0 Ju l 2 00 7 PERIOD POLYNOMIALS AND EXPLICIT FORMULAS FOR HECKE OPERATORS ON Γ

Let Sw+2(Γ0(N)) be the vector space of cusp forms of weight w+2 on the congruence subgroup Γ0(N). We first determine explicit formulas for period polynomials of elements in Sw+2(Γ0(N)) by means of Bernoulli polynomials. When N = 2, from these explicit formulas we obtain new bases for Sw+2(Γ0(2)), and extend the Eichler-Shimura-Manin isomorphism theorem to Γ0(2). This implies that there are natural correspondences between the spaces of cusp forms on Γ0(2) and the spaces of period polynomials. Based on these results, we will find explicit form of Hecke operators on Sw+2(Γ0(2)). As an application of our main theorems, we will also give an affirmative answer to a speculation of Imamoḡlu and Kohnen on a basis of Sw+2(Γ0(2)). 0. Introduction Let Γ be a congruence subgroup of SL2(Z). One of the most important problems in the theory of modular forms is to obtain explicit formulas for Hecke operators on cusp forms for Γ . When Γ is the full modular group SL2(Z), this was done in [8], where we gave explicit formulas in terms of Bernoulli numbers Bk and divisor functions σk. Here we briefly recall the approach in [8]. For a cusp form f of weight w + 2 on SL2(Z) with w ≥ 2 even we consider the n-th period rn(f) = ∫ i∞ 0 f(z)z dz, 0 ≤ n ≤ w. Since rn : Sw+2(SL2(Z)) → C is a linear functional, there exists a unique cusp form Rn of weight w + 2 such that rn(f) = (f,Rn), where (f, g) is the Petersson inner product. In [8], we first showed that a certain subsets of {Rn} forms a basis for the space Sw+2(SL2(Z)) of cusp forms of weight w+2. We then studied the action of Hecke operators on this basis and obtained an explicit matrix representation of Hecke operators. The Dedekind symbols ([9]) played a central role in our argument. (Note that periods and the Hecke operators on periods have been studied by a number of mathematicians. To name a few, see [1, 5, 8, 12, 14, 15, 17, 18, 19, 20].) The main goal in this article is to extend our formulas from SL2(Z) to Γ0(N). Our starting point is the same as the case for SL2(Z). Namely, for a cusp form f of weight w + 2 on Γ0(N), we consider the n-th period rn(f) = ∫ i∞ 0 f(z)z n dz for 0 ≤ n ≤ w, and let RΓ0(N),w,n be the unique cusp form determined by the property rn(f) = 2 (2i)(f,RΓ0(N),w,n) 2000 Mathematics Subject Classification. Primary 11F25; Secondary 11F11, 11F67.