Period polynomials and explicit formulas for Hecke operators on Γ₀(2)

2006/08/15 by Fukuhara, Shinji, Yang, Yifan
#11F11 #11F25 #11F67 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.math/0608372

Abstract

Let S_w+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 S_w+2(Γ₀(N)) by means of Bernoulli polynomials. When N=2, from these explicit formulas we obtain new bases for S_w+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 S_w+2(Γ₀(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(Γ₀(2)).

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• Hecke Operators for Period Functions for Congruence Subgroups
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