顶点形式的标量值深度二Eichler-Shimura积分

IF 1.1 Q1 MATHEMATICS
Tobias Magnusson, Martin Raum
{"title":"顶点形式的标量值深度二Eichler-Shimura积分","authors":"Tobias Magnusson, Martin Raum","doi":"10.1112/tlm3.12055","DOIUrl":null,"url":null,"abstract":"Given cusp forms and of integral weight , the depth two holomorphic iterated Eichler–Shimura integral  is defined by , where is the Eichler integral of and are formal variables. We provide an explicit vector‐valued modular form whose top components are given by . We show that this vector‐valued modular form gives rise to a scalar‐valued iterated Eichler integral of depth two, denoted by , that can be seen as a higher depth generalization of the scalar‐valued Eichler integral of depth one. As an aside, our argument provides an alternative explanation of an orthogonality relation satisfied by period polynomials originally due to Paşol–Popa. We show that can be expressed in terms of sums of products of components of vector‐valued Eisenstein series with classical modular forms after multiplication with a suitable power of the discriminant modular form . This allows for effective computation of .","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":"16 1","pages":"0"},"PeriodicalIF":1.1000,"publicationDate":"2023-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Scalar‐valued depth two Eichler–Shimura integrals of cusp forms\",\"authors\":\"Tobias Magnusson, Martin Raum\",\"doi\":\"10.1112/tlm3.12055\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Given cusp forms and of integral weight , the depth two holomorphic iterated Eichler–Shimura integral  is defined by , where is the Eichler integral of and are formal variables. We provide an explicit vector‐valued modular form whose top components are given by . We show that this vector‐valued modular form gives rise to a scalar‐valued iterated Eichler integral of depth two, denoted by , that can be seen as a higher depth generalization of the scalar‐valued Eichler integral of depth one. As an aside, our argument provides an alternative explanation of an orthogonality relation satisfied by period polynomials originally due to Paşol–Popa. We show that can be expressed in terms of sums of products of components of vector‐valued Eisenstein series with classical modular forms after multiplication with a suitable power of the discriminant modular form . This allows for effective computation of .\",\"PeriodicalId\":41208,\"journal\":{\"name\":\"Transactions of the London Mathematical Society\",\"volume\":\"16 1\",\"pages\":\"0\"},\"PeriodicalIF\":1.1000,\"publicationDate\":\"2023-09-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Transactions of the London Mathematical Society\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1112/tlm3.12055\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Transactions of the London Mathematical Society","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1112/tlm3.12055","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

摘要

给定顶点形式和积分权值,定义深度二全纯迭代Eichler - shimura积分为,其中为和为形式变量的Eichler积分。我们提供了一个显式的向量值模形式,它的上分量由。我们证明了这个向量值模形式产生了深度2的标量值迭代Eichler积分,记为,它可以看作是深度1的标量值Eichler积分的更高深度推广。作为题外话,我们的论点提供了一个由周期多项式满足的正交关系的另一种解释,最初是由于pa ol - popa。我们证明了它可以用具有经典模形式的向量值爱森斯坦级数的分量的乘积的和来表示,这些分量是用判别模形式的一个合适的幂次相乘后表示的。这允许有效的计算。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Scalar‐valued depth two Eichler–Shimura integrals of cusp forms
Given cusp forms and of integral weight , the depth two holomorphic iterated Eichler–Shimura integral  is defined by , where is the Eichler integral of and are formal variables. We provide an explicit vector‐valued modular form whose top components are given by . We show that this vector‐valued modular form gives rise to a scalar‐valued iterated Eichler integral of depth two, denoted by , that can be seen as a higher depth generalization of the scalar‐valued Eichler integral of depth one. As an aside, our argument provides an alternative explanation of an orthogonality relation satisfied by period polynomials originally due to Paşol–Popa. We show that can be expressed in terms of sums of products of components of vector‐valued Eisenstein series with classical modular forms after multiplication with a suitable power of the discriminant modular form . This allows for effective computation of .
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
CiteScore
1.40
自引率
0.00%
发文量
8
审稿时长
41 weeks
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信