{"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":null,"pages":null},"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\":null,\"pages\":null},\"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。
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 .