{"title":"On Drinfeld modular forms of higher rank VII: Expansions at the boundary","authors":"Ernst-Ulrich Gekeler","doi":"10.1016/j.jnt.2024.09.015","DOIUrl":null,"url":null,"abstract":"<div><div>We study expansions of Drinfeld modular forms of rank <span><math><mi>r</mi><mo>≥</mo><mn>2</mn></math></span> along the boundary of moduli varieties. Product formulas for the discriminant forms <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> are developed, which are analogous with Jacobi's formula for the classical elliptic discriminant. The vanishing orders are described through values at <span><math><mi>s</mi><mo>=</mo><mn>1</mn><mo>−</mo><mi>r</mi></math></span> of partial zeta functions of the underlying Drinfeld coefficient ring <em>A</em>. We show linear independence properties for Eisenstein series, which allow to split spaces of modular forms into the subspaces of cusp forms and of Eisenstein series, and give various characterizations of the boundary condition for modular forms.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"269 ","pages":"Pages 260-340"},"PeriodicalIF":0.6000,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Number Theory","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022314X24002269","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We study expansions of Drinfeld modular forms of rank along the boundary of moduli varieties. Product formulas for the discriminant forms are developed, which are analogous with Jacobi's formula for the classical elliptic discriminant. The vanishing orders are described through values at of partial zeta functions of the underlying Drinfeld coefficient ring A. We show linear independence properties for Eisenstein series, which allow to split spaces of modular forms into the subspaces of cusp forms and of Eisenstein series, and give various characterizations of the boundary condition for modular forms.
期刊介绍:
The Journal of Number Theory (JNT) features selected research articles that represent the broad spectrum of interest in contemporary number theory and allied areas. A valuable resource for mathematicians, the journal provides an international forum for the publication of original research in this field.
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