论文标题
在达尔蒙（Darmon
On Darmon's program for the generalized Fermat equation, I
论文作者
论文摘要
在2000年，达尔蒙（Darmon）描述了一项计划，使用$ \ mathrm {gl} _2 $ -Type在完全真实领域的Abelian品种的模块化研究中进行了广义的Fermat方程。原始方法是基于坚硬的猜想，这使得在实践中很难应用。在本文中，基于过去二十年来模块化方法的进度，我们通过开发所有必要的成分来使用Frey Abelian品种用于新的Dioophantine应用程序来分析和扩大该程序的当前限制。特别是，除了在几乎完全通用的情况下，我们处理模块化方法的第五和最后一步。 作为一个应用程序，对于所有整数$ n \ geq 2 $，我们给出了通用的fermat方程$ x^{11} + y^{11} = z^n $的solutions $（a，b，c）$，以便$ a + a + a + a + b $满足一定的$ 2 $ - 或$ 11 $ - ad-adic-adic-adic-adic-adic-adic-adic-adic-adiC。 此外，开发的工具可以被视为解决达尔蒙（Darmon）原始程序中未治疗的难度的进步：即使假设“大图像”关于残留的galois表示形式的猜想，仍然需要找到一种方法来消除没有复杂乘法的Serre级别的Hilbert Newforms。实际上，我们能够减少将$ x^5 + y^5 = z^p $求解到达尔森的“大图像猜想”的问题，从而完成了他的原始程序中建议的一系列思想，特别是只需要他的猜想的货架。
In 2000, Darmon described a program to study the generalized Fermat equation using modularity of abelian varieties of $\mathrm{GL}_2$-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. In particular, we deal with all but the fifth and last step in the modular method for Fermat equations of signature $(r,r,p)$ in almost full generality. As an application, for all integers $n \geq 2$, 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. Moreover, the tools developed can be viewed as an advance in addressing a difficulty not treated in Darmon's original program: even assuming `big image' conjectures about residual Galois representations, one still needs to find a method to eliminate Hilbert newforms at the Serre level which do not have complex multiplication. In fact, we are able to reduce the problem of solving $x^5 + y^5 = z^p$ to 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.
