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Period Polynomials

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On the Theory of Maass Wave Forms

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Abstract

In this chapter, we discuss period polynomials that emerge from Eichler integrals on the full modular group. Period polynomials and their coefficients have significant impact on the theory of modular forms. In fact, the coefficients of these polynomials are certain values of L-functions. Moreover, by defining the cohomology of period polynomials, one can show that there is an isomorphism between two copies of the space of modular forms and the cohomology group. This is known as the Eichler-Shimura isomorphism. We give all the basic properties of period polynomials and prove the Eichler-Shimura isomorphism theorem. We continue by describing the action of Hecke operators on period polynomials, following an approach by Choie and Zagier. We omit some of the proofs due to the abundance of references in the literature or, in many cases, because we prove their analogues in the Maass forms case.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, FernUniversität in Hagen, Hagen, Germany

    Tobias Mühlenbruch

  2. Department of Mathematics, American University of Beirut, Beirut, Lebanon

    Wissam Raji

Authors
  1. Tobias Mühlenbruch
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  2. Wissam Raji
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Mühlenbruch, T., Raji, W. (2020). Period Polynomials. In: On the Theory of Maass Wave Forms. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-030-40475-8_2

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