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 39 条

[1] Failures of weak approximation in families [J].

Bright, M. J. ;

Browning, T. D. ;

Loughran, D. .

COMPOSITIO MATHEMATICA, 2016, 152 (07) :1435-1475

[2]

Bright M. J., 2002, THESIS U CAMBRIDGE

[3] Obstructions to the Hasse principle in families [J].

Bright, Martin .

MANUSCRIPTA MATHEMATICA, 2018, 157 (3-4) :529-550

[4] Bad reduction of the Brauer Manin obstruction [J].

Bright, Martin .

JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES, 2015, 91 :643-666

[5] The Brauer-Manin obstruction on a general diagonal quartic surface [J].

Bright, Martin .

ACTA ARITHMETICA, 2011, 147 (03) :291-302

[6]

Colliot-Th?l?ne J.-L., 2021, BRAUER GROTHENDIECK, V71

[7] 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

[8]

Deligne P., 1980, II. Publ. Math. IHS, V49, P137, DOI [10.1007/BF02684780, DOI 10.1007/BF02684780]

[9]

Fogels E., 1962, ACTA ARITH, V8, P307

[10]

Gille P., 2006, Cambridge Studies in Advanced Mathematics, DOI DOI 10.1017/CBO9780511607219

← 1 2 3 4 →