Generalized automorphic sheaves and the proportionality principle of Hirzebruch-Mumford

TL;DR

This paper introduces a new algebraic framework for automorphic sheaves on Shimura varieties, enabling a concise proof of the Hirzebruch-Mumford proportionality theorem.

Contribution

It proposes a generalized notion of automorphic sheaves and applies it to provide an algebraic proof of a classical proportionality principle.

Findings

Unified algebraic approach to automorphic sheaves

Inclusion of Jacobi forms and weakly holomorphic modular forms

Simplified proof of Hirzebruch-Mumford proportionality theorem

Abstract

We axiomatize the algebraic properties of toroidal compactifications of (mixed) Shimura varieties and their automorphic vector bundles. A notion of generalized automorphic sheaf is proposed which includes sheaves of (meromorphic) sections of automorphic vector bundles with prescribed vanishing and pole orders along strata in the compactification, and their quotients. These include, for instance, sheaves of Jacobi forms and weakly holomorphic modular forms. Using this machinery we give a short and purely algebraic proof of the proportionality theorem of Hirzebruch and Mumford.