Modular forms and period polynomials

TL;DR

This paper explores the structure of period polynomials linked to modular forms, extending classical formulas and operators to broader contexts, and providing new insights into their algebraic and analytical properties.

Contribution

It generalizes Haberland's formula to non-cuspidal modular forms for finite index subgroups and extends Hecke operator actions to period polynomials.

Findings

Haberland's pairing is nondegenerate for these polynomials

Hecke operators' adjoints are explicitly determined

New formulas for Fourier coefficients of Hecke eigenforms

Abstract

We study the space of period polynomials associated with modular forms of integral weight for finite index subgroups of the modular group. For the modular group, this space is endowed with a pairing, corresponding to the Petersson inner product on modular forms via a formula of Haberland, and with an action of Hecke operators, defined algebraically by Zagier. We generalize Haberland's formula to (not necessarily cuspidal) modular forms for finite index subgroups, and we show that it conceals two stronger formulas. We extend the action of Hecke operators to period polynomials of modular forms, we show that the pairing on period polynomials appearing in Haberland's formula is nondegenerate, and we determine the adjoints of Hecke operators with respect to it. We give a few applications for $\Gamma_1(N)$: an extension of the Eichler-Shimura isomorphism to the entire space of modular forms; the determination of the relations satisfied by the even and odd parts of period polynomials associated with cusp forms, which are independent of the period relations; and an explicit formula for Fourier coefficients of Hecke eigenforms in terms of their period polynomials, generalizing the Coefficients Theorem of Manin.