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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Simple ${\mathcal L}$-invariants for GL$_n$


Author: Yiwen Ding
Journal: Trans. Amer. Math. Soc. 372 (2019), 7993-8042
MSC (2010): Primary 11S80; Secondary 22D12
DOI: https://doi.org/10.1090/tran/7859
Published electronically: July 2, 2019
MathSciNet review: 4029688
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Abstract: Let $L$ be a finite extension of ${\mathbb Q}_p$, and $\rho _L$ be an $n$-dimensional semistable noncrystalline $p$-adic representation of ${\mathrm {Gal}}_L$ with full monodromy rank. Via a study of Breuil’s (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 field $F^+$ 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 subrepresentation 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 Breuil’s simple ${\mathcal L}$-invariants and Fontaine–Mazur simple $L$-invariants.


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Additional Information

Yiwen Ding
Affiliation: BICMR, Peking University, 5 Yiheyuan Road, Haidian District, Beijing 100871, People’s Republic of China
MR Author ID: 984199
Email: yiwen.ding@bicmr.pku.edu.cn

Received by editor(s): August 31, 2018
Received by editor(s) in revised form: March 4, 2019
Published electronically: July 2, 2019
Additional Notes: This work was supported by EPSRC Grant No. EP/L025485/1 and by Grant No. 8102600240 from BICMR
Article copyright: © Copyright 2019 American Mathematical Society