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Enumerative geometry for complex geodesics on quasi-hyperbolic 4-spaces with cusps. (English) Zbl 1053.14060
Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 4th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 6–15, 2002. Sofia: Coral Press Scientific Publishing (ISBN 954-90618-4-1/pbk). 42-87 (2003).
An orbital surface $\stackrel{^}{𝕏}=\left(\stackrel{^}{X},{\stackrel{^}{B}}^{1}\right)$ is a complex normal algebraic surface $\stackrel{^}{X}$ with at most quotient and cusp singularities together with an orbital Weil divisor ${\stackrel{^}{B}}^{1}$. Orbital functionals can be defined simultaneously for each commensurability class of orbital surfaces, and they are realized on infinite-dimensional orbital divisor spaces spanned by orbital curves on an orbital surface. In this paper the author finds infinitely many orbital functionals 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 $\left(\int {h}_{N}\right)$ of such functionals he studies the the corresponding formal orbital $q$-series ${\sum }_{N=0}^{\infty }\left(\int {h}_{N}\right){q}^{N}$. After substitution $q={e}^{2\pi i\tau }$ and the application to arithmetic orbital curves $\stackrel{^}{𝕐}$ on a fixed Picard surface class, the author proves that the series ${\sum }_{N=0}^{\infty }\left({\int }_{\stackrel{^}{𝕐}}{h}_{N}\right){e}^{2\pi iN\tau }$ defines modular forms of well-determined fixed weight, level and Nebentypus. The proof uses the notion of orbital heights and Mumford-Fulton’s rational intersection theory on singular surfaces in the Riemann-Roch-Hirzebruch style.
##### MSC:
 14N20 Configurations and arrangements of linear subspaces 11F55 Groups and their modular and automorphic forms (several variables)
##### Keywords:
orbital curves; orbital surfaces; modular forms