Geometric generalizations of the square sieve, with an application to cyclic covers

IF 0.8 3区 数学 Q2 MATHEMATICS
Mathematika Pub Date : 2022-12-10 DOI:10.1112/mtk.12180
Alina Bucur, Alina Carmen Cojocaru, Matilde N. Lalín, Lillian B. Pierce
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引用次数: 2

Abstract

We formulate a general problem: Given projective schemes Y $\mathbb {Y}$ and X $\mathbb {X}$ over a global field K and a K-morphism η from Y $\mathbb {Y}$ to X $\mathbb {X}$ of finite degree, how many points in X ( K ) $\mathbb {X}(K)$ of height at most B have a pre-image under η in Y ( K ) $\mathbb {Y}(K)$ ? 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 K = F q ( T ) $K=\mathbb {F}_q(T)$ and Y $\mathbb {Y}$ is a prime degree cyclic cover of X = P K n $\mathbb {X}=\mathbb {P}_{K}^n$ . Our tool is a new geometric sieve, which generalizes the polynomial sieve to a geometric setting over global function fields.

方筛的几何推广,并应用于循环盖
我们公式化了一个一般问题:给定全局域K上的投影方案Y$\mathbb{Y}$和X$\mathbb{X}$,以及从Y$\math bb{Y}$到X$\math BB{X}$的有限度K态射η,在Y(K)$\mathbb{Y}(K)$中,至多有多少个高度为B的点在η下具有预像?这个问题的灵感来自Serre关于在数域上定义的不可约投影变种上有界高度的点的数量上限的一个众所周知的猜想。当K=Fq(T)$K=\mathbb时,我们给出了一般问题的一个非平凡答案{F}_q(T) $和Y$\mathbb{Y}$是X=PKn$\mathbb{X}=\mathbb的素数循环覆盖{P}_{K} ^n$。我们的工具是一个新的几何筛,它将多项式筛推广到全局函数域上的几何设置。
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来源期刊
Mathematika
Mathematika MATHEMATICS, APPLIED-MATHEMATICS
CiteScore
1.40
自引率
0.00%
发文量
60
审稿时长
>12 weeks
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.
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