Rolf-Peter Holzapfel
Preprint series:
Institut für Mathematik, Humboldt-Universität zu Berlin (ISSN 0863-0976), 45
MSC 2000
- 11F11 Modular forms, one variable
-
11F27 Theta series; Weil representation
ZDM: 11F30
Abstract
We introduce orbital functionals $\int \boldsymbol{\beta} $ simultaneously
for each commensurability class of orbital surfaces. They are realized on infinitely
dimensional \emph{orbital} divisor spaces spanned by (arithmetic-geodesic real
$2$-dimensional) orbital curves on any orbital surface. We discover infinitely many
of them on each commensurability class of orbital Picard surfaces, which are real
$4$-spaces with cusps and negative constant Kähler-Einstein metric degenerated
along an orbital cycle. For a suitable (Heegner) sequence $\int \mathbf{h}_N$,
$N \in \mathbb{N}$, of them we investigate the corresponding formal orbital
$q$-series $\mathop{\sum}\limits_{N=0}^\infty (\int \mathbf{\mathbf{h}}_N)q^N$.
We show that after substitution $q = \re^{2\pi\ri\tau}$ and application to
arithmetic orbital curves $\mathbf{\hat{C}}$ on a fixed Picard surface class the
series $\mathop{\sum}\limits_{N=0}^{\infty} (\int_{\mathbf{\hat{C}}}
\mathbf{\mathbf{h}}_N)\re^{2\pi\ri\tau}$ define modular forms of well-determined
fixed weight, level and Nebentypus. The proof needs a new orbital understan-ding
of orbital hights introduced in \cite{Ho1} and Mumford-Fulton's rational intersection
theory on singular surfaces in Riemann-Roch-Hirzebruch style. It has to be
connected with Zeta and Theta functions of hermitian lines, indefinit quaternionic
fields and of a matrix algebra along a research marathon over $75$ years
represented by Cogdell, Kudla, Hirzebruch, Zagier, Shimura, Schoeneberg and Hecke.
Our aim is to open a door to an effective enumerative geometry for
complex geodesics on orbital varieties with nice metrics.
Keywords:
Arithmetic groups, unit ball, Picard modular surfaces,
Kähler-Einstein metric, constant curvature, arithmetic curves,
Shimura curves, modular curves, geodesics, modular form, surface
singularities, orbital divisors, rational intersections, orbital
functionals, Zeta functions, Theta functions, Mellin transform