Semistable deformation rings in even Hodge-Tate weights
type: Item
instance of: scholarly article
title: Semistable deformation rings in even Hodge-Tate weights
zbMATH Open document ID: 1469.11157
author: Lucio Guerberoff
author: Chol Park
published in: Pacific Journal of Mathematics
publication date: 2019-05-03T00:00:00Z
review text: Let \(\bar{\rho} : \mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q}) \to \mathrm{GL}_2(\bar{\mathbb{F}}_p)\) be a continuous representation. \textit{C. Breuil} and \textit{A. Mézard} [Duke Math. J. 115, No. 2, 205--310 (2002; Zbl 1042.11030)] computed the irreducible components of the semistable deformation ring for \(\bar{\rho}\) with Hodge-Tate weights \((0,r)\), where \(0 < r < p-1\) is odd. They also computed the mod \(p\) reductions of semistable representations with these Hodge-Tate weights in terms of their parameters: a Frobenius eigenvalue \(\lambda\) and the \(\mathfrak{L}\)-invariant. The present paper carries out this calculation in the case that \(0 < r< p-1\) is even. The method is as in [loc. cit.]: to write down `strongly divisible modules' corresponding to Galois-stable lattices in semistable Galois representations with given parameters. This is a very involved calculation, even more so when \(r\) is even. One novelty is that the strongly divisible modules in the even case require coefficients that are not simply rational functions of \(\lambda\) and \(\mathfrak{L}\), but instead are limits of such functions satisfying certain quadratic equations.
reviewed by: Jack Shotton
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 11F33
zbMATH DE Number: 7050832
zbMATH Keywords: semistable representations
zbMATH Keywords: strongly divisible modules
zbMATH Keywords: Breuil modules
zbMATH Keywords: semistable deformation rings
MaRDI profile type: Publication
full work available at URL: pjm.2019.298.299
OpenAlex ID: W2931245211
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Wikidata QID: Q128078346
DOI: 10.2140/PJM.2019.298.299
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description: scientific article; zbMATH DE number 7050832