Enumerative Geometry for Complex Geodesics on Quasi-hyperbolic 4-spaces with Cusps
نویسندگان
چکیده
We introduce orbital functionals ∫ β 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 ∫ hN , N ∈ N, of them we investigate the corresponding formal orbital q-series ∑∞ N=0( ∫ hN )q . We show that after substitution q = e and application to arithmetic orbital curves Ĉ on a fixed Picard surface class, the series ∑∞ N=0( ∫ Ĉ hN ) e 2πiτ 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.
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