Relative Proportionality on Picard and Hilbert Modular Surfaces

Abstract

We introduce “orbital categories”. The background objects are compactified quotient varieties of bounded symmetric domains $\mathbb{B}$ by lattice subgroups of the complex automorphism group of $\mathbb{B}$. Additionally, we endow some subvarieties of a given compact complex normal variety V with a natural weight > 1, imitating ramifications. They define an “orbital cycle” Z. The pairs V = (V,Z) are orbital varieties. These objects — also understood as an explicit approach to stacks — allow to introduce “orbital invariants” in a functorial manner. Typical are the orbital categories of Hilbert and Picard modular spaces. From the finite orbital data (e.g. the orbital Apollonius cycle on ℙ²) we read off “orbital Heegner series” as orbital invariants with the help of “orbital intersection theory”. We demonstrate for Hilbert and Picard surface F how their Fourier coefficients can be used to count Shimura curves of given norm on F. On recently discovered orbital projective planes the Shimura curves are joined with well-known classical elliptic modular forms.