Higher order modular forms and mixed Hodge theory

TL;DR

This paper introduces higher order modular forms with Hodge structures derived from the fundamental group of modular curves, generalizing classical structures and constructing new higher weight forms with mixed Hodge structures.

Contribution

It develops a new framework for higher order modular forms with Hodge structures, extending classical cohomological interpretations and constructing higher weight forms via Poincare series.

Findings

Higher order modular forms have a Hodge structure from the fundamental group.

Construction of higher order Poincare series for higher weight forms.

Higher weight, higher order forms possess a mixed Hodge structure.

Abstract

In this paper we introduce a certain space of higher order modular forms of weight 0 and show that it has a Hodge structure coming from the geometry of the fundamental group of a modular curve. This generalizes the usual structure on classical weight 2 forms coming from the cohomology of the modular curve. Further we construct some higher order Poincare series to get higher order higher weight forms and using them we define a space of higher weight, higher order forms which has a mixed Hodge structure as well.