Deforming Galois representations and the conjectures of Serre and Fontaine-Mazur.
type: Item
instance of: scholarly article
title: Deforming Galois representations and the conjectures of Serre and Fontaine-Mazur.
zbMATH Open document ID: 1076.11035
DOI: 10.2307/3597186
author: Ravi Ramakrishna
published in: Annals of Mathematics. Second Series
publication date: 2003-03-09T00:00:00Z
review text: The author achieves a major breakthrough in the theory of deformations of \(\bmod p\) representations of the absolute Galois group \(G_{\mathbb Q}\) of \(\mathbb Q\). He proves that a continuous representation \(\rho: G_{\mathbb Q} \rightarrow \text{GL}_2(\mathbb F)\), with \(\mathbb F\) a finite field of characteristic \(p>5\), and with some mild assumptions on the behaviour of \(\rho\) when restricted to a decomposition group at \(p\) (we refer to this as the local behaviour of \(\rho\) at \(p\)), lifts to a representation \(\tilde{\rho}: G_ {\mathbb Q} \rightarrow \text{GL}_2(W(\mathbb F))\), with \(W(F)\) the Witt vectors of \(\mathbb F\), that is ramified at finitely many primes. For the precise assumptions on the local behaviour of \(\rho\) at \(p\) we refer to Theorem 1 of the paper: for instance, when \(\rho\) is odd, if \(\rho\) is ramified at \(p\), then the assumptions on \(\rho\) in Theorem 1 of the paper are fulfilled. In this case the author in fact shows that there is a lift \(\tilde{\rho}\) as above that is ramified at finitely many primes and is potentially semistable at \(p\) in the sense of Fontaine. Such \(p\)-adic representations \(\tilde{\rho}\), with these 2 additional properties, are said to be geometric in the sense of \textit{J.-M. Fontaine} and \textit{B. C. Mazur} [in Coates, John (ed.) et al., Elliptic curves, modular forms, \& Fermat's last theorem. Proceedings ot the conference on elliptic curves and modular forms held at the Chinese University of Hong Kong, 1993. Cambridge, MA: International Press. Ser. Number Theory, 41--78 (1995; Zbl 0839.14011)]. The paper builds on earlier work of the author [Invent. Math. 138, No. 3, 537--562 (1999; Zbl 0968.11024)], but there are new ideas in this paper which are needed to get the results here that there exist geometric lifts for a wide class of 2-dimensional odd mod \(p\) Galois representations of \(G_{\mathbb Q}\). To put this paper in context it is useful to recall a very important conjecture…
Mathematics Subject Classification ID: 11F80
zbMATH DE Number: 1850541
Wikidata QID: Q123197598
reviewed by: Chandrashekhar Khare
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/3597186
OpenAlex ID: W1999167167
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description: scientific article; zbMATH DE number 1850541