Algebra & Number Theory

On the locus of $2$-dimensional crystalline representations with a given reduction modulo $p$

Sandra Rozensztajn

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We consider the family of irreducible crystalline representations of dimension 2 of Gal( ̄pp) given by the Vk,ap for a fixed weight k2. We study the locus of the parameter ap where these representations have a given reduction modulo p. We give qualitative results on this locus and show that for a fixed p and k it can be computed by determining the reduction modulo p of Vk,ap for a finite number of values of the parameter ap. We also generalize these results to other Galois types.

Article information

Algebra Number Theory, Volume 14, Number 3 (2020), 643-700.

Received: 13 September 2018
Revised: 26 August 2019
Accepted: 30 September 2019
First available in Project Euclid: 2 July 2020

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 11F80: Galois representations
Secondary: 14G22: Rigid analytic geometry

Galois representations p-adic representations


Rozensztajn, Sandra. On the locus of $2$-dimensional crystalline representations with a given reduction modulo $p$. Algebra Number Theory 14 (2020), no. 3, 643--700. doi:10.2140/ant.2020.14.643.

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  • R. Bellovin, “$p$-adic Hodge theory in rigid analytic families”, Algebra Number Theory 9:2 (2015), 371–433.
  • R. L. Benedetto, “Attaining potentially good reduction in arithmetic dynamics”, Int. Math. Res. Not. 2015:22 (2015), 11828–11846.
  • L. Berger, “Local constancy for the reduction $\bmod p$ of 2-dimensional crystalline representations”, Bull. Lond. Math. Soc. 44:3 (2012), 451–459.
  • L. Berger and P. Colmez, “Familles de représentations de de Rham et monodromie $p$-adique”, pp. 303–337 in Représentations $p$-adiques de groupes $p$-adiques, vol. I: Représentations galoisiennes et $(\phi,\Gamma)$-modules, edited by L. Berger et al., Astérisque 319, Société Mathématique de France, Paris, 2008.
  • L. Berger, H. Li, and H. J. Zhu, “Construction of some families of 2-dimensional crystalline representations”, Math. Ann. 329:2 (2004), 365–377.
  • S. Bhattacharya and E. Ghate, “Reductions of Galois representations for slopes in $(1,2)$”, Doc. Math. 20 (2015), 943–987.
  • S. Bhattacharya, E. Ghate, and S. Rozensztajn, “Reductions of Galois representations of slope 1”, J. Algebra 508 (2018), 98–156.
  • S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean analysis: a systematic approach to rigid analytic geometry, Grundlehren der Mathematischen Wissenschaften 261, Springer, 1984.
  • C. Breuil and A. Mézard, “Multiplicités modulaires et représentations de ${\rm GL}_2({\bf Z}_p)$ et de ${\rm Gal}(\overline{\bf Q}_p/{\bf Q}_p)$ en $l=p$”, Duke Math. J. 115:2 (2002), 205–310.
  • K. Buzzard and T. Gee, “Explicit reduction modulo $p$ of certain two-dimensional crystalline representations”, Int. Math. Res. Not. 2009:12 (2009), 2303–2317.
  • K. Buzzard and T. Gee, “Explicit reduction modulo $p$ of certain 2-dimensional crystalline representations, II”, Bull. Lond. Math. Soc. 45:4 (2013), 779–788.
  • K. Buzzard and T. Gee, “Slopes of modular forms”, pp. 93–109 in Families of automorphic forms and the trace formula, edited by W. Müller et al., Springer, 2016.
  • M. Emerton and T. Gee, “A geometric perspective on the Breuil–Mézard conjecture”, J. Inst. Math. Jussieu 13:1 (2014), 183–223.
  • J.-M. Fontaine, “Représentations $l$-adiques potentiellement semi-stables”, pp. 321–347 in Périodes $p$-adiques (Bures-sur-Yvette, 1988), Astérisque 223, Société Mathématique de France, Paris, 1994.
  • A. Ganguli and E. Ghate, “Reductions of Galois representations via the ${\rm mod}\, p$ local Langlands correspondence”, J. Number Theory 147 (2015), 250–286.
  • E. Ghate and A. Mézard, “Filtered modules with coefficients”, Trans. Amer. Math. Soc. 361:5 (2009), 2243–2261.
  • E. Ghate and V. Rai, “Reductions of Galois representations of slope $\frac{3}{2}$”, preprint, 2019.
  • A. J. de Jong, “Crystalline Dieudonné module theory via formal and rigid geometry”, Inst. Hautes Études Sci. Publ. Math. 82 (1995), 5–96.
  • C. Kappen, “Uniformly rigid spaces”, Algebra Number Theory 6:2 (2012), 341–388.
  • M. Kisin, “Potentially semi-stable deformation rings”, J. Amer. Math. Soc. 21:2 (2008), 513–546.
  • M. Kisin, “The Fontaine–Mazur conjecture for ${\rm GL}_2$”, J. Amer. Math. Soc. 22:3 (2009), 641–690.
  • M. Kisin, “The structure of potentially semi-stable deformation rings”, pp. 294–311 in Proceedings of the International Congress of Mathematicians, vol. II, edited by R. Bhatia et al., Hindustan Book Agency, New Delhi, 2010.
  • M.-A. Knus and M. Ojanguren, Théorie de la descente et algèbres d'Azumaya, Lecture Notes in Mathematics 389, Springer, 1974.
  • C. Kratzer, “Rationalité des représentations de groupes finis”, J. Algebra 81:2 (1983), 390–402.
  • L. Lipshitz and Z. Robinson, “Rigid subanalytic subsets of the line and the plane”, Amer. J. Math. 118:3 (1996), 493–527.
  • L. Lipshitz and Z. Robinson, Rings of separated power series and quasi-affinoid geometry, Astérisque 264, Société Mathématique de France, Paris, 2000.
  • Q. Liu, “Ouverts analytiques d'une courbe algébrique en géométrie rigide”, Ann. Inst. Fourier $($Grenoble$)$ 37:3 (1987), 39–64.
  • H. Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics 8, Cambridge University Press, 1986.
  • B. Mazur, “On monodromy invariants occurring in global arithmetic, and Fontaine's theory”, pp. 1–20 in $p$-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991), edited by B. Mazur and G. Stevens, Contemp. Math. 165, Amer. Math. Soc., Providence, RI, 1994.
  • V. Paškūnas, “On the Breuil–Mézard conjecture”, Duke Math. J. 164:2 (2015), 297–359.
  • V. Paškūnas, “On 2-dimensional 2-adic Galois representations of local and global fields”, Algebra Number Theory 10:6 (2016), 1301–1358.
  • R. Rouquier, “Caractérisation des caractères et pseudo-caractères”, J. Algebra 180:2 (1996), 571–586.
  • S. Rozensztajn, “Potentially semi-stable deformation rings for discrete series extended types”, J. Éc. Polytech. Math. 2 (2015), 179–211.
  • S. Rozensztajn, “An algorithm for computing the reduction of 2-dimensional crystalline representations of ${\rm Gal}(\overline{\mathbb{Q}}_p/\Bbb{Q}_p)$”, Int. J. Number Theory 14:7 (2018), 1857–1894.
  • W. Stein et al., “Sage Mathematics Software (Version 7.0)”,
  • D. Savitt, “On a conjecture of Conrad, Diamond, and Taylor”, Duke Math. J. 128:1 (2005), 141–197.
  • L. C. Washington, Introduction to cyclotomic fields, 2nd ed., Graduate Texts in Mathematics 83, Springer, 1997.