Rational points on a class of cubic hypersurfaces

IF 1 3区数学 Q1 MATHEMATICS

Forum Mathematicum Pub Date : 2024-07-26 DOI:10.1515/forum-2023-0394

Yujiao Jiang, Tingting Wen, Wenjia Zhao

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引用次数: 0

Abstract

Let r ⩾ 3

r\geqslant 3

be an integer and 𝑄 any positive definite quadratic form in 𝑟 variables. We establish asymptotic formulae with power-saving error terms for the number of rational points of bounded height on singular hypersurfaces S Q

S_{Q}

defined by x 3 = Q ⁢ ( y 1 , … , y r ) ⁢ z

x^{3}=Q(y_{1},\dots,y_{r})z

. This confirms Manin’s conjecture for any S Q

S_{Q}

. Our proof is based on analytic methods, and uses some estimates for character sums and moments of 𝐿-functions. In particular, one of the ingredients is Siegel’s mass formula in the argument for the case r = 3

r=3

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一类立方超曲面上的有理点

设 r ⩾ 3 r\geqslant 3 为整数，𝑄 为 𝑟 变量中的任意正定二次型。我们用省力误差项建立了奇异超曲面 S Q S_{Q} 上有界高的有理点数的渐近公式，定义为 x 3 = Q ( y 1 , ... , y r ) z x^{3}=Q(y_{1},\dots,y_{r})z 。这证实了马宁对任意 S Q S_{Q} 的猜想。我们的证明基于分析方法，并使用了𝐿 函数的特征和与矩的一些估计值。特别是，其中一个要素是西格尔的质量公式，它是针对 r = 3 r=3 情况的论证。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Forum Mathematicum 数学-数学

CiteScore

1.60

自引率

0.00%

发文量

审稿时长

6-12 weeks

期刊介绍： Forum Mathematicum is a general mathematics journal, which is devoted to the publication of research articles in all fields of pure and applied mathematics, including mathematical physics. Forum Mathematicum belongs to the top 50 journals in pure and applied mathematics, as measured by citation impact.