On the reductions of certain two-dimensional crystabelline representations

Abstract

Crystabelline representations are representations of the absolute Galois group \(\smash {G_{\mathbb {Q}_p}}\) over \(\smash {\smash {\overline{\mathbb {Q}}_p}}\) that become crystalline on \(\smash {G_{F}}\) for some abelian extension \(\smash {F/\mathbb {Q}_p}\). Their relation to modular forms is that the representation associated with a finite slope newform of level divisible by \(\smash {p^2}\) is crystabelline. In this article, we study the connection between the slopes of two-dimensional crystabelline representations and the reducibility of their modulo \(\smash {p}\) reductions. This question is inspired by a theorem by Buzzard and Kilford which implies that the slopes on the boundary of the \(\smash {2}\)-adic eigencurve of tame level \(\smash {1}\) are integers (and in arithmetic progression); an analogous theorem by Roe which says that the same is true for the \(\smash {3}\)-adic eigencurve; Coleman’s halo conjecture and the ghost conjecture which give predictions about the slopes on the \(\smash {p}\)-adic eigencurve of general tame level; and Hodge theoretic conjectures by Breuil, Buzzard, Emerton, and Gee which indicate that there is a connection between all of these and the slopes of two-dimensional crystabelline representations whose reductions modulo \(\smash {p}\) are reducible. We prove that the reductions of certain two-dimensional crystabelline representations with slopes in \(\smash {(0,\frac{p-1}{2})\backslash \mathbb {Z}}\) are usually irreducible, with the exception of a small region where the slopes are half-integers and reducible representations do occur.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Arsovski, B.: On the reductions of certain two-dimensional crystalline representations. arXiv:1711.03057

  2. 2.

    Arsovski, B.: On the reductions of certain two-dimensional crystalline representations, II. arXiv:1808.03224

  3. 3.

    Ash, A., Stevens, G.: Modular forms in characteristic \(\ell \) and special values of their \(L\)-functions. Duke Math. J. 53(3), 849–868 (1986)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bergdall, J., Pollack, R.: Slopes of modular forms and the ghost conjecture. International Mathematics Research Notices, (to appear)

  5. 5.

    Bergdall, J., Pollack, R.: Slopes of modular forms and the ghost conjecture, II. Transactions of the American Mathematical Society, (to appear)

  6. 6.

    Berger, L.: Représentations modulaires de \({{\rm GL}}_2({\mathbb{Q}}_p)\) et représentations galoisiennes de dimension 2. Astérisque, Société Mathématique de France 330, 263–279 (2010)

    Google Scholar 

  7. 7.

    Berger, L.: Trianguline representations. Bull. Lond. Math. Soc. 43(4), 619–635 (2011)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Berger, L., Breuil, C.: Sur quelques représentations potentiellement cristallines de \({{\rm GL}}_2({\mathbb{Q}}_p)\). Astérisque, Société Mathématique de France 330, 155–211 (2010)

    MATH  Google Scholar 

  9. 9.

    Bhattacharya, S., Ghate, E.: Reductions of Galois representations for slopes in \((1,2)\). Docum. Math. 20, 943–987 (2015)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Breuil, C.: Sur quelques représentations modulaires et \(p\)-adiques de \({{\rm GL}}_2({\mathbb{Q}}_p)\), I. Compos. Math. 138(2), 165–188 (2003)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Breuil, C.: Sur quelques représentations modulaires et \(p\)-adiques de \({{\rm GL}}_2(\mathbb{Q}_p)\), II. J. Instit. Math. Jussieu 2(1), 23–58 (2003)

    MATH  Google Scholar 

  12. 12.

    Breuil, C.: Invariant \(\mathscr {L}\) et série spéciale \(p\)-adique. Annales Scientifiques de l’École Normale Supérieure 37, 559–610 (2004)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Buzzard, K.: Questions about slopes of modular forms. Astérisque, Société Mathématique de France 298, 1–15 (2005)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Buzzard, K.: Eigenvarieties. Lond. Math. Soc. Lecture Note Ser. 320, 59–120 (2007)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Buzzard, K., Gee, T.: Explicit reduction modulo \(p\) of certain two-dimensional crystalline representations. Int. Math. Res. Notices 12, 2303–2317 (2009)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Buzzard, K., Gee, T.: Explicit reduction modulo \(p\) of certain two-dimensional crystalline representations. II. Bull. Lond. Math. Soc. 45(4), 779–788 (2013)

    Article  Google Scholar 

  17. 17.

    Buzzard, K., Gee, T.: Slopes of modular forms. Families of Automorphic Forms and the Trace Formula 93–109, (2016)

  18. 18.

    Buzzard, K., Kilford, L.J.P.: The 2-adic eigencurve at the boundary of weight space. Compos. Math. 141(3), 605–619 (2005)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Chenevier, G.: Sur la densité des représentations cristallines du groupe de Galois absolu de \({\mathbb{Q}}_p\). Math. Ann. 335, 1469–1525 (2013)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Coleman, R.F., Mazur, B.: The eigencurve. Lond. Math. Soc. Lecture Note Ser. 254, 1–113 (1998)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Colmez, P.: Représentations triangulines de dimension 2. Astérisque, Société Mathématique de France 319, 213–258 (2008)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Ghate, E., Rai, V.: Reductions of Galois representations of slope \(\frac{3}{2}\). https://arxiv.org/abs/1901.01728

  23. 23.

    Ghate, E., Mézard, A.: Filtered modules with coefficients. Trans. Am. Math. Soc. 361(5), 2243–2261 (2009)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Kilford, L.J.P.: On the slopes of the \(U_5\) operator acting on overconvergent modular forms. J. Théorie des Nombres de Bordeaux 20(1), 165–182 (2008)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Kilford, L.J.P.: McMurdy, Ken, Slopes of the \(U_7\) operator acting on a space of overconvergent modular forms. Lond. Math. Soc. J. Comput. Math. 15, 113–139 (2012)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Liu, R., Wan, D., Xiao, L.: The eigencurve over the boundary of weight space. Duke Math. J. 166(9), 1739–1787 (2017)

    MathSciNet  Article  Google Scholar 

  27. 27.

    book Ribet, K. A., Stein, W.A.: Lectures on modular forms and Hecke operators. https://github.com/williamstein/ribet-stein-modforms

  28. 28.

    Roe, D.: The 3-adic eigencurve at the boundary of weight space. Int. J. Number Theory 10(7), 1791–1806 (2014)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

I would like to thank professor Kevin Buzzard for his helpful remarks, suggestions, and support. This work was supported by Imperial College London and its President’s PhD Scholarship.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Bodan Arsovski.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Arsovski, B. On the reductions of certain two-dimensional crystabelline representations. Res Math Sci 8, 12 (2021). https://doi.org/10.1007/s40687-020-00231-6

Download citation

Keywords

  • Galois representations
  • Modular forms