Algebra & Number Theory

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

Sandra Rozensztajn

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/ant.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ant/1593655267

Digital Object Identifier
doi:10.2140/ant.2020.14.643

Mathematical Reviews number (MathSciNet)
MR4113777

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

Keywords
Galois representations p-adic representations

Citation

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. https://projecteuclid.org/euclid.ant/1593655267


Export citation

References

  • 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)”, http://www.sagemath.org.
  • 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.