Toroidal Compactifications and Dimension Formulas for Spaces of Modular Forms for Orthogonal Shimura Varieties

TL;DR

This paper develops the theory of toroidal compactifications for orthogonal Shimura varieties and derives dimension formulas for modular forms, focusing on quadratic forms of signature (2,n).

Contribution

It provides a detailed framework for constructing compactifications and computing dimensions for spaces of modular forms on orthogonal locally symmetric spaces.

Findings

Explicit descriptions of toroidal compactifications for orthogonal spaces.

Dimension formulas for modular forms in the (2,n) case.

Foundational results without explicit cone decompositions or cusp counts.

Abstract

In this paper we describe the general theory of constructing toroidal compactifications of locally symmetric spaces and using these to compute dimension formulas for spaces of modular forms. We focus explicitly on the case of the orthogonal locally symmetric spaces arising from quadratic forms of signature $(2,n)$, giving explicit details of the constructions, structures and results in these cases. This article does not give explicit cone decompositions, compute explicit intersection pairings, or count cusps and thus does not give any complete formulas for the dimensions. This article is still `in preparation'.