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

Published online by Cambridge University Press:  01 March 2009

SHINJI FUKUHARA  and
YIFAN YANG
SHINJI FUKUHARA
Affiliation:
Department of Mathematics, Tsuda College, Tsuda-machi 2-1-1, Kodaira-shi, Tokyo 187-8577, Japan. e-mail: fukuhara@tsuda.ac.jp
YIFAN YANG
Affiliation:
Department of Applied Mathematics, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, Taiwan300. e-mail: yfyang@math.nctu.edu.tw
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Abstract

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 main theorems, we will also give an affirmative answer to a speculation of Imamoglu and Kohnen on a basis of Sw+2(Γ0(2)).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 146 , Issue 2 , March 2009 , pp. 321 - 350
Copyright
Copyright © Cambridge Philosophical Society 2008

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