{"title":"模块形式、穿刺球变形和对称张量表示的扩展","authors":"Gabriele Bogo","doi":"10.4310/mrl.2023.v30.n5.a2","DOIUrl":null,"url":null,"abstract":"Let $X = \\mathbb{H}/\\Gamma$ be an $n$-punctured sphere, $n \\gt 3$. We introduce and study $n-3$ deformation operators on the space of modular forms $M_\\ast (\\Gamma)$ based on the classical theory of uniformizing differential equations and accessory parameters. When restricting to modular functions, we recover a construction in Teichmüller theory related to the deformation of the complex structure of $X$. We describe the deformation operators in terms of derivations with respect to Eichler integrals of weight-four cusp forms, and in terms of vector-valued modular forms attached to extensions of symmetric tensor representations.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":"17 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Modular forms, deformation of punctured spheres, and extensions of symmetric tensor representations\",\"authors\":\"Gabriele Bogo\",\"doi\":\"10.4310/mrl.2023.v30.n5.a2\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $X = \\\\mathbb{H}/\\\\Gamma$ be an $n$-punctured sphere, $n \\\\gt 3$. We introduce and study $n-3$ deformation operators on the space of modular forms $M_\\\\ast (\\\\Gamma)$ based on the classical theory of uniformizing differential equations and accessory parameters. When restricting to modular functions, we recover a construction in Teichmüller theory related to the deformation of the complex structure of $X$. We describe the deformation operators in terms of derivations with respect to Eichler integrals of weight-four cusp forms, and in terms of vector-valued modular forms attached to extensions of symmetric tensor representations.\",\"PeriodicalId\":49857,\"journal\":{\"name\":\"Mathematical Research Letters\",\"volume\":\"17 1\",\"pages\":\"\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2024-05-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Research Letters\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.4310/mrl.2023.v30.n5.a2\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Research Letters","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4310/mrl.2023.v30.n5.a2","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Modular forms, deformation of punctured spheres, and extensions of symmetric tensor representations
Let $X = \mathbb{H}/\Gamma$ be an $n$-punctured sphere, $n \gt 3$. We introduce and study $n-3$ deformation operators on the space of modular forms $M_\ast (\Gamma)$ based on the classical theory of uniformizing differential equations and accessory parameters. When restricting to modular functions, we recover a construction in Teichmüller theory related to the deformation of the complex structure of $X$. We describe the deformation operators in terms of derivations with respect to Eichler integrals of weight-four cusp forms, and in terms of vector-valued modular forms attached to extensions of symmetric tensor representations.
期刊介绍:
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