A RESULT ON THE EQUATION xp + yp = zr USING FREY ABELIAN VARIETIES

被引：4

作者：

Billerey, Nicolas ^{[1
,2
]}

Chen, Imin ^{[3
]}

Dieulefait, Luis ^{[4
]}

Freitas, Nuno ^{[5
]}

机构：

[1] Univ Blaise Pascal, Univ Clermont Auvergne, Lab Math, BP 10448, F-63000 Clermont Ferrand, France

[2] CNRS, UMR 6620, LM, F-63171 Aubiere, France

[3] Simon Fraser Univ, Dept Math, Burnaby, BC V5A 1S6, Canada

[4] Univ Barcelona, Dept Algebra & Geometria, GV De Les Corts Catalanes 585, E-08007 Barcelona, Spain

[5] Univ British Columbia, Dept Math, Vancouver, BC V6T 1Z2, Canada

来源：

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY | 2017年 / 145卷 / 10期

基金：

加拿大自然科学与工程研究理事会;

关键词：

STABLE ELLIPTIC-CURVES; ABELIAN-VARIETIES; REPRESENTATIONS; CRITERIA; POINTS; FIELDS;

D O I：

10.1090/proc/13475

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

We prove a Diophantine result on generalized Fermat equations of the form x(p) + y(p) = z(r) which for the first time requires the use of Frey abelian varieties of dimension >= 2 in Darmon's program. More precisely, for r >= 5 a regular prime we prove that there exists a constant C(r) such that for every prime number p > C(r) the equation x(p) + y(p) = z(r) has no non-trivial primitive integer solutions (a, b, c) satisfying r vertical bar ab and 2 vertical bar ab. For the proof, we complement Darmon's ideas in a particular case by providing an irreducibility criterion for the mod p representations attached to certain families of abelian varieties of GL2-type over totally real fields.

引用

页码：4111 / 4117

页数：7

共 12 条

[1] IRREDUCIBLE CRITERIA FOR THE REPRESENTATION OF ELLIPTIC CURVES [J].