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

Abstract In 2000, Darmon described a program to study the generalized Fermat equation using modularity of abelian varieties of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>GL</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> \operatorname{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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> n\geq 2 , we give a resolution of the generalized Fermat equation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msup> <m:mi>x</m:mi> <m:mn>11</m:mn> </m:msup> <m:mo>+</m:mo> <m:msup> <m:mi>y</m:mi> <m:mn>11</m:mn> </m:msup> </m:mrow> <m:mo>=</m:mo> <m:msup> <m:mi>z</m:mi> <m:mi>n</m:mi> </m:msup> </m:mrow> </m:math> x^{11}+y^{11}=z^{n} for solutions <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>a</m:mi> <m:mo>,</m:mo> <m:mi>b</m:mi> <m:mo>,</m:mo> <m:mi>c</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> (a,b,c) such that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>a</m:mi> <m:mo>+</m:mo> <m:mi>b</m:mi> </m:mrow> </m:math> a+b satisfies certain 2- or 11-adic conditions. We are also able to reduce the problem of solving <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msup> <m:mi>x</m:mi> <m:mn>5</m:mn> </m:msup> <m:mo>+</m:mo> <m:msup> <m:mi>y</m:mi> <m:mn>5</m:mn> </m:msup> </m:mrow> <m:mo>=</m:mo> <m:msup> <m:mi>z</m:mi> <m:mi>p</m:mi> </m:msup> </m:mrow> </m:math> 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.