We start with a brief survey on the Northcott property for subfields of the algebraic numbers $Qbar$. Then we introduce a new criterion for its validity (refining the author's previous criterion), addressing a problem of Bombieri. We show that Bombieri and Zannier's theorem, stating that the maximal abelian extension of a number field $K$ contained in $K^{(d)}$ has the Northcott property, follows very easily from this refined criterion. Here $K^{(d)}$ denotes the composite field of all extensions of $K$ of degree at most $d$.
{"title":"On the Northcott property for infinite extensions","authors":"Martin Widmer","doi":"10.2140/ent.2023.2.1","DOIUrl":"https://doi.org/10.2140/ent.2023.2.1","url":null,"abstract":"We start with a brief survey on the Northcott property for subfields of the algebraic numbers $Qbar$. Then we introduce a new criterion for its validity (refining the author's previous criterion), addressing a problem of Bombieri. We show that Bombieri and Zannier's theorem, stating that the maximal abelian extension of a number field $K$ contained in $K^{(d)}$ has the Northcott property, follows very easily from this refined criterion. Here $K^{(d)}$ denotes the composite field of all extensions of $K$ of degree at most $d$.","PeriodicalId":338657,"journal":{"name":"Essential Number Theory","volume":"89 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139317805","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Upper and lower bounds are given for the number of rational points of bounded height on a double cover of projective space ramified over a Kummer surface.
给出了在Kummer曲面上分形的双投影空间上有界高度有理点的数目的上界和下界。
{"title":"A Diophantine problem about Kummer surfaces","authors":"W. Duke","doi":"10.2140/ent.2022.1.51","DOIUrl":"https://doi.org/10.2140/ent.2022.1.51","url":null,"abstract":"Upper and lower bounds are given for the number of rational points of bounded height on a double cover of projective space ramified over a Kummer surface.","PeriodicalId":338657,"journal":{"name":"Essential Number Theory","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130662021","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. Some upper bounds for the number of monogenizations of quartic orders are established by considering certain classical Diophantine equations, namely index form equations in quartic number fields, and cubic and quartic Thue equations.
{"title":"Quartic index form equations and monogenizations of quartic orders","authors":"S. Akhtari","doi":"10.2140/ent.2022.1.57","DOIUrl":"https://doi.org/10.2140/ent.2022.1.57","url":null,"abstract":". Some upper bounds for the number of monogenizations of quartic orders are established by considering certain classical Diophantine equations, namely index form equations in quartic number fields, and cubic and quartic Thue equations.","PeriodicalId":338657,"journal":{"name":"Essential Number Theory","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124708432","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Updated version of 2013 Arizona WInter School notes on modularity lifting theorems for for two-dimensional p-adic representations, using wherever possible arguments that go over to the n-dimensional (self-dual) case.
{"title":"Modularity lifting theorems","authors":"Toby Gee","doi":"10.2140/ent.2022.1.73","DOIUrl":"https://doi.org/10.2140/ent.2022.1.73","url":null,"abstract":"Updated version of 2013 Arizona WInter School notes on modularity lifting theorems for for two-dimensional p-adic representations, using wherever possible arguments that go over to the n-dimensional (self-dual) case.","PeriodicalId":338657,"journal":{"name":"Essential Number Theory","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126045337","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. In this mostly expository note, we explain a proof of Tate’s two conjectures [Tat65] for algebraic cycles of arbitrary codimension on certain products of elliptic curves and abelian surfaces over number fields.
{"title":"A note on Tate’s conjectures for abelian\u0000varieties","authors":"Chao Li, Wei Zhang","doi":"10.2140/ent.2022.1.41","DOIUrl":"https://doi.org/10.2140/ent.2022.1.41","url":null,"abstract":". In this mostly expository note, we explain a proof of Tate’s two conjectures [Tat65] for algebraic cycles of arbitrary codimension on certain products of elliptic curves and abelian surfaces over number fields.","PeriodicalId":338657,"journal":{"name":"Essential Number Theory","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127877097","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that the tame 'etale fundamental group of a connected normal finite type separated scheme remains invariant upon base change between algebraically closed fields of characteristic $p geq 0$.
{"title":"Invariance of the tame fundamental group under base change between algebraically closed fields","authors":"Aaron Landesman","doi":"10.2140/ent.2024.3.1","DOIUrl":"https://doi.org/10.2140/ent.2024.3.1","url":null,"abstract":"We show that the tame 'etale fundamental group of a connected normal finite type separated scheme remains invariant upon base change between algebraically closed fields of characteristic $p geq 0$.","PeriodicalId":338657,"journal":{"name":"Essential Number Theory","volume":"55 21","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141204169","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This is an expository paper, giving a simplified proof of the cubic case of the main conjecture for Vinogradov's mean value theorem.
本文是一篇说明性的论文，给出了维诺格拉多夫中值定理主猜想的三次情形的简化证明。
{"title":"The cubic case of Vinogradov’s mean value\u0000theorem","authors":"D. R. Heath-Brown","doi":"10.2140/ent.2022.1.1","DOIUrl":"https://doi.org/10.2140/ent.2022.1.1","url":null,"abstract":"This is an expository paper, giving a simplified proof of the cubic case of the main conjecture for Vinogradov's mean value theorem.","PeriodicalId":338657,"journal":{"name":"Essential Number Theory","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125766510","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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