Vector-valued orthogonal modular forms

TL;DR

This monograph develops the theory of vector-valued orthogonal modular forms for signature (2,n), establishing foundational concepts, exploring advanced aspects, and deriving vanishing theorems with applications to orthogonal modular varieties.

Contribution

It introduces the second Hodge bundle and its role in vector-valued modular forms, providing new insights and results in the geometry and analysis of orthogonal modular forms.

Findings

Vanishing theorems for small weight vector-valued modular forms

Optimal bounds for vanishing of holomorphic tensors on modular varieties

Development of geometric and analytical tools for orthogonal modular forms

Abstract

This monograph is devoted to the theory of vector-valued modular forms for orthogonal groups of signature (2,n). Our purpose is multi-layered: (1) to lay a foundation of the theory of vector-valued orthogonal modular forms; (2) to develop some aspects of the theory in more depth such as geometry of the Siegel operators, filtrations associated to 1-dimensional cusps, decomposition of vector-valued Jacobi forms, square integrability etc; and (3) as applications derive several types of vanishing theorems for vector-valued modular forms of small weight. Our vanishing theorems imply in particular vanishing of holomorphic tensors of degree <n/2-1 on orthogonal modular varieties, which is optimal as a general bound. The fundamental ingredients of the theory are the two Hodge bundles. The first is the Hodge line bundle which already appears in the theory of scalar-valued modular forms. The second Hodge bundle emerges in the vector-valued theory and plays a central role. It corresponds to the non-abelian part O(n,R) of the maximal compact subgroup of O(2,n). The main focus of this monograph is centered around the properties and the role of the second Hodge bundle in the theory of vector-valued orthogonal modular forms.