Diagonal quartic surfaces with a Brauer-Manin obstruction

被引：2

作者：

Santens, Tim ^{[1
]}

机构：

[1] Katholieke Univ Leuven, Dept wiskunde, Celestijnenlaan 200B, B-3001 Leuven, Belgium

来源：

COMPOSITIO MATHEMATICA | 2023年 / 159卷 / 04期

关键词：

quartic surfaces; rational points; Brauer-Manin obstruction; Artin L-series; HASSE PRINCIPLE; POINTS; NUMBER;

D O I：

10.1112/S0010437X22007916

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

In this paper we investigate the quantity of diagonal quartic surfaces a(0)X(0)(4) + a(1)X(1)(4) + a(2)X(2)(4) + a(3)X(3)(4) = 0 which have a Brauer-Manin obstruction to the Hasse principle. We are able to find an asymptotic formula for the quantity of such surfaces ordered by height. The proof uses a generalization of a method of Heath-Brown on sums over linked variables. We also show that there exists no uniform formula for a generic generator in this family.

引用

页码：659 / 710

页数：53

共 50 条

[41] The Brauer-Manin obstruction for subvarieties of abelian varieties over function fields [J].

Poonen, Bjorn ;

Voloch, Jose Felipe .

ANNALS OF MATHEMATICS, 2010, 171 (01) :511-532

[42] An example of a Brauer-Manin obstruction to weak approximation at a prime with good reduction [J].

Pagano, Margherita .

RESEARCH IN NUMBER THEORY, 2022, 8 (03)

[43] The Brauer group and the Brauer-Manin set of products of varieties [J].

Skorobogatov, Alexei N. ;

Zarhin, Yuri G. .

JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY, 2014, 16 (04) :749-768

[44] Evidence for the Dynamical Brauer-Manin Criterion [J].

Amerik, Ekaterina ;

Kurlberg, Par ;

Nguyen, Khoa D. ;

Towsley, Adam ;

Viray, Bianca ;

Voloch, Jose Felipe .

EXPERIMENTAL MATHEMATICS, 2016, 25 (01) :54-65

[45] Brauer-Manin obstruction for integral points of homogeneous spaces and representation by integral quadratic forms [J].

Colliot-Thelene, Jean-Louis ;

Xu, Fei .

COMPOSITIO MATHEMATICA, 2009, 145 (02) :309-363

[46] Pathologies of the Brauer–Manin obstruction [J].

Jean-Louis Colliot-Thélène ;

Ambrus Pál ;

Alexei N. Skorobogatov .

Mathematische Zeitschrift, 2016, 282 :799-817

[47] Integral Brauer-Manin obstructions for sums of two squares and a power [J].

Gundlach, Fabian .

JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES, 2013, 88 :599-618

[48] ORDER 5 BRAUER-MANIN OBSTRUCTIONS TO THE INTEGRAL HASSE PRINCIPLE ON LOG K3 SURFACES [J].

Lyczak, Julian .

ANNALES DE L INSTITUT FOURIER, 2023, 73 (02) :447-478

[49] Applications of the fibration method to the Brauer-Manin obstruction to the existence of zero-cycles on certain varieties [J].

Ieronymou, Evis .

INTERNATIONAL JOURNAL OF NUMBER THEORY, 2023, 19 (02) :409-417

[50] The triviality of Brauer-Manin obstruction for subvarieties of semi-abelian varieties over function fields of characteristic zero [J].

Sun, Chia-Liang .

MANUSCRIPTA MATHEMATICA, 2019, 160 (3-4) :385-389

← 1 2 3 4 5 →