# On the reductions of certain two-dimensional crystabelline representations

On the reductions of certain two-dimensional crystabelline representations Crystabelline representations are representations of the absolute Galois group GQp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\smash {G_{\mathbb {Q}_p}}$$\end{document} over Q¯p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\smash {\smash {\overline{\mathbb {Q}}_p}}$$\end{document} that become crystalline on GF\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\smash {G_{F}}$$\end{document} for some abelian extension F/Qp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\smash {F/\mathbb {Q}_p}$$\end{document}. Their relation to modular forms is that the representation associated with a finite slope newform of level divisible by p2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\smash {p^2}$$\end{document} is crystabelline. In this article, we study the connection between the slopes of two-dimensional crystabelline representations and the reducibility of their modulo p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\smash {p}$$\end{document} reductions. This question is inspired by a theorem by Buzzard and Kilford which implies that the slopes on the boundary of the 2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\smash {2}$$\end{document}-adic eigencurve of tame level 1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\smash {1}$$\end{document} are integers (and in arithmetic progression); an analogous theorem by Roe which says that the same is true for the 3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\smash {3}$$\end{document}-adic eigencurve; Coleman’s halo conjecture and the ghost conjecture which give predictions about the slopes on the p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\smash {p}$$\end{document}-adic eigencurve of general tame level; and Hodge theoretic conjectures by Breuil, Buzzard, Emerton, and Gee which indicate that there is a connection between all of these and the slopes of two-dimensional crystabelline representations whose reductions modulo p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\smash {p}$$\end{document} are reducible. We prove that the reductions of certain two-dimensional crystabelline representations with slopes in (0,p-12)\Z\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\smash {(0,\frac{p-1}{2})\backslash \mathbb {Z}}$$\end{document} are usually irreducible, with the exception of a small region where the slopes are half-integers and reducible representations do occur. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Research in the Mathematical Sciences Springer Journals

# On the reductions of certain two-dimensional crystabelline representations

, Volume 8 (1) – Feb 5, 2021
50 pages

/lp/springer-journals/on-the-reductions-of-certain-two-dimensional-crystabelline-cMEbg02VPk
Publisher
Springer Journals
eISSN
2197-9847
DOI
10.1007/s40687-020-00231-6
Publisher site
See Article on Publisher Site

### Abstract

Crystabelline representations are representations of the absolute Galois group GQp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\smash {G_{\mathbb {Q}_p}}$$\end{document} over Q¯p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\smash {\smash {\overline{\mathbb {Q}}_p}}$$\end{document} that become crystalline on GF\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\smash {G_{F}}$$\end{document} for some abelian extension F/Qp\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\smash {F/\mathbb {Q}_p}$$\end{document}. Their relation to modular forms is that the representation associated with a finite slope newform of level divisible by p2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\smash {p^2}$$\end{document} is crystabelline. In this article, we study the connection between the slopes of two-dimensional crystabelline representations and the reducibility of their modulo p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\smash {p}$$\end{document} reductions. This question is inspired by a theorem by Buzzard and Kilford which implies that the slopes on the boundary of the 2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\smash {2}$$\end{document}-adic eigencurve of tame level 1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\smash {1}$$\end{document} are integers (and in arithmetic progression); an analogous theorem by Roe which says that the same is true for the 3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\smash {3}$$\end{document}-adic eigencurve; Coleman’s halo conjecture and the ghost conjecture which give predictions about the slopes on the p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\smash {p}$$\end{document}-adic eigencurve of general tame level; and Hodge theoretic conjectures by Breuil, Buzzard, Emerton, and Gee which indicate that there is a connection between all of these and the slopes of two-dimensional crystabelline representations whose reductions modulo p\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\smash {p}$$\end{document} are reducible. We prove that the reductions of certain two-dimensional crystabelline representations with slopes in (0,p-12)\Z\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\smash {(0,\frac{p-1}{2})\backslash \mathbb {Z}}$$\end{document} are usually irreducible, with the exception of a small region where the slopes are half-integers and reducible representations do occur.

### Journal

Research in the Mathematical SciencesSpringer Journals

Published: Feb 5, 2021