The dimension growth conjecture, polynomial in the degree and without logarithmic factors
The dimension growth conjecture, polynomial in the degree and without logarithmic factors
复制标题
维数增长猜想，次数多项式且无对数因子
DOI：
10.2140/ant.2020.14.2261
复制
发表时间：
2019
影响因子：
1.3
通讯作者：
K. Nguyen
中科院分区：
文献类型：
--
作者：
W. Castryck;R. Cluckers;Philip Dittmann;K. Nguyen
我们解决希思布朗和塞尔的尺寸增长猜想（证明Salberger），当度$d$增长。回想一下Salberger的维数增长结果给出了形式为O_{X，\varepoch}的界（B^{\dim X+\varepoch}）$对于在次数$d\geq 2$的${\martbb P}^n_{\martbb Q}$的任何整数子簇$X$上高度至多为$B$的有理点的数目，其中可以写$O_{d，n，\varepoch}$而不是$O_{X，$d\geq 4$。我们的主要贡献是删除的因素$B^\vareps $尽快$d \geq 5$，而不引入一个因素$\log B$，同时还获得多项式依赖于$d$的隐含常数。在$d$多项式工作允许我们给一个独立的和稍微简化的处理维度增长的程度$d \geq 16$，而在范围$5 \leq d \leq 15$，我们调用的结果由布朗宁，希斯布朗和Salberger。沿着的方式，我们改进了著名的边界，由于Besieri和皮拉的数目的整数点的有界高度仿射曲线和那些由沃尔什的数目的有理点的有界高度射影曲线。前一个改进导致轻微锐化最近的估计由于Bhargava，Shankar，谷口，索恩，Tsimerman和赵的大小$2$-扭转子群的类组的一个度$d$数域。我们的治疗建立在最近的工作由Salberger带来了许多素数在希斯-布朗的变种的行列式方法，并在最近的工作由沃尔什和Ellenberg-Venkatesh，谁带来的大小的定义多项式。我们还得到下界表明，一个不能做得比多项式依赖$d$。
We address Heath-Brown's and Serre's dimension growth conjecture (proved by Salberger), when the degree $d$ grows. Recall that Salberger's dimension growth results give bounds of the form $O_{X, \varepsilon} (B^{\dim X+\varepsilon})$ for the number of rational points of height at most $B$ on any integral subvariety $X$ of ${\mathbb P}^n_{\mathbb Q}$ of degree $d\geq 2$, where one can write $O_{d,n, \varepsilon}$ instead of $O_{X, \varepsilon}$ as soon as $d\geq 4$. Our main contribution is to remove the factor $B^\varepsilon$ as soon as $d \geq 5$, without introducing a factor $\log B$, while moreover obtaining polynomial dependence on $d$ of the implied constant. Working polynomially in $d$ allows us to give a self-contained and slightly simplified treatment of dimension growth for degree $d \geq 16$, while in the range $5 \leq d \leq 15$ we invoke results by Browning, Heath-Brown and Salberger. Along the way we improve the well-known bounds due to Bombieri and Pila on the number of integral points of bounded height on affine curves and those by Walsh on the number of rational points of bounded height on projective curves. The former improvement leads to a slight sharpening of a recent estimate due to Bhargava, Shankar, Taniguchi, Thorne, Tsimerman and Zhao on the size of the $2$-torsion subgroup of the class group of a degree $d$ number field. Our treatment builds on recent work by Salberger which brings in many primes in Heath-Brown's variant of the determinant method, and on recent work by Walsh and Ellenberg--Venkatesh, who bring in the size of the defining polynomial. We also obtain lower bounds showing that one cannot do better than polynomial dependence on $d$.
DOI：
10.48550/arxiv.1605.05916
发表时间：
2016
期刊：
arXiv e-prints
影响因子：
--
作者：
Cluckers Raf
通讯作者：
Cluckers Raf