SOLUTIONS OF THE CUBIC FERMAT EQUATION IN QUADRATIC FIELDS

被引：2

作者：

Jones, Marvin ^{[1
]}

Rouse, Jeremy ^{[2
]}

机构：

[1] Univ S Carolina, Dept Math, Columbia, SC 29208 USA

[2] Wake Forest Univ, Dept Math, Winston Salem, NC 27109 USA

来源：

INTERNATIONAL JOURNAL OF NUMBER THEORY | 2013年 / 9卷 / 06期

关键词：

Elliptic curve; Fermat equation; modular form; half-integer weight modular form; Waldspurger's theorem; MODULAR-FORMS; ELLIPTIC-CURVES; DERIVATIVES; POINTS;

D O I：

10.1142/S1793042113500449

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We give necessary and sufficient conditions on a squarefree integer d for there to be non-trivial solutions to x(3) + y(3) = z(3) in Q(root d), conditional on the Birch and Swinnerton-Dyer conjecture. These conditions are similar to those obtained by J. Tunnell in his solution to the congruent number problem.

引用

页码：1579 / 1591

页数：13

共 50 条

[1] THE FERMAT EQUATION OVER QUADRATIC FIELDS
HAO, FH
PARRY, CJ
JOURNAL OF NUMBER THEORY, 1984, 19 (01) : 115 - 130
[2] Solutions of the cubic Fermat equation in ring class fields of imaginary quadratic fields (as periodic points of a 3-adic algebraic function)
Morton, Patrick
INTERNATIONAL JOURNAL OF NUMBER THEORY, 2016, 12 (04) : 853 - 902
[3] The quartic Fermat equation in Hilbert class fields of imaginary quadratic fields
Lynch, Rodney
Morton, Patrick
INTERNATIONAL JOURNAL OF NUMBER THEORY, 2015, 11 (06) : 1961 - 2017
[4] Solutions of cubic equations in quadratic fields
Chakraborty, K
Kulkarni, MV
ACTA ARITHMETICA, 1999, 89 (01) : 37 - 43
[5] On Fermat’s equation over some quadratic imaginary number fields
Ţurcaş G.C.
Research in Number Theory, 2018, 4 (2)
[6] The Fermat cubic and quartic curves over cyclic fields
Bremner, Andrew
Choudhry, Ajai
PERIODICA MATHEMATICA HUNGARICA, 2020, 80 (02) : 147 - 157
[7] The Fermat cubic and quartic curves over cyclic fields
Andrew Bremner
Ajai Choudhry
Periodica Mathematica Hungarica, 2020, 80 : 147 - 157
[8] On a cubic equation and a Jensen-quadratic equation
Bae, Jae-Hyeong
Park, Won-Gil
ABSTRACT AND APPLIED ANALYSIS, 2007,
[9] THE GENERALIZED FERMAT EQUATION IN FUNCTION-FIELDS
BOMBIERI, E
MUELLER, J
JOURNAL OF NUMBER THEORY, 1991, 39 (03) : 339 - 350
[10] Peaked Traveling Wave Solutions to a Generalized Novikov Equation with Cubic and Quadratic Nonlinearities
李国芹
屈长征
Communications in Theoretical Physics, 2014, 61 (06) : 742 - 750

← 1 2 3 4 5 →