On Darmon's program for the generalized Fermat equation, I

被引:0
作者
Billerey, Nicolas [1 ,2 ]
Chen, Imin [3 ]
Dieulefait, Luis [4 ,5 ]
Freitas, Nuno [6 ]
机构
[1] Univ Clermont Auvergne, Lab Math Blaise Pascal, Campus Univ Cezeaux 3,Pl Vasarely, F-63178 Aubiere, France
[2] CNRS, Campus Univ Cezeaux 3,Pl Vasarely, F-63178 Aubiere, France
[3] Simon Fraser Univ, Dept Math, Burnaby, BC V5A 1S6, Canada
[4] Univ Barcelona UB, Dept Matemat & Informat, Gran Via De Les Corts Catalanes 585, Barcelona 08007, Spain
[5] Ctr Recerca Matemat CRM, Edif C,Campus Bellaterra, Bellaterra 08193, Spain
[6] CSIC, Inst Ciencias Matemat, Calle Nicolas Cabrera 13-15, Madrid 28049, Spain
来源
JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK | 2025年 / 2025卷 / 822期
关键词
ELLIPTIC-CURVES; SIGNATURE; MODULARITY; AUTOMORPHY; NUMBER;
D O I
10.1515/crelle-2025-0014
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
In 2000, Darmon described a program to study the generalized Fermat equation using modularity of abelian varieties of GL( 2) -type over totally real fields. The original approach was based on hard open conjectures, which have made it difficult to apply in practice. In this paper, building on the progress surrounding the modular method from the last two decades, we analyze and expand the current limits of this program by developing all the necessary ingredients to use Frey abelian varieties for new Diophantine applications. As an application, for all integers n >= 2 , we give a resolution of the generalized Fermat equation x (11) + y (11) = z (n )for solutions (a,b,c) such that a+b satisfies certain 2- or 11-adic conditions. We are also able to reduce the problem of solving x (5) + y( 5 )= z (p) to a weaker version of Darmon's "big image conjecture", thus completing a line of ideas suggested in his original program, and notably only needing the Cartan case of his conjecture.
引用
收藏
页码:107 / 146
页数:40
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