We show that the largest prime factor of $n^2+1$ is infinitely often greater than $n^{1.279}$. This improves the result of de la Bret`eche and Drappeau (2019) who obtained this with $1.2182$ in place of $1.279.$ The main new ingredients in the proof are a new Type II estimate and using this estimate by applying Harman's sieve method. To prove the Type II estimate we use the bounds of Dehouillers and Iwaniec on linear forms of Kloosterman sums. We also show that conditionally on Selberg's eigenvalue conjecture the exponent $1.279$ may be increased to $1.312.$
我们证明了n^2+1$的最大素数因子通常无限大于n^{1.279}$。这改善了de la Bret ' eche和Drappeau(2019)的结果，他们用1.2182美元代替1.279美元获得了这个结果。证明中的主要新成分是一种新的II型估计，并通过哈曼筛子法使用这种估计。为了证明II型估计，我们使用了Kloosterman和的线性形式上的Dehouillers和Iwaniec的界。我们还证明了在Selberg特征值猜想的条件下，指数$1.279$可以增加到$1.312.$
{"title":"On the largest prime factor of $n^2+1$","authors":"Jori Merikoski","doi":"10.4171/jems/1216","DOIUrl":"https://doi.org/10.4171/jems/1216","url":null,"abstract":"We show that the largest prime factor of $n^2+1$ is infinitely often greater than $n^{1.279}$. This improves the result of de la Bret`eche and Drappeau (2019) who obtained this with $1.2182$ in place of $1.279.$ The main new ingredients in the proof are a new Type II estimate and using this estimate by applying Harman's sieve method. To prove the Type II estimate we use the bounds of Dehouillers and Iwaniec on linear forms of Kloosterman sums. We also show that conditionally on Selberg's eigenvalue conjecture the exponent $1.279$ may be increased to $1.312.$","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"5 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2022-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138503874","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove the last open case of the conjecture on the possible values of the first isotropy index of an anisotropic quadratic form over a field. It was initially stated by Detlev Hoffmann for fields of characteristic ̸= 2 and then extended to arbitrary characteristic by Burt Totaro. The initial statement was proven by the author in 2002. In characteristic 2, the case of a totally singular quadratic form was done by Stephen Scully in 2015 and the nonsingular case by Eric Primozic in early 2019.
{"title":"An ultimate proof of Hoffmann–Totaro’s conjecture","authors":"N. Karpenko","doi":"10.4171/jems/1211","DOIUrl":"https://doi.org/10.4171/jems/1211","url":null,"abstract":"We prove the last open case of the conjecture on the possible values of the first isotropy index of an anisotropic quadratic form over a field. It was initially stated by Detlev Hoffmann for fields of characteristic ̸= 2 and then extended to arbitrary characteristic by Burt Totaro. The initial statement was proven by the author in 2002. In characteristic 2, the case of a totally singular quadratic form was done by Stephen Scully in 2015 and the nonsingular case by Eric Primozic in early 2019.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"384 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2022-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74741293","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Erratum to “Uniform K-stability and asymptotics of energy functionals in Kähler geometry”","authors":"S. Boucksom, Tomoyuki Hisamoto, Mattias Jonsson","doi":"10.4171/jems/1215","DOIUrl":"https://doi.org/10.4171/jems/1215","url":null,"abstract":"","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"40 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2022-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81451536","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Local convergence of random planar graphs","authors":"Benedikt Stufler","doi":"10.4171/jems/1174","DOIUrl":"https://doi.org/10.4171/jems/1174","url":null,"abstract":"","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"10 2","pages":""},"PeriodicalIF":2.6,"publicationDate":"2021-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138503873","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We classify the irreducible representations of smooth, connected affine algebraic groups over a field, by tackling the case of pseudo-reductive groups. We reduce the problem of calculating the dimension for pseudo-split pseudo-reductive groups to the split reductive case and the pseudo-split pseudo-reductive commutative case. Moreover, we give the first results on the latter, including a rather complete description of the rank one case.
{"title":"Irreducible modules for pseudo-reductive groups","authors":"Michael Bate, David I. Stewart","doi":"10.4171/jems/1153","DOIUrl":"https://doi.org/10.4171/jems/1153","url":null,"abstract":"We classify the irreducible representations of smooth, connected affine algebraic groups over a field, by tackling the case of pseudo-reductive groups. We reduce the problem of calculating the dimension for pseudo-split pseudo-reductive groups to the split reductive case and the pseudo-split pseudo-reductive commutative case. Moreover, we give the first results on the latter, including a rather complete description of the rank one case.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"11 4","pages":""},"PeriodicalIF":2.6,"publicationDate":"2021-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138503872","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Bogdan Gheorghe, Daniel Isaksen, Achim Krause, N. Ricka
{"title":"$mathbb{C}$-motivic modular forms","authors":"Bogdan Gheorghe, Daniel Isaksen, Achim Krause, N. Ricka","doi":"10.4171/jems/1171","DOIUrl":"https://doi.org/10.4171/jems/1171","url":null,"abstract":"","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"19 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2021-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85334215","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Landau–Ginzburg mirror symmetry conjecture","authors":"Weiqiang He, Si Li, Yefeng Shen, Rachel Webb","doi":"10.4171/jems/1155","DOIUrl":"https://doi.org/10.4171/jems/1155","url":null,"abstract":"","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"5 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2021-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79568127","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Summability of the coefficients of a multilinear form","authors":"F. Bayart","doi":"10.4171/JEMS/1109","DOIUrl":"https://doi.org/10.4171/JEMS/1109","url":null,"abstract":"","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"8 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2021-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72713098","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the distribution of ${Cl}(K)[l^infty]$ for degree $l$ cyclic fields","authors":"P. Koymans, Carlo Pagano","doi":"10.4171/JEMS/1112","DOIUrl":"https://doi.org/10.4171/JEMS/1112","url":null,"abstract":"","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"12 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2021-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85013265","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In a little-known paper Hurwitz gave an infinite series representation of the class number for positive definite binary quadratic forms. In this paper we give a similar formula in the indefinite case. We also give a simple proof of Hurwitz’s formula and indicate some extensions.
{"title":"On a class number formula of Hurwitz","authors":"W. Duke, Ö. Imamoḡlu, Á. Tóth","doi":"10.4171/JEMS/1097","DOIUrl":"https://doi.org/10.4171/JEMS/1097","url":null,"abstract":"In a little-known paper Hurwitz gave an infinite series representation of the class number for positive definite binary quadratic forms. In this paper we give a similar formula in the indefinite case. We also give a simple proof of Hurwitz’s formula and indicate some extensions.","PeriodicalId":50003,"journal":{"name":"Journal of the European Mathematical Society","volume":"29 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2021-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75556520","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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