{"title":"The fake monster algebra and singular Borcherds products","authors":"Haowu Wang , Brandon Williams","doi":"10.1016/j.aim.2024.110083","DOIUrl":null,"url":null,"abstract":"<div><div>In this paper we consider several problems in the theory of automorphic products and generalized Kac–Moody algebras proposed by Borcherds in 1995. We show that the denominator of the fake monster algebra defines the unique holomorphic Borcherds product of singular weight on a maximal lattice. We give a full classification of symmetric holomorphic Borcherds products of singular weight on lattices of prime level. Finally we prove that all twisted denominator identities of the fake monster algebra arise as the Fourier expansions of Borcherds products of singular weight at a certain cusp. The proofs rely on an identification between modular forms for the Weil representation attached to lattices of type <span><math><mi>U</mi><mo>(</mo><mi>N</mi><mo>)</mo><mo>⊕</mo><mi>U</mi><mo>⊕</mo><mi>L</mi></math></span> and certain tuples of Jacobi forms of level <em>N</em>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"461 ","pages":"Article 110083"},"PeriodicalIF":1.5000,"publicationDate":"2025-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0001870824005991","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper we consider several problems in the theory of automorphic products and generalized Kac–Moody algebras proposed by Borcherds in 1995. We show that the denominator of the fake monster algebra defines the unique holomorphic Borcherds product of singular weight on a maximal lattice. We give a full classification of symmetric holomorphic Borcherds products of singular weight on lattices of prime level. Finally we prove that all twisted denominator identities of the fake monster algebra arise as the Fourier expansions of Borcherds products of singular weight at a certain cusp. The proofs rely on an identification between modular forms for the Weil representation attached to lattices of type and certain tuples of Jacobi forms of level N.
期刊介绍:
Emphasizing contributions that represent significant advances in all areas of pure mathematics, Advances in Mathematics provides research mathematicians with an effective medium for communicating important recent developments in their areas of specialization to colleagues and to scientists in related disciplines.