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تسجيل مستخدم جديد$mathcal{L}$-invariants and local-global compatibility for the group $mathrm{GL}_2/F$
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Yiwen Ding
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Let $F$ be a totally real number field, $wp$ a place of $F$ above $p$. Let $rho$ be a $2$-dimensional $p$-adic representation of $mathrm{Gal}(bar{F}/F)$ which appears in the etale cohomology of quaternion Shimura curves (thus $rho$ is associated to Hilbert eigenforms). When the restriction $rho_{wp}:=rho|_{D_{wp}}$ at the decomposition group of $wp$ is semi-stable non-crystalline, one can associate to $rho_{wp}$ the so-called Fontaine-Mazur $mathcal{L}$-invariants, which are however invisible in the classical local Langlands correspondence. In this paper, we prove one can find these $mathcal{L}$-invariants in the completed cohomology group of quaternion Shimura curves, which generalizes some of Breuils results in $mathrm{GL}_2/mathbb{Q}$-case.
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