花环附近扭转点的伽罗瓦轨道

IF 0.9 1区 数学 Q2 MATHEMATICS Algebra & Number Theory Pub Date : 2024-10-18 DOI:10.2140/ant.2024.18.1945
Vesselin Dimitrov, Philipp Habegger
{"title":"花环附近扭转点的伽罗瓦轨道","authors":"Vesselin Dimitrov, Philipp Habegger","doi":"10.2140/ant.2024.18.1945","DOIUrl":null,"url":null,"abstract":"<p>We prove that the Galois equidistribution of torsion points of the algebraic torus <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msubsup><mrow><mi mathvariant=\"double-struck\">𝔾</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math> extends to the singular test functions of the form <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi> log</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo>|</mo><mi>P</mi><mo>|</mo></math>, where <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>P</mi></math> is a Laurent polynomial having algebraic coefficients that vanishes on the unit real <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>d</mi></math>-torus in a set whose Zariski closure in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msubsup><mrow><mi mathvariant=\"double-struck\">𝔾</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math> has codimension at least <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>2</mn></math>. Our result includes a power-saving quantitative estimate of the decay rate of the equidistribution. It refines an ergodic theorem of Lind, Schmidt, and Verbitskiy, of which it also supplies a purely Diophantine proof. As an application, we confirm Ih’s integrality finiteness conjecture on torsion points for a class of atoral divisors of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msubsup><mrow><mi mathvariant=\"double-struck\">𝔾</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math>. </p>","PeriodicalId":50828,"journal":{"name":"Algebra & Number Theory","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Galois orbits of torsion points near atoral sets\",\"authors\":\"Vesselin Dimitrov, Philipp Habegger\",\"doi\":\"10.2140/ant.2024.18.1945\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We prove that the Galois equidistribution of torsion points of the algebraic torus <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><msubsup><mrow><mi mathvariant=\\\"double-struck\\\">𝔾</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math> extends to the singular test functions of the form <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi> log</mi><mo> ⁡<!--FUNCTION APPLICATION--> </mo><!--nolimits--><mo>|</mo><mi>P</mi><mo>|</mo></math>, where <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>P</mi></math> is a Laurent polynomial having algebraic coefficients that vanishes on the unit real <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>d</mi></math>-torus in a set whose Zariski closure in <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><msubsup><mrow><mi mathvariant=\\\"double-struck\\\">𝔾</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math> has codimension at least <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mn>2</mn></math>. Our result includes a power-saving quantitative estimate of the decay rate of the equidistribution. It refines an ergodic theorem of Lind, Schmidt, and Verbitskiy, of which it also supplies a purely Diophantine proof. As an application, we confirm Ih’s integrality finiteness conjecture on torsion points for a class of atoral divisors of <math display=\\\"inline\\\" xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><msubsup><mrow><mi mathvariant=\\\"double-struck\\\">𝔾</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math>. </p>\",\"PeriodicalId\":50828,\"journal\":{\"name\":\"Algebra & Number Theory\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-10-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Algebra & Number Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.2140/ant.2024.18.1945\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebra & Number Theory","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.2140/ant.2024.18.1945","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

摘要

我们证明了代数环𝔾md 的扭转点的伽罗华等差数列扩展到 log |P|形式的奇异检验函数,其中 P 是具有代数系数的劳伦多项式,它在单位实数 d 环上消失在一个集合中,该集合在𝔾md 中的扎里斯基闭合至少有 2 个开元维。它完善了林德、施密特和韦尔比茨基的一个遍历定理,并提供了一个纯粹的 Diophantine 证明。作为应用,我们证实了 Ih 关于𝔾md 的一类口角除数的扭转点的积分有限性猜想。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
Galois orbits of torsion points near atoral sets

We prove that the Galois equidistribution of torsion points of the algebraic torus 𝔾md extends to the singular test functions of the form log |P|, where P is a Laurent polynomial having algebraic coefficients that vanishes on the unit real d-torus in a set whose Zariski closure in 𝔾md has codimension at least 2. Our result includes a power-saving quantitative estimate of the decay rate of the equidistribution. It refines an ergodic theorem of Lind, Schmidt, and Verbitskiy, of which it also supplies a purely Diophantine proof. As an application, we confirm Ih’s integrality finiteness conjecture on torsion points for a class of atoral divisors of 𝔾md.

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
CiteScore
1.80
自引率
7.70%
发文量
52
审稿时长
6-12 weeks
期刊介绍: ANT’s inclusive definition of algebra and number theory allows it to print research covering a wide range of subtopics, including algebraic and arithmetic geometry. ANT publishes high-quality articles of interest to a broad readership, at a level surpassing all but the top four or five mathematics journals. It exists in both print and electronic forms. The policies of ANT are set by the editorial board — a group of working mathematicians — rather than by a profit-oriented company, so they will remain friendly to mathematicians'' interests. In particular, they will promote broad dissemination, easy electronic access, and permissive use of content to the greatest extent compatible with survival of the journal. All electronic content becomes free and open access 5 years after publication.
期刊最新文献
Separating G2-invariants of several octonions Scattering diagrams for generalized cluster algebras Moduli of linear slices of high degree smooth hypersurfaces Matrix Kloosterman sums Rooted tree maps for multiple L-values from a perspective of harmonic algebras
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1