Derivations of Siegel Modular Forms from Connections

TL;DR

The paper develops a differential geometric approach to derive both holomorphic and non-holomorphic derivative operators for Siegel modular forms using invariant connections on the Siegel upper half plane.

Contribution

It introduces the concept of modular connection to obtain holomorphic derivatives of Siegel modular forms, extending the differential geometric framework.

Findings

Constructed a non-holomorphic derivative operator using Levi-Civita connection.

Defined a weaker notion called modular connection for holomorphic derivatives.

Proved uniqueness of the holomorphic modular connection under certain conditions.

Abstract

We introduce a method in differential geometry to study the derivative operators of Siegel modular forms. By determining the coefficients of the invariant Levi-Civita connection on a Siegel upper half plane, and further by calculating the expressions of the differential forms under this connection, we get a non-holomorphic derivative operator of the Siegel modular forms. In order to get a holomorphic derivative operator, we introduce a weaker notion, called modular connection, on the Siegel upper half plane than a connection in differential geometry. Then we show that on a Siegel upper half plane there exists at most one holomorphic modular connection in some sense, and get a possible holomorphic derivative operator of Siegel modular forms.