Modular forms and period polynomials

Details Modular forms and period polynomials Modular forms and period polynomials

2012-02-26

arxiv.org/abs/1202.5802v3

Mathematics

Number Theory

33 pages. Ver. 3: Included the case of odd weight and made minor corrections

Scientific paper

We study the space of period polynomials associated with modular forms of integral weight for finite index subgroups of the modular group. For the full 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 extend Haberland's formula to arbitrary modular forms for finite index subgroups, and we show that it conceals two stronger formulas. One application is an extension of the Eichler-Shimura isomorphism to the entire space of modular forms. We extend the action of Hecke operators to $\Gamma_0(N)$ and $\Gamma_1(N)$, and we prove algebraically that the pairing on period polynomials appearing in Haberland's formula is Hecke equivariant. As a consequence of this proof, we derive two indefinite theta series identities which can be seen as analogues of Jacobi's formula for the theta series associated with the sum of four squares. We give two ways of determining the extra relations satisfied by the even and odd parts of period polynomials associated with cusp forms, which are independent of the period relations.

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