Humboldt-Universität zu Berlin - Mathematisch-Naturwissenschaftliche Fakultät - Institut für Mathematik

Rolf-Peter Holzapfel

Indefinit Eisenstein Lattices: A Modern Ball-Rendevous with Poincaré, Picard, Hecke, Shimura, Mumford, Deligne, Hirzebruch.

Abstract: In [Ho02] we counted indefinit metrics (two-dimensional, integrally defined, over Gauß numbers) with a fixed norm (discriminant). We would like to call them also indefinit class numbers. In this article we change from Gauß to Eisenstein numbers. 1 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 P2 . In the first part we present a new proof of this fact.
In the second part we construct explicitly a Heegner series with help of Legendre-symbol coefficients. They can be interpreted as ”indefinit class numbers” we look for. Geometrically they appear also as number of plane curves with (normed) Eisenstein disc uniformization. 2

Preprint series: Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 2018-01

40 pp.