Algebraic de Rham theory for weakly holomorphic modular forms of level one

TL;DR

This paper develops an algebraic de Rham theory framework for weakly holomorphic modular forms of level one, connecting them to the cohomology of modular curves and deriving formulas for their periods.

Contribution

It establishes an Eichler-Shimura isomorphism for weakly modular forms, linking Fourier coefficients to algebraic de Rham cohomology and period formulas.

Findings

Eichler-Shimura isomorphism for weakly modular forms

Explicit formulas for periods and quasi-periods

Connection between Fourier coefficients and cohomology

Abstract

We establish an Eichler-Shimura isomorphism for weakly modular forms of level one. We do this by relating weakly modular forms with rational Fourier coefficients to the algebraic de Rham cohomology of the modular curve with twisted coefficients. This leads to formulae for the periods and quasi-periods of modular forms.