论文标题
使用Frey Arethelliptic曲线的签名$（P，P，5）$的模块化方法
A modular approach to Fermat equations of signature $(p,p,5)$ using Frey hyperelliptic curves
论文作者
论文摘要
在本文中，我们执行了Darmon计划的“广义费用方程” $$ x^n + y^n = z^5的步骤。尤其是$$，我们开发了必要的机械，以证明满足某些$ 2 $ - adiC和$ 5 $ - 亚种条件的解决方案的最佳限制，从该方法的角度来看。我们还将解决该方程式解决的问题减少到了“大图像猜想”，完成了他的原始程序中建议的一系列想法。 上面的方程是一个广义的Fermat方程的一个示例，预测的Frey Abelian品种具有尺寸$> 1 $，因此它代表了Darmon的程序有趣的测试用例。
In this paper we carry out the steps of Darmon's program for the generalized Fermat equation $$ x^n + y^n = z^5. $$ In particular, we develop the machinery necessary to prove an optimal bound on the exponent $n$ for solutions satisfying certain $2$-adic and $5$-adic conditions which are natural from the point of view of the method. We also reduce the problem of resolving this equation to a `big image conjecture', completing a line of ideas suggested in his original program. The above equation is an example of a generalized Fermat equation for which the predicted Frey abelian varieties have dimension $ > 1$ and thus it represents an interesting test case for Darmon's program.
