Indefinite Eisenstein Lattices: A Modern Ball-Rendevous with Poincarè, Picard, Hecke, Shimura, Mumford, Deligne and Hirzebruch

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.