{"title":"Generic Diagonal Conic Bundles Revisited","authors":"Alexei N Skorobogatov, Efthymios Sofos","doi":"10.1093/qmath/haae022","DOIUrl":null,"url":null,"abstract":"We prove a stronger form of our previous result that Schinzel’s Hypothesis holds for 100% of n-tuples of integer polynomials satisfying the usual necessary conditions, where the primes represented by the polynomials are subject to additional constraints in terms of Legendre symbols, as well as upper and lower bounds. We establish the triviality of the Brauer group of generic diagonal conic bundles over the projective line. Finally, we give an explicit lower bound for the probability that diagonal conic bundles in certain natural families have rational points.","PeriodicalId":54522,"journal":{"name":"Quarterly Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Quarterly Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1093/qmath/haae022","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We prove a stronger form of our previous result that Schinzel’s Hypothesis holds for 100% of n-tuples of integer polynomials satisfying the usual necessary conditions, where the primes represented by the polynomials are subject to additional constraints in terms of Legendre symbols, as well as upper and lower bounds. We establish the triviality of the Brauer group of generic diagonal conic bundles over the projective line. Finally, we give an explicit lower bound for the probability that diagonal conic bundles in certain natural families have rational points.
期刊介绍:
The Quarterly Journal of Mathematics publishes original contributions to pure mathematics. All major areas of pure mathematics are represented on the editorial board.