## Proceedings of the International Conference on Geometry, Integrability and Quantization

- Geom. Integrability & Quantization
- Proceedings of the Fourth International Conference on Geometry, Integrability and Quantization, Ivaïlo M. Mladenov and Gregory L. Naber, eds. (Sofia: Coral Press Scientific Publishing, 2003), 42 - 87

### Enumerative Geometry on Quasi-Hyperbolic 4-Spaces with Cusps

#### Abstract

We introduce orbital functionals $\int \beta$ simultaneously for each commensurability class of orbital surfaces. They are realized on infinitely dimensional *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 $\sum_{N=0}^{\infty} (\int \mathbf{h}_N)q^N$. We show that after substitution $q = {\rm e}^{2 \pi \mathbf{i} \tau}$ and application to arithmetic orbital curves **Ĉ** on a fixed Picard surface class, the series $\sum_{N=0}^{\infty}(\int_{Ĉ} {\mathbb{h}_N})e^{2 \pi \mathbf{i} \tau}$ define modular forms of well-determined fixed weight, level and Nebentypus. The proof needs a new orbital understanding of orbital heights introduced in [12] 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, indefinite 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.

#### Article information

**Dates**

First available in Project Euclid:
12 June 2015

**Permanent link to this document**

https://projecteuclid.org/
euclid.pgiq/1434148596

**Digital Object Identifier**

doi:10.7546/giq-4-2003-11-41

**Mathematical Reviews number (MathSciNet)**

MR1977559

**Zentralblatt MATH identifier**

1039.53104

#### Citation

Holzapfel, Rolf-Peter. Enumerative Geometry on Quasi-Hyperbolic 4-Spaces with Cusps. Proceedings of the Fourth International Conference on Geometry, Integrability and Quantization, 42--87, Coral Press Scientific Publishing, Sofia, Bulgaria, 2003. doi:10.7546/giq-4-2003-11-41. https://projecteuclid.org/euclid.pgiq/1434148596

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