This is a brief introduction to the study of growth in groups of Lie type, with $SL_2(mathbb{F}_q)$ and some of its subgroups as the key examples. They are an edited version of the notes I distributed at the Arizona Winter School in 2016. Those notes were, in turn, based in part on my survey in Bull. Am. Math. Soc. (2015) and in part on the notes for courses I gave on the subject in Cusco (AGRA) and G"ottingen.
{"title":"Growth and expansion in algebraic groups over\u0000 finite fields","authors":"H. Helfgott","doi":"10.1090/conm/740/14902","DOIUrl":"https://doi.org/10.1090/conm/740/14902","url":null,"abstract":"This is a brief introduction to the study of growth in groups of Lie type, with $SL_2(mathbb{F}_q)$ and some of its subgroups as the key examples. They are an edited version of the notes I distributed at the Arizona Winter School in 2016. Those notes were, in turn, based in part on my survey in Bull. Am. Math. Soc. (2015) and in part on the notes for courses I gave on the subject in Cusco (AGRA) and G\"ottingen.","PeriodicalId":344558,"journal":{"name":"Analytic Methods in Arithmetic\n Geometry","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116965226","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 describe how a systematic use the deep methods from $ell$-adic cohomology pioneered by Grothendieck and Deligne and further developed by Katz, Laumon allow to make progress on various classical questions from analytic number theory. This text is an extended version of a series of lectures given by the third and fourth authors during the 2016 Arizona Winter School.
{"title":"Lectures on applied ℓ-adic cohomology","authors":"É. Fouvry, E. Kowalski, P. Michel, W. Sawin","doi":"10.1090/conm/740/14903","DOIUrl":"https://doi.org/10.1090/conm/740/14903","url":null,"abstract":"We describe how a systematic use the deep methods from $ell$-adic cohomology pioneered by Grothendieck and Deligne and further developed by Katz, Laumon allow to make progress on various classical questions from analytic number theory. This text is an extended version of a series of lectures given by the third and fourth authors during the 2016 Arizona Winter School.","PeriodicalId":344558,"journal":{"name":"Analytic Methods in Arithmetic\n Geometry","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132137635","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 expository article we explore the relationship between Galois representations, motivic L-functions, Mumford-Tate groups, and Sato-Tate groups, and we give an explicit formulation of the Sato-Tate conjecture for abelian varieties as an equidistribution statement relative to the Sato-Tate group. We then discuss the classification of Sato-Tate groups of abelian varieties of dimension g <= 3 and compute some of the corresponding trace distributions. This article is based on a series of lectures presented at the 2016 Arizona Winter School held at the Southwest Center for Arithmetic Geometry.
{"title":"Sato-Tate distributions","authors":"Andrew V. Sutherland","doi":"10.1090/conm/740/14904","DOIUrl":"https://doi.org/10.1090/conm/740/14904","url":null,"abstract":"In this expository article we explore the relationship between Galois representations, motivic L-functions, Mumford-Tate groups, and Sato-Tate groups, and we give an explicit formulation of the Sato-Tate conjecture for abelian varieties as an equidistribution statement relative to the Sato-Tate group. We then discuss the classification of Sato-Tate groups of abelian varieties of dimension g <= 3 and compute some of the corresponding trace distributions. This article is based on a series of lectures presented at the 2016 Arizona Winter School held at the Southwest Center for Arithmetic Geometry.","PeriodicalId":344558,"journal":{"name":"Analytic Methods in Arithmetic\n Geometry","volume":"104 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115654664","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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