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.