Overconvergent modular forms and the Fontaine-Mazur conjecture
type: Item
instance of: scholarly article
title: Overconvergent modular forms and the Fontaine-Mazur conjecture
zbMATH Open document ID: 1045.11029
author: Mark Kisin
published in: Inventiones Mathematicae
publication date: 2003-11-04T00:00:00Z
review text: Let \(\rho: G_{\mathbb{Q}}\to \text{GL}_m(\mathbb{Q}_p)\) be a continuous irreducible representation of \(G_{\mathbb{Q}}= \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\), which is unramified outside finitely many primes and whose restriction to a decomposition group at \(p\) is potentially semistable. \textit{J.-M. Fontaine} and \textit{B. Mazur} [Elliptic curves, modular forms, and Fermat's last theorem, Ser. Number Theory 1, 41--78 (1995; Zbl 0839.14011)] made the following conjectures: (i) \(\rho\) appears (up to a twist) as a subquotient in the cohomology of a finite type \(\mathbb{Q}\)-scheme (i.e. \(\rho\) is ``geometric''), (ii) Let \(E/\mathbb{Q}_p\) be a finite extension, and let \(\rho: G_{\mathbb{Q}}\to \text{GL}_2(E)\) be a continuous, odd, irreducible representation, unramified outside finitely many primes, whose restriction to a decomposition group at \(p\) is potentially semistable when regarded as a \(\mathbb{Q}_p\)-representation. Then, up to a twist, \(\rho\) arises from a modular form. The author proves that, apart from a certain exceptional case, the Fontaine-Mazur conjecture is true for representations coming from finite slope, overconvergent eigenforms of weight \(k\neq 1\) (Theorem 6.6). One of the key ingredients in the proof is the theory of the ``eigencurve'' of \textit{R. Coleman} and \textit{B. Mazur} [Galois representations in arithmetic algebraic geometry, Lond. Math. Soc. Lect. Note Ser. 254, 1--113 (1998; Zbl 0932.11030)]. As an applicaton of this theorem (more precisely, the technical result Theorem 6.3), he answers a question of \textit{F. Q. Gouvêa} [Arithmetic of \(p\)-adic modular forms, Lectures Notes in Mathematics 1304, Springer-Verlag, Berlin (1988; Zbl 0641.10024)], who asked to what extent the \(u_p\)-eigenvalue of an overconvergent modular form \(f\) is determined by its Fourier coefficients \(a_n(f)\) with \(pX_n\). As a second application of Theorem 6.3 he recovers the result of \textit{B. Mazur} and \textit{A. Wiles} [Co…
Mathematics Subject Classification ID: 11F33
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11F30
zbMATH DE Number: 2001023
zbMATH Keywords: overconvergent modular form
zbMATH Keywords: Fontaine-Mazur conjecture
zbMATH Keywords: Galois representations
zbMATH Keywords: \(p\)-adic modular form
zbMATH Keywords: deformation theory
reviewed by: Andrzej Dąbrowski
MaRDI profile type: Publication
full work available at URL: s00222-003-0293-8
OpenAlex ID: W2075141681
Wikidata QID: Q123333483
DOI: 10.1007/S00222-003-0293-8
Recommended article: On locally analytic vectors of the completed cohomology of modular curves
Recommended article: Q4672834
Recommended article: Familles p-adiques de formes automorphes pour GLn
Recommended article: Q5423240
Recommended article: Extended eigenvarieties for overconvergent cohomology
Recommended article: The Coleman-Mazur eigencurve is proper at integral weights
Recommended article: Local to global compatibility on the eigencurve
Recommended article: The Eisenstein family
Recommended article: Overconvergent modular forms
Recommended article: On the modularity of 2-adic potentially semi-stable deformation rings
description: scientific article; zbMATH DE number 2001023