Finite-dimensional period spaces for the spaces of cusp forms

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• 作者：Dohoon Choi ; Subong Lim
• 刊名：Israel Journal of Mathematics
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：216
• 期：2
• 页码：507-543
• 全文大小：404 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Algebra
  Group Theory and Generalizations
  Analysis
  Applications of Mathematics
  Mathematical and Computational Physics
  
• 出版者：Hebrew University Magnes Press
• ISSN：1565-8511
• 卷排序：216

文摘

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 A1(C) of images of mixed mock modular forms under these linear maps in terms of their weights.