Finite-dimensional period spaces for the spaces of cusp forms

Dohoon Choi, Subong Lim

Research output: Contribution to journalArticle

Abstract

Let k be a positive integer and Pk ⊂ 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 Pk-2. Moreover, Pk-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.

Original languageEnglish
Pages (from-to)507-543
Number of pages37
JournalIsrael Journal of Mathematics
Volume216
Issue number2
DOIs
Publication statusPublished - 2016 Oct 1
Externally publishedYes

Fingerprint

Cusp Form
Period Function
Isomorphism
Modular Forms
Cohomology Group
Linear map
Module
Coefficient
Less than or equal to
Growth Conditions
Half-plane
Injective
Vector space
Analytic function
Closure
Classify
Analogue
Polynomial
Integer

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Finite-dimensional period spaces for the spaces of cusp forms. / Choi, Dohoon; Lim, Subong.

In: Israel Journal of Mathematics, Vol. 216, No. 2, 01.10.2016, p. 507-543.

Research output: Contribution to journalArticle

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