von Pippich, Anna-Maria1  Schwagenscheidt, Markus1  Voelz, Fabian1  | |
[1] Tech Univ Darmstadt, Fachbereich Math, Schlossgartenstr 7, D-64289 Darmstadt, Germany | |
关键词: CM-values; Harmonic Maass forms; Meromorphic modular forms; Theta lifts; Weakly holomorphic modular forms; Regularized Petersson inner products; | |
DOI : 10.1016/j.jnt.2021.01.010 | |
来源: Elsevier | |
【 摘 要 】
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 Gamma 0(N) as regularized theta lifts of a single type of Poincare series, due to Selberg. Using this realization and properties of the Poincare 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 Poincare series. (c) 2021 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
【 授权许可】
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