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Period polynomials and explicit formulas for Hecke operators on Γ_0(2)

Shinji Fukuhara·Yifan Yang
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Abstract

Let S_{w+2}(Γ_0(N)) be the vector space of cusp forms of weight w+2 on the congruence subgroup Γ_0(N). We first determine explicit formulas for period polynomials of elements in S_{w+2}(Γ_0(N)) by means of Bernoulli polynomials. When N=2, from these explicit formulas we obtain new bases for S_{w+2}(Γ_0(2)), and extend the Eichler-Shimura-Manin isomorphism theorem to Γ_0(2). This implies that there are natural correspondences between the spaces of cusp forms on Γ_0(2) and the spaces of period polynomials. Based on these results, we will find explicit form of Hecke operators on S_{w+2}(Γ_0(2)). As an application of our main theorems, we will also give an affirmative answer to a speculation of Imamoglu and Kohnen on a basis of S_{w+2}(Γ_0(2)).

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Shinji Fukuhara, Yifan Yang. 2007-07-10. Period polynomials and explicit formulas for Hecke operators on Γ_0(2). https://arxiv.org/abs/math/0608372

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