On a conjecture of Conrad, Diamond, and Taylor
type: Item
instance of: scholarly article
title: On a conjecture of Conrad, Diamond, and Taylor
zbMATH Open document ID: 1101.11017
DOI: 10.1215/S0012-7094-04-12816-7
author: David Savitt
published in: Duke Mathematical Journal
publication date: 2005-11-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0404327
review text: The author proves a special case of a conjecture of \textit{C.~Breuil} and \textit{A.~Mézard} [Duke Math. J. 115, No. 2, 205--310 (2002; Zbl 1042.11030)], a part of which was already conjectured by \textit{B. Conrad, F. Diamond} and \textit{R. Taylor} [J. Am. Math. Soc. 12, No. 2, 521--567 (1999; Zbl 0923.11085)], hence the title. Let \(p\neq2\) be a prime number. Fix an algebraic closure \(\overline{\mathbb Q}_p\) of \({\mathbb Q}_p\) and denote by \(G_p\) the absolute Galois group, by \(W_p\) the Weil group, by \(I_p\) the inertia subgroup and by \(\overline{\mathbb F}_p\) the residue field. Let \(\overline{\mathbb Q}\) be the algebraic closure \({\mathbb Q}\) in \(\overline{\mathbb Q}_p\). Let \(E\) be a finite extension of \({\mathbb Q}_p\) in \(\overline{\mathbb Q}_p\), with residue field \(F\), and let \(\rho:G_p\rightarrow\text{GL}_2(F)\) be a representation with ``trivial endomorphisms'', i.e.~the ring of \(F\)-linear \(G_p\)-equivariant endomorphisms of \(F^2\) is reduced to \(F\). Let \(\tau:I_p\rightarrow\text{GL}_2(E)\) (the ``type'') be a representation which extends to \(G_p\) and has finite image. Finally, let \(k\) (the ``weight'') be an integer \(\geq2\). To the triple \((k,\tau,\rho)\), Breuil and Mézard attach two numbers, the ``automorphic multiplicity'' \(\alpha(k,\tau,\rho)\) and the ``Galois multiplicity'' \(\gamma(k,\tau,\rho)\). \(\alpha(k,\tau,\rho)\) : There is a unique (G.~Henniart) \(\overline{\mathbb Q}_p\)-representation \(\sigma(\tau)\) of \(\text{GL}_2({\mathbb Z}_p)\) with finite image which occurs in the restriction to \(\text{GL}_2({\mathbb Z}_p)\) of every (\(\infty\)-dimensional) irreducible admissible representation of \(\text{GL}_2({\mathbb Q}_p)\) for which the restriction to \(I_p\) of the associated representation of \(W_p\) (R. P. Langlands' correspondence) is isomorphic to \(\tau\). Let \(a(m,n)\) (with \(0\leq m\leq p-1\) and \(0\leq n\leq p-2\)) be the multiplicity of the irreducible representation \(\text{ Sym}^m\overl…
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 14L15
zbMATH DE Number: 2223079
zbMATH Keywords: Galois representations
zbMATH Keywords: modularity lifting theorems
zbMATH Keywords: Breuil-Mézard conjecture
reviewed by: Chandan Singh Dalawat
MaRDI profile type: Publication
arXiv ID: math/0404327
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Wikidata QID: Q123278309
OpenAlex ID: W2102356356
description: scientific article