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Finite-dimensional period spaces for the spaces of cusp forms

机译：尖形形式空间的有限维周期空间

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摘要

Let k be a positive integer and P (k) aS, 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.
机译：令k为正整数，令P（k）aS，C [X]为度数小于或等于k的多项式集。在积分权重为k的尖峰形式的空间与系数模块为P（k-2）的第一抛物线同调群之间存在同构，称为Eichler-Shimura同构。此外，P（k-2）包含权重k的尖峰形式的周期函数。 Eichler-Shimura同构用系数模块P扩展到实际权重的尖点形式的空间。在这里，与积分权重P的情况相反，它是由复态上半平面上的全纯函数组成的无穷维矢量空间，具有一定的生长条件。但是，即使是半整数权重的情况，也没有就有限维空间描述实际权重的尖点形式的周期函数。在本文中，我们在实际权重的尖峰形式的空间与具有系数模块P的Eichler-Shimura同调群之间构造了一个新的同构，以便获得包含尖峰形式的那些到共界的周期函数的有限维空间。作为应用，我们证明了真重情况下的Haberland公式的类似物，并构造了从混合模拟模态形式的空间到量子模态形式的空间的内射线性映射，并在A（1）（C）中对Zariski闭包进行分类这些线性映射下混合模拟模块形式的图像的权重。

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