Archer Clayton , Helen Dai , Tianyu Ni , Hui Xue , Jake Zummo
{"title":"Nonvanishing of second coefficients of Hecke polynomials","authors":"Archer Clayton , Helen Dai , Tianyu Ni , Hui Xue , Jake Zummo","doi":"10.1016/j.jnt.2024.03.014","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><mi>N</mi><mo>,</mo><mn>2</mn><mi>k</mi><mo>)</mo></math></span> be the <em>m</em>th Hecke operator on the space <span><math><mi>S</mi><mo>(</mo><mi>N</mi><mo>,</mo><mn>2</mn><mi>k</mi><mo>)</mo></math></span> of cuspforms of weight 2<em>k</em> and level <em>N</em>. This paper shows that in all but finitely many cases, which we list, the second coefficient of the characteristic polynomial of <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>,</mo><mn>2</mn><mi>k</mi><mo>)</mo></math></span> does not vanish when 2 and <em>N</em> are coprime.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-04-22","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/S0022314X24000891","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
Let be the mth Hecke operator on the space of cuspforms of weight 2k and level N. This paper shows that in all but finitely many cases, which we list, the second coefficient of the characteristic polynomial of does not vanish when 2 and N are coprime.
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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