{"title":"On Shimura curves generated by families of Galois G-covers of curves","authors":"Abolfazl Mohajer","doi":"10.1016/j.aim.2024.109855","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper we prove that there are no families of cyclic <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>-covers of elliptic curves which generate non-compact Shimura (special) curves that lie generically in the Torelli locus <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span> of abelian varieties with <span><math><mi>g</mi><mo>≥</mo><mn>8</mn></math></span> when <em>n</em> has a proper prime factor <span><math><mi>p</mi><mo>≥</mo><mn>7</mn></math></span>. This non-existence is also shown for families of <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>-covers of curves of any genus <em>s</em> provided that <em>n</em> has a large enough prime factor <em>p</em> (depending on <em>s</em>). We achieve these results by applying the theory of Higgs bundles and the Viehweg-Zuo characterization of Shimura curves in the moduli space of principally polarized abelian varieties.</p></div>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0001870824003700","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
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Abstract
In this paper we prove that there are no families of cyclic -covers of elliptic curves which generate non-compact Shimura (special) curves that lie generically in the Torelli locus of abelian varieties with when n has a proper prime factor . This non-existence is also shown for families of -covers of curves of any genus s provided that n has a large enough prime factor p (depending on s). We achieve these results by applying the theory of Higgs bundles and the Viehweg-Zuo characterization of Shimura curves in the moduli space of principally polarized abelian varieties.
期刊介绍：
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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