On the proportion of locally soluble superelliptic curves

被引：0

作者：

Beneish, Lea ^{[1
]}

Keyes, Christopher ^{[2
]}

机构：

[1] Univ Calif Berkeley, Dept Math, 970 Evans Hall, Berkeley, CA 94720 USA

[2] Emory Univ, Dept Math, 400 Dowman Dr, Atlanta, GA 30322 USA

来源：

FINITE FIELDS AND THEIR APPLICATIONS | 2023年 / 85卷

关键词：

Superelliptic curves; Local solubility; HYPERELLIPTIC CURVES; RATIONAL-POINTS; VARIETIES; NUMBER;

D O I：

10.1016/j.ffa.2022.102128

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

We investigate the proportion of superelliptic curves that have a Op point for every place p of O. We show that this proportion is positive and given by the product of local densities, we provide lower bounds for this proportion in general, and for superelliptic curves of the form y3 = f (x, z) for an integral binary form f of degree 6, we determine this proportion to be 96.94%. More precisely, we give explicit rational functions in p for the proportion of such curves over Zp having a Op-point. (c) 2022 Elsevier Inc. All rights reserved.

引用

页数：85

共 33 条

[1] The case for superelliptic curves
Beshaj, L.
Shaska, T.
Zhupa, E.
ADVANCES ON SUPERELLIPTIC CURVES AND THEIR APPLICATIONS, 2015, 41 : 1 - 14
[2] Weierstrass points of superelliptic curves
Shor, C.
Shaska, T.
ADVANCES ON SUPERELLIPTIC CURVES AND THEIR APPLICATIONS, 2015, 41 : 15 - 46
[3] A POSITIVE PROPORTION OF LOCALLY SOLUBLE HYPERELLIPTIC CURVES OVER Q HAVE NO POINT OVER ANY ODD DEGREE EXTENSION
Bhargava, Manjul
Gross, Benedict H.
Wang, Xiaoheng
Dokchitser, Tim
Dokchitser, Vladimir
JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY, 2017, 30 (02) : 451 - 493
[4] Tropical superelliptic curves
Brandt, Madeline
Helminck, Paul Alexander
ADVANCES IN GEOMETRY, 2020, 20 (04) : 527 - 551
[5] RATIONAL POINTS ON THREE SUPERELLIPTIC CURVES
Shen, Zhongyan
Cai, Tianxin
BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY, 2012, 85 (01) : 105 - 113
[6] Theta functions of superelliptic curves
Beshaj, Lubjana
Elezi, Artur
Shaska, Tony
ADVANCES ON SUPERELLIPTIC CURVES AND THEIR APPLICATIONS, 2015, 41 : 47 - 69
[7] On the field of moduli of superelliptic curves
Hidalgo, Ruben
Shaska, Tony
HIGHER GENUS CURVES IN MATHEMATICAL PHYSICS AND ARITHMETIC GEOMETRY, 2018, 703 : 47 - 62
[8] On the arithmetic of twists of superelliptic curves
Chang, Sungkon
ACTA ARITHMETICA, 2006, 124 (04) : 371 - 389
[9] Galois representations of superelliptic curves
Pacetti, Ariel
Villanueva, Angel
GLASGOW MATHEMATICAL JOURNAL, 2023, 65 (02) : 356 - 382
[10] Triangular Schlesinger systems and superelliptic curves
Dragovic, Vladimir
Gontsov, Renat
Shramchenko, Vasilisa
PHYSICA D-NONLINEAR PHENOMENA, 2021, 424

← 1 2 3 4 →