### Abstract

Let k be a positive integer and P_{k} ⊂ C[X] the set of polynomials of degree less than or equal to k. There exists an isomorphism, called the Eichler–Shimura isomorphism, between the space of cusp forms of integral weight k and a first parabolic cohomology group with coefficient module P_{k-2}. Moreover, P_{k-2} contains period functions of cusp forms of weight k. The Eichler–Shimura isomorphism was extended to the space of cusp forms of real weight with coefficient module P. Here, in contrast to the case of integral weight P is an infinite-dimensional vector space consisting of holomorphic functions on the complex upper half plane with a certain growth condition. However, period functions of cusp forms of real weight have not been described in terms of a finite-dimensional space even for the case of half-integral weight. In this paper, we construct a new isomorphism between the space of cusp forms of real weight and an Eichler–Shimura cohomology group with coefficient module P so that we obtain a finite-dimensional space of period functions containing those of cusp forms up to coboundaries. As applications, we prove analogues of the Haberland formula for the case of real weight, and we construct injective linear maps from the spaces of mixed mock modular forms to the spaces of quantum modular forms and classify Zariski closures in A^{1}(C) of images of mixed mock modular forms under these linear maps in terms of their weights.

Original language | English |
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Pages (from-to) | 507-543 |

Number of pages | 37 |

Journal | Israel Journal of Mathematics |

Volume | 216 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2016 Oct 1 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Israel Journal of Mathematics*,

*216*(2), 507-543. https://doi.org/10.1007/s11856-016-1418-2