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2018 Indefinite Eisenstein Lattices: A Modern Ball-Rendevous with Poincarè, Picard, Hecke, Shimura, Mumford, Deligne and Hirzebruch
Rolf-Peter Holzapfel
J. Geom. Symmetry Phys. 47(none): 1-40 (2018). DOI: 10.7546/jgsp-47-2018-1-40

Abstract

In [21] we have counted indefinite metrics (two-dimensional, integrally defined, over Gauss numbers) with a fixed norm (discriminant). We would like to call them also indefinite class numbers. In this article we change from Gauss to Eisenstein numbers. We have to work on the complex two-dimensional unit ball, an Eisenstein lattice on it and the quotient surface. It turns out that the compactified quotient is the complex plane ${\mathbb P}^2$. In the first part we present a new proof of this fact. In the second part we construct explicitly a Heegner series with the help of Legendre-symbol coefficients. They can be interpreted as “indefinite class numbers” we look for. Geometrically they appear also as number of plane curves with (normed) Eisenstein disc uniformization.

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Rolf-Peter Holzapfel. "Indefinite Eisenstein Lattices: A Modern Ball-Rendevous with Poincarè, Picard, Hecke, Shimura, Mumford, Deligne and Hirzebruch." J. Geom. Symmetry Phys. 47 1 - 40, 2018. https://doi.org/10.7546/jgsp-47-2018-1-40

Information

Published: 2018
First available in Project Euclid: 10 May 2018

zbMATH: 06944464
MathSciNet: MR3822058
Digital Object Identifier: 10.7546/jgsp-47-2018-1-40

Rights: Copyright © 2018 Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences

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