Kronecker limit formulas for parabolic, hyperbolic and elliptic Eisenstein series via Borcherds products
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Date
2021-08Type
- Journal Article
ETH Bibliography
yes
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Abstract
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 Γ0(N) 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. Show more
Permanent link
https://doi.org/10.3929/ethz-b-000473525Publication status
publishedExternal links
Journal / series
Journal of Number TheoryVolume
Pages / Article No.
Publisher
ElsevierOrganisational unit
08799 - Imamoglu, Oezlem (Tit.-Prof.)
08799 - Imamoglu, Oezlem (Tit.-Prof.)
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ETH Bibliography
yes
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