Abstract:We prove new cases of Fontaine-Mazur conjecture on two-dimensional Galois representations over $\mathbb {Q}$ when the residual representation is reducible. Our approach is via a semi-simple local-global compatibility of the completed cohomology and a Taylor-Wiles patching argument for the completed homology in this case. As a key input, we generalize the work of Skinner-Wiles in the ordinary case. In addition, we also treat the residually irreducible case at the end of the paper. Combining with people’s earlier work, we can prove the Fontaine-Mazur conjecture completely in the regular case when $p\geq 5$.

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中文翻译：

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剩余可约情况下的 Fontaine-Mazur 猜想

摘要：当残差表示可约时，我们证明了在 $\mathbb {Q}$ 上的二维伽罗瓦表示上的 Fontaine-Mazur 猜想的新案例。我们的方法是通过完整的上同调的半简单局部-全局兼容性和在这种情况下完成的同调的 Taylor-Wiles 修补参数。作为关键输入，我们在普通情况下概括了 Skinner-Wiles 的工作。此外，我们还在论文末尾处理了剩余不可约的情况。结合人们早期的工作，我们可以在$p\geq 5$ 的正则情况下完全证明Fontaine-Mazur 猜想。

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