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

[21] Insufficiency of the Brauer-Manin obstruction applied to etale covers [J].

Poonen, Bjorn .

ANNALS OF MATHEMATICS, 2010, 171 (03) :2157-2169

[22] The Brauer-Manin obstruction for sections of the fundamental group [J].

Stix, Jakob .

JOURNAL OF PURE AND APPLIED ALGEBRA, 2011, 215 (06) :1371-1397

[23] Non-invariance of Weak Approximation with Brauer-Manin Obstruction [J].

Wu, Han .

ALGEBRA COLLOQUIUM, 2024, 31 (04) :595-608

[24] Brauer-Manin obstruction for zero-cycles on certain varieties [J].

Ieronymou, Evis .

JOURNAL DE THEORIE DES NOMBRES DE BORDEAUX, 2023, 35 (01) :151-166

[25] Brauer-Manin obstructions on degree 2 K3 surfaces [J].

Corn, Patrick ;

Nakahara, Masahiro .

RESEARCH IN NUMBER THEORY, 2018, 4

[26] Transcendental Brauer-Manin obstruction on a pencil of elliptic curves [J].

Wittenberg, O .

ARITHMETIC OF HIGHER-DIMENSIONAL ALGEBRAIC VARIETIES, 2004, 226 :259-267

[27] EMBEDDING FUNCTOR FOR CLASSICAL GROUPS AND BRAUER-MANIN OBSTRUCTION [J].

Bayer-Fluckiger, Eva ;

Lee, Ting-Yu ;

Parimala, Raman .

PACIFIC JOURNAL OF MATHEMATICS, 2015, 279 (1-2) :87-100

[28] On the Brauer-Manin obstruction for zero-cycles on curves [J].

Eriksson, Dennis ;

Scharaschkin, Victor .

ACTA ARITHMETICA, 2008, 135 (02) :99-110

[29] STRONG APPROXIMATION WITH BRAUER-MANIN OBSTRUCTION FOR TORIC VARIETIES [J].

Cao, Yang ;

Xu, Fei .

ANNALES DE L INSTITUT FOURIER, 2018, 68 (05) :1879-1908

[30] THE BRAUER-MANIN OBSTRUCTION FOR CONSTANT CURVES OVER GLOBAL FUNCTION FIELDS [J].

Brendan, Creutz ;

Felipe, Voloch Jose .

ANNALES DE L INSTITUT FOURIER, 2022, 72 (01) :43-58

← 1 2 3 4 5 →