The average number of integral points on the congruent number curves

被引：0
作者：
Chan, Stephanie [1 ,2 ]
机构：
[1] Univ Michigan, Dept Math, 530 Church St, Ann Arbor, MI 48109 USA
[2] IST Austria, Campus 1, A-3400 Klosterneuburg, Austria
来源：
ADVANCES IN MATHEMATICS | 2024年 / 457卷
关键词：
Elliptic curve; Quadratic twist; Integral point; SELMER GROUPS; THEOREM; SIZE;
D O I：
10.1016/j.aim.2024.109946
中图分类号：
O1 [数学];
学科分类号：
0701 ; 070101 ;
摘要：
We show that the total number of non-torsion integral points on the elliptic curves epsilon(D): y(2) = x(3) -D(2)x, where D ranges over positive squarefree integers less than N , is O ( N (log N ) (- 1/4 +E )). The proof involves a discriminant-lowering procedure on integral binary quartic forms and an application of HeathBrown's method on estimating the average size of the 2-Selmer groups of the curves in this family. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).
引用
收藏
页数：31
相关论文
共 50 条
[31]   ON θ-CONGRUENT NUMBERS OVER REAL NUMBER FIELDS [J].
Das, Shamik ;
Saikia, Anupam .
BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY, 2021, 103 (02) :218-229
[32]   On the class number divisibility of pairs of quadratic fields obtained from points on elliptic curves [J].
Iizuka, Yoshichika ;
Konomi, Yutaka ;
Nakano, Shin .
JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN, 2016, 68 (02) :899-915
[33]   Uniformity of stably integral points on elliptic curves [J].
Abramorich, D .
INVENTIONES MATHEMATICAE, 1997, 127 (02) :307-317
[34]   Uniformity of stably integral points on elliptic curves [J].
Dan Abramovich .
Inventiones mathematicae, 1997, 127 :307-317
[35]   Invitation to integral and rational points on curves and surfaces [J].
Das, Pranabesh ;
Turchet, Amos .
RATIONAL POINTS, RATIONAL CURVES, AND ENTIRE HOLOMORPHIC CURVES ON PROJECTIVE VARIETIES, 2015, 654 :53-73
[36]   Holomorphic curves and integral points off divisors [J].
Junjiro Noguchi ;
Jörg Winkelmann .
Mathematische Zeitschrift, 2002, 239 :593-610
[37]   Bounding the j-invariant of integral points on certain modular curves [J].
Sha, Min .
INTERNATIONAL JOURNAL OF NUMBER THEORY, 2014, 10 (06) :1545-1551
[38]   On the number of points in general position in the plane [J].
Balogh, Jozsef ;
Solymosi, Jozsef .
DISCRETE ANALYSIS, 2018,
[39]   The number of extreme points of tropical polyhedra [J].
Allamigeon, Xavier ;
Gaubert, Stephane ;
Katz, Ricardo D. .
JOURNAL OF COMBINATORIAL THEORY SERIES A, 2011, 118 (01) :162-189
[40]   Bounds for the size of integral points on curves of genus zero [J].
D. Poulakis .
Acta Mathematica Hungarica, 2001, 93 :327-346
12345