Orbital Functional Series on Picard Surfaces

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.