{"title":"Refinements of strong multiplicity one for $\\mathrm{GL}(2)$","authors":"PENG-JIE Wong","doi":"10.4310/mrl.2022.v29.n2.a11","DOIUrl":null,"url":null,"abstract":"For distinct unitary cuspidal automorphic representations π1 and π2 for GL(2) over a number field F and any α ∈ R, let Sα be the set of primes v of F for which λπ1(v) 6= eλπ2(v), where λπi(v) is the Fourier coefficient of πi at v. In this article, we show that the lower Dirichlet density of Sα is at least 1 16 . Moreover, if π1 and π2 are not twist-equivalent, we show that the lower Dirichlet densities of Sα and ∩α Sα are at least 2 13 and 1 11 , respectively. Furthermore, for non-twist-equivalent π1 and π2, if each πi corresponds to a non-CM newform of weight ki ≥ 2 and with trivial nebentypus, we obtain various upper bounds for the number of primes p ≤ x such that λπ1(p) 2 = λπ2(p) . These present refinements of the works of Murty-Pujahari, Murty-Rajan, Ramakrishnan, and Walji.","PeriodicalId":49857,"journal":{"name":"Mathematical Research Letters","volume":" ","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2022-03-20","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.2022.v29.n2.a11","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
For distinct unitary cuspidal automorphic representations π1 and π2 for GL(2) over a number field F and any α ∈ R, let Sα be the set of primes v of F for which λπ1(v) 6= eλπ2(v), where λπi(v) is the Fourier coefficient of πi at v. In this article, we show that the lower Dirichlet density of Sα is at least 1 16 . Moreover, if π1 and π2 are not twist-equivalent, we show that the lower Dirichlet densities of Sα and ∩α Sα are at least 2 13 and 1 11 , respectively. Furthermore, for non-twist-equivalent π1 and π2, if each πi corresponds to a non-CM newform of weight ki ≥ 2 and with trivial nebentypus, we obtain various upper bounds for the number of primes p ≤ x such that λπ1(p) 2 = λπ2(p) . These present refinements of the works of Murty-Pujahari, Murty-Rajan, Ramakrishnan, and Walji.
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