Alina Bucur, Alina Carmen Cojocaru, Matilde N. Lalín, Lillian B. Pierce
{"title":"方筛的几何推广,并应用于循环盖","authors":"Alina Bucur, Alina Carmen Cojocaru, Matilde N. Lalín, Lillian B. Pierce","doi":"10.1112/mtk.12180","DOIUrl":null,"url":null,"abstract":"<p>We formulate a general problem: Given projective schemes <math>\n <semantics>\n <mi>Y</mi>\n <annotation>$\\mathbb {Y}$</annotation>\n </semantics></math> and <math>\n <semantics>\n <mi>X</mi>\n <annotation>$\\mathbb {X}$</annotation>\n </semantics></math> over a global field <i>K</i> and a <i>K</i>-morphism η from <math>\n <semantics>\n <mi>Y</mi>\n <annotation>$\\mathbb {Y}$</annotation>\n </semantics></math> to <math>\n <semantics>\n <mi>X</mi>\n <annotation>$\\mathbb {X}$</annotation>\n </semantics></math> of finite degree, how many points in <math>\n <semantics>\n <mrow>\n <mi>X</mi>\n <mo>(</mo>\n <mi>K</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$\\mathbb {X}(K)$</annotation>\n </semantics></math> of height at most <i>B</i> have a pre-image under η in <math>\n <semantics>\n <mrow>\n <mi>Y</mi>\n <mo>(</mo>\n <mi>K</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$\\mathbb {Y}(K)$</annotation>\n </semantics></math>? This problem is inspired by a well-known conjecture of Serre on quantitative upper bounds for the number of points of bounded height on an irreducible projective variety defined over a number field. We give a nontrivial answer to the general problem when <math>\n <semantics>\n <mrow>\n <mi>K</mi>\n <mo>=</mo>\n <msub>\n <mi>F</mi>\n <mi>q</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <mi>T</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$K=\\mathbb {F}_q(T)$</annotation>\n </semantics></math> and <math>\n <semantics>\n <mi>Y</mi>\n <annotation>$\\mathbb {Y}$</annotation>\n </semantics></math> is a prime degree cyclic cover of <math>\n <semantics>\n <mrow>\n <mi>X</mi>\n <mo>=</mo>\n <msubsup>\n <mi>P</mi>\n <mi>K</mi>\n <mi>n</mi>\n </msubsup>\n </mrow>\n <annotation>$\\mathbb {X}=\\mathbb {P}_{K}^n$</annotation>\n </semantics></math>. Our tool is a new geometric sieve, which generalizes the polynomial sieve to a geometric setting over global function fields.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2022-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Geometric generalizations of the square sieve, with an application to cyclic covers\",\"authors\":\"Alina Bucur, Alina Carmen Cojocaru, Matilde N. Lalín, Lillian B. Pierce\",\"doi\":\"10.1112/mtk.12180\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We formulate a general problem: Given projective schemes <math>\\n <semantics>\\n <mi>Y</mi>\\n <annotation>$\\\\mathbb {Y}$</annotation>\\n </semantics></math> and <math>\\n <semantics>\\n <mi>X</mi>\\n <annotation>$\\\\mathbb {X}$</annotation>\\n </semantics></math> over a global field <i>K</i> and a <i>K</i>-morphism η from <math>\\n <semantics>\\n <mi>Y</mi>\\n <annotation>$\\\\mathbb {Y}$</annotation>\\n </semantics></math> to <math>\\n <semantics>\\n <mi>X</mi>\\n <annotation>$\\\\mathbb {X}$</annotation>\\n </semantics></math> of finite degree, how many points in <math>\\n <semantics>\\n <mrow>\\n <mi>X</mi>\\n <mo>(</mo>\\n <mi>K</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\mathbb {X}(K)$</annotation>\\n </semantics></math> of height at most <i>B</i> have a pre-image under η in <math>\\n <semantics>\\n <mrow>\\n <mi>Y</mi>\\n <mo>(</mo>\\n <mi>K</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\mathbb {Y}(K)$</annotation>\\n </semantics></math>? This problem is inspired by a well-known conjecture of Serre on quantitative upper bounds for the number of points of bounded height on an irreducible projective variety defined over a number field. We give a nontrivial answer to the general problem when <math>\\n <semantics>\\n <mrow>\\n <mi>K</mi>\\n <mo>=</mo>\\n <msub>\\n <mi>F</mi>\\n <mi>q</mi>\\n </msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>T</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$K=\\\\mathbb {F}_q(T)$</annotation>\\n </semantics></math> and <math>\\n <semantics>\\n <mi>Y</mi>\\n <annotation>$\\\\mathbb {Y}$</annotation>\\n </semantics></math> is a prime degree cyclic cover of <math>\\n <semantics>\\n <mrow>\\n <mi>X</mi>\\n <mo>=</mo>\\n <msubsup>\\n <mi>P</mi>\\n <mi>K</mi>\\n <mi>n</mi>\\n </msubsup>\\n </mrow>\\n <annotation>$\\\\mathbb {X}=\\\\mathbb {P}_{K}^n$</annotation>\\n </semantics></math>. Our tool is a new geometric sieve, which generalizes the polynomial sieve to a geometric setting over global function fields.</p>\",\"PeriodicalId\":18463,\"journal\":{\"name\":\"Mathematika\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2022-12-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematika\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/mtk.12180\",\"RegionNum\":3,\"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":"Mathematika","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/mtk.12180","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Geometric generalizations of the square sieve, with an application to cyclic covers
We formulate a general problem: Given projective schemes and over a global field K and a K-morphism η from to of finite degree, how many points in of height at most B have a pre-image under η in ? This problem is inspired by a well-known conjecture of Serre on quantitative upper bounds for the number of points of bounded height on an irreducible projective variety defined over a number field. We give a nontrivial answer to the general problem when and is a prime degree cyclic cover of . Our tool is a new geometric sieve, which generalizes the polynomial sieve to a geometric setting over global function fields.
期刊介绍:
Mathematika publishes both pure and applied mathematical articles and has done so continuously since its founding by Harold Davenport in the 1950s. The traditional emphasis has been towards the purer side of mathematics but applied mathematics and articles addressing both aspects are equally welcome. The journal is published by the London Mathematical Society, on behalf of its owner University College London, and will continue to publish research papers of the highest mathematical quality.