关于剩余二面体表示自同构的注记
Remarks on Automorphy of Residually Dihedral Representations
摘要：
证明了几何表示$ρ:G_F \rightarrow GL_2(\mathcal{O})$的自同构提升结果，其中$F$是全实数域，$\mathcal{O}$是$\mathbb{Q}_p$的有限扩张的整数环，$p$是奇素数，使得剩余表示$\barρ$是全奇的，并且是由$F(ζ_p)/F$的二次子域$K$的绝对伽罗瓦群的一个特征标导出的.这样的表示不符合泰勒-怀尔斯假设，证明自同构的修补技术也不起作用。我们应用这一点的椭圆曲线$E$的$F$上的自同构，当$E$没有$F$合理的7-等距和这样的图像$G_F$作用于$E[7]$规范化分裂Cartan子群的$GL_2(\mathbb{F}_7)$。
英文摘要：
We prove automorphy lifting results for geometric representations $ρ:G_F \rightarrow GL_2(\mathcal{O})$, with $F$ a totally real field, and $\mathcal{O}$ the ring of integers of a finite extension of $\mathbb{Q}_p$ with $p$ an odd prime, such that the residual representation $\barρ$ is totally odd and induced from a character of the absolute Galois group of the quadratic subfield $K$ of $F(ζ_p)/F$. Such representations fail the Taylor-Wiles hypothesis and the patching techniques to prove automorphy do not work. We apply this to automorphy of elliptic curves $E$ over $F$, when $E$ has no $F$ rational 7-isogeny and such that the image of $G_F$ acting on $E[7]$ normalizes a split Cartan subgroup of $GL_2(\mathbb{F}_7)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
