Algebra & Number Theory

Multiplicity one for wildly ramified representations

Daniel Le

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Let F be a totally real field in which p is unramified. Let r̄:GF GL2(F¯p) be a modular Galois representation which satisfies the Taylor–Wiles hypotheses and is generic at a place v above p. Let m be the corresponding Hecke eigensystem. We show that the m-torsion in the modp cohomology of Shimura curves with full congruence level at v coincides with the GL2(kv)-representation D0(r̄|GFv) constructed by Breuil and Paškūnas. In particular, it depends only on the local representation r̄|GFv, and its Jordan–Hölder factors appear with multiplicity one. This builds on and extends work of the author with Morra and Schraen and, independently, Hu–Wang, which proved these results when r̄|GFv was additionally assumed to be tamely ramified. The main new tool is a method for computing Taylor–Wiles patched modules of integral projective envelopes using multitype tamely potentially Barsotti–Tate deformation rings and their intersection theory.

Article information

Algebra Number Theory, Volume 13, Number 8 (2019), 1807-1827.

Received: 19 October 2017
Revised: 13 February 2019
Accepted: 27 May 2019
First available in Project Euclid: 29 October 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Galois deformations mod p Langlands program


Le, Daniel. Multiplicity one for wildly ramified representations. Algebra Number Theory 13 (2019), no. 8, 1807--1827. doi:10.2140/ant.2019.13.1807.

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  • C. Breuil, “Sur un problème de compatibilité local-global modulo $p$ pour ${\rm GL}_2$”, J. Reine Angew. Math. 692 (2014), 1–76.
  • C. Breuil and V. Paškūnas, Towards a modulo $p$ Langlands correspondence for ${\rm GL}_2$, Mem. Amer. Math. Soc. 1016, Amer. Math. Soc., Providence, RI, 2012.
  • K. Buzzard, F. Diamond, and F. Jarvis, “On Serre's conjecture for mod $\ell$ Galois representations over totally real fields”, Duke Math. J. 155:1 (2010), 105–161.
  • F. Diamond, “The Taylor–Wiles construction and multiplicity one”, Invent. Math. 128:2 (1997), 379–391.
  • F. Diamond, “A correspondence between representations of local Galois groups and Lie-type groups”, pp. 187–206 in $L$-functions and Galois representations (Durham, UK, 2004), edited by D. Burns et al., London Math. Soc. Lecture Note Ser. 320, Cambridge Univ. Press, 2007.
  • M. Emerton, T. Gee, and D. Savitt, “Lattices in the cohomology of Shimura curves”, Invent. Math. 200:1 (2015), 1–96.
  • K. Fujiwara, “Deformation rings and Hecke algebras in the totally real case”, submitted, 2006.
  • G. Henniart, “Sur l'unicité des types pour ${\rm GL}_2$”, (2002). Appendix to C. Breuil and A. Mézard, “Multiplicités modulaires et représentations de ${\rm GL}_2(\mathbb{Z}_p)$ et de ${\rm Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ en $l=p$”, Duke Math. J. 115:2 (2002), 205–310.
  • F. Herzig, “The weight in a Serre-type conjecture for tame $n$-dimensional Galois representations”, Duke Math. J. 149:1 (2009), 37–116.
  • Y. Hu and H. Wang, “Multiplicity one for the $p$ cohomology of Shimura curves: the tame case”, Math. Res. Lett. 25:3 (2018), 843–873.
  • J. C. Jantzen, Representations of algebraic groups, Pure Appl. Math. 131, Academic Press, Boston, 1987.
  • M. Kisin, “Potentially semi-stable deformation rings”, J. Amer. Math. Soc. 21:2 (2008), 513–546.
  • D. Le, “Lattices in the cohomology of $U(3)$ arithmetic manifolds”, Math. Ann. 372:1-2 (2018), 55–89.
  • D. Le, B. V. Le Hung, B. Levin, and S. Morra, “Serre weights and Breuil's lattice conjecture in dimension three”, preprint, 2016.
  • D. Le, S. Morra, and B. Schraen, “Multiplicity one at full congruence level”, preprint, 2016.
  • D. Le, B. V. Le Hung, B. Levin, and S. Morra, “Potentially crystalline deformation rings and Serre weight conjectures: shapes and shadows”, Invent. Math. 212:1 (2018), 1–107.
  • D. Le, B. V. Le Hung, and S. Morra, “Cohomology of $U(3)$ arithmetic manifolds at full congruence level”, in preparation.