Orbits in Lattices

TL;DR

This paper presents algorithms for computing orbits in lattices and Tits' buildings, aiding the study of boundary components in orthogonal modular varieties and enhancing computational methods for modular forms.

Contribution

The paper introduces new algorithms for calculating lattice orbits and Tits' buildings, with applications to orthogonal modular varieties and modular form computations.

Findings

Algorithms successfully compute lattice orbits and Tits' buildings.

Applications improve understanding of boundary components in modular varieties.

Enhanced computational performance for orthogonal modular forms.

Abstract

We exhibit algorithms for calculating Tits' buildings and orbits of vectors in a lattice $L$ for certain subgroups of $\operatorname{O}(L)$. We discuss how these algorithms can be applied to understand the configuration of boundary components in the Baily-Borel compactification of orthogonal modular varieties and to improve the performance of computer arithmetic of orthogonal modular forms.