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Register a new user$mathcal{L}$-invariants and local-global compatibility for the group $mathrm{GL}_2/F$
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Authors
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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Let $rho_p$ be a $3$-dimensional $p$-adic semi-stable representation of $mathrm{Gal}(overline{mathbb{Q}_p}/mathbb{Q}_p)$ with Hodge-Tate weights $(0,1,2)$ (up to shift) and such that $N^2 e 0$ on $D_{mathrm{st}}(rho_p)$. When $rho_p$ comes from an automorphic representation $pi$ of $G(mathbb{A}_{F^+})$ (for a unitary group $G$ over a totally real field $F^+$ which is compact at infinite places and $mathrm{GL}_3$ at $p$-adic places), we show under mild genericity assumptions that the associated Hecke-isotypic subspaces of the Banach spaces of $p$-adic automorphic forms on $G(mathbb{A}_{F^+}^infty)$ of arbitrary fixed tame level contain (copies of) a unique admissible finite length locally analytic representation of $mathrm{GL}_3(mathbb{Q}_p)$ which only depends on and completely determines $rho_p$.
Let $L$ be a finite extension of $mathbb{Q}_p$, and $rho_L$ be an $n$-dimensional semi-stable non crystalline $p$-adic representation of $mathrm{Gal}_L$ with full monodromy rank. Via a study of Breuils (simple) $mathcal{L}$-invariants, we attach to $rho_L$ a locally $mathbb{Q}_p$-analytic representation $Pi(rho_L)$ of $mathrm{GL}_n(L)$, which carries the exact information of the Fontaine-Mazur simple $mathcal{L}$-invariants of $rho_L$. When $rho_L$ comes from an automorphic representation of $G(mathbb{A}_{F^+})$ (for a unitary group $G$ over a totally real filed $F^+$ which is compact at infinite places and $mathrm{GL}_n$ at $p$-adic places), we prove under mild hypothesis that $Pi(rho_L)$ is a subrerpresentation of the associated Hecke-isotypic subspaces of the Banach spaces of $p$-adic automorphic forms on $G(mathbb{A}_{F^+})$. In other words, we prove the equality of Breuils simple $mathcal{L}$-invariants and Fontaine-Mazur simple $mathcal{L}$-invariants.
Let $F_{wp}$ be a finite extension of $mathbb{Q}_p$. By considering partially de Rham families, we establish a Colmez-Greenberg-Stevens formula (on Fontaine-Mazur $mathcal{L}$-invariants) for (general) $2$-dimensional semi-stable non-crystalline $mathrm{Gal}(overline{mathbb{Q}_p}/F_{wp})$-representations. As an application, we prove local-global compatibility results for completed cohomology of quaternion Shimura curves, and in particular the equality of Fontaine-Mazur $mathcal{L}$-invariants and Breuils $mathcal{L}$-invariants, in critical case.
Let $p>2$ be a prime number, and $L$ be a finite extension of $mathbb{Q}_p$, we prove Breuils locally analytic socle conjecture for $mathrm{GL}_2(L)$, showing the existence of all the companion points on the definite (patched) eigenvariety. This work relies on infinitesimal R=T results for the patched eigenvariety and the comparison of (partially) de Rham families and (partially) Hodge-Tate families. This method allows in particular to find companion points of non-classical points.
We prove automorphy lifting results for certain essentially conjugate self-dual $p$-adic Galois representations $rho$ over CM imaginary fields $F$, which satisfy in particular that $p$ splits in $F$, and that the restriction of $rho$ on any decomposition group above $p$ is reducible with all the Jordan-Holder factors of dimension at most $2$. We also show some results on Breuils locally analytic socle conjecture in certain non-trianguline case. The main results are obtained by establishing an $R=mathbb{T}$-type result over the $mathrm{GL}_2(mathbb{Q}_p)$-ordinary families considered by Breuil-Ding.
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