Algebra & Number Theory

Local deformation rings for $\operatorname{GL}_2$ and a Breuil–Mézard conjecture when $l \neq p$

Jack Shotton

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We compute the deformation rings of two dimensional mod l representations of Gal(F¯F) with fixed inertial type for l an odd prime, p a prime distinct from l, and Fp a finite extension. We show that in this setting an analogue of the Breuil–Mézard conjecture holds, relating the special fibres of these deformation rings to the mod l reduction of certain irreducible representations of GL2(OF).

Article information

Algebra Number Theory, Volume 10, Number 7 (2016), 1437-1475.

Received: 28 April 2015
Revised: 14 April 2016
Accepted: 18 July 2016
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11S37: Langlands-Weil conjectures, nonabelian class field theory [See also 11Fxx, 22E50]

Galois representations deformation rings local Langlands Breuil–Mézard


Shotton, Jack. Local deformation rings for $\operatorname{GL}_2$ and a Breuil–Mézard conjecture when $l \neq p$. Algebra Number Theory 10 (2016), no. 7, 1437--1475. doi:10.2140/ant.2016.10.1437.

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  • T. Barnet-Lamb, T. Gee, D. Geraghty, and R. Taylor, “Potential automorphy and change of weight”, Ann. of Math. $(2)$ 179:2 (2014), 501–609.
  • C. Breuil and A. Mézard, “Multiplicités modulaires et représentations de ${\rm GL}\sb 2({\mathbb Z}\sb p)$ et de ${\rm Gal}(\overline{\mathbb Q}\sb p/{\mathbb Q}\sb p)$ en $l=p$”, Duke Math. J. 115:2 (2002), 205–310.
  • C. J. Bushnell and G. Henniart, The local Langlands conjecture for $\rm GL(2)$, Grundlehren der Mathematischen Wissenschaften 335, Springer, Berlin, 2006.
  • S. H. Choi, Local deformation lifting spaces of mod $l$ Galois representations, Ph.D. thesis, Harvard University, 2009.
  • L. Clozel, M. Harris, and R. Taylor, “Automorphy for some $l$-adic lifts of automorphic mod $l$ Galois representations”, Publ. Math. Inst. Hautes Études Sci. 108 (2008), 1–181.
  • D. Eisenbud, Commutative algebra, Graduate Texts in Mathematics 150, Springer, New York, 1995.
  • M. Emerton and T. Gee, “A geometric perspective on the Breuil–Mézard conjecture”, J. Inst. Math. Jussieu 13:1 (2014), 183–223.
  • T. Gee and M. Kisin, “The Breuil–Mézard conjecture for potentially Barsotti–Tate representations”, Forum Math. Pi 2 (2014), e1, 56.
  • D. Helm, “On $l$-adic families of cuspidal representations of ${\rm GL}\sb 2(\mathbb Q\sb p)$”, Math. Res. Lett. 17:5 (2010), 805–822.
  • G. Henniart, appendix to [breuil2002?], 2002.
  • Y. Hu and F. Tan, “The Breuil–Mézard conjecture for non-scalar split residual representations”, preprint, 2013.
  • M. Kisin, “The Fontaine–Mazur conjecture for ${\rm GL}\sb 2$”, J. Amer. Math. Soc. 22:3 (2009), 641–690.
  • M. Kisin, “Moduli of finite flat group schemes, and modularity”, Ann. of Math. $(2)$ 170:3 (2009), 1085–1180.
  • V. Paškūnas, “On the Breuil–Mézard conjecture”, Duke Math. J. 164:2 (2015), 297–359.
  • V. Pilloni, “The study of 2-dimensional $p$-adic Galois deformations in the $l$ not $p$ case”, 2008, hook \posturlhook.
  • D. Reduzzi, “On the number of irreducible components of local deformation rings in the unequal characteristic case”, 2013, hook \posturlhook.
  • A. Snowden, “Singularities of ordinary deformation rings”, 2011.
  • R. Taylor, “Automorphy for some $l$-adic lifts of automorphic mod $l$ Galois representations II”, Publ. Math. Inst. Hautes Études Sci. 108 (2008), 183–239.
  • R. Taylor, “Modularity lifting theorems”, Lecture notes, Harvard University, 2009.