.svg){kind=logo}
ON SOME PROPERTY OF $p$-ADIC FAMILIES OF POTENTIALLY SEMI-STABLE REPRESENTATIONS
Keywords:
potentially semi-stable representations, Weil group.DOI:
https://doi.org/10.17654/0972555523021Abstract
Let $p$ be an odd prime number and $\left\{\rho_k\right\}_{k \in \mathcal{K}}$ be a $p$-adic analytic family of potentially semi-stable representations $\rho_k$ with an arithmetic progression $\mathcal{K}$. Under some assumptions, we shall prove in this article that if there exists some $k \in \mathcal{K}$ such that the $W_{\mathbb{Q}_p}$ representation $D_{\mathrm{pst}}\left(\rho_k\right)$ associated to $\rho_k$ is reducible, then for any $k^{\prime} \in \mathcal{K}, D_{\mathrm{pst}}\left(\rho_{k^{\prime}}\right)$ is also reducible, where $W_{\mathbb{Q}_p}$ is the Weil group of the field $\mathbb{Q}_p$ of $p$-adic numbers.
Received: June 27, 2023
Accepted: August 1, 2023
References
S. Bosch, U. Güntzer and R. Remmert, Non-Archimedean Analysis, Grundlehren der Math. Wissenschaften 261, 1984.
K. Conrad, Exterior Powers.
https://kconrad.math.uconn.edu/blurbs/linmultialg/extmod.pdf.
J.-M. Fontaine, Le corps des périodes $p$-adiques, Exposé II, Astérisque 223 (1994), 59-101.
J.-M. Fontaine, Représentations $p$-adiques semi-stables, Exposé III, Astérisque 223 (1994), 113-184.
J.-M. Fontaine, Représentations $l$-adiques potentiellement semi-stables, Exposé VIII, Astérisque 223 (1994), 321-347.
I. Kaplansky, Elementary divisions and modules, Trans. Amer. Math. Soc. 66 (1949), 464-491.
H. Matsumura, Commutative Ring Theory, Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, 1986.
Downloads
Published
Issue
Section
License
Copyright (c) 2023 PUSHPA PUBLISHING HOUSE, PRAYAGRAJ, INDIA
This work is licensed under a Creative Commons Attribution 4.0 International License.
_________________________________
Attribution: Credit Pusha Publishing House as the original publisher, including title and author(s) if applicable.
Non-Commercial Use: For non-commercial purposes only. No commercial activities without explicit permission.
No Derivatives: Modifying or creating derivative works not allowed without written permission.
Contact Pusha Publishing House for more info or permissions.
