Skip to main content
×
×
Home

UNRAMIFIEDNESS OF GALOIS REPRESENTATIONS ATTACHED TO HILBERT MODULAR FORMS MOD  $p$ OF WEIGHT 1

  • Mladen Dimitrov (a1) and Gabor Wiese (a2)
Abstract

The main result of this article states that the Galois representation attached to a Hilbert modular eigenform defined over  $\overline{\mathbb{F}}_{p}$ of parallel weight 1 and level prime to $p$ is unramified above  $p$ . This includes the important case of eigenforms that do not lift to Hilbert modular forms in characteristic 0 of parallel weight 1. The proof is based on the observation that parallel weight 1 forms in characteristic  $p$ embed into the ordinary part of parallel weight  $p$ forms in two different ways per prime dividing  $p$ , namely via ‘partial’ Frobenius operators.

Copyright
References
Hide All
1. Andreatta, F. and Goren, E. Z., Hilbert modular forms: mod p and p-adic aspects, Mem. Amer. Math. Soc. 173(819) (2005), vi+100.
2. Buzzard, K., Diamond, F. and Jarvis, F., On Serre’s conjecture for mod  Galois representations over totally real fields, Duke Math. J. 55 (2010), 105161.
3. Buzzard, K. and Taylor, R., Companion forms and weight 1 forms, Ann. of Math. (2) 149 (1999), 905919.
4. Calegari, F. and Geraghty, D., Modularity lifting theorems beyond the Taylor–Wiles method, Invent. Math. 211 (2018), 297433.
5. Carayol, H., Formes modulaires et représentations galoisiennes à valeurs dans un anneau local complet, in p-adic Monodromy and the Birch and Swinnerton-Dyer Conjecture (Boston, MA, 1991), Contemp. Math., Volume 165, pp. 213237. (1994).
6. Chai, C.-L., Arithmetic minimal compactification of the Hilbert–Blumenthal moduli space, Ann. of Math. (2) 131 (1990), 541554.
7. Coleman, R. F. and Voloch, J. F., Companion forms and Kodaira–Spencer theory, Invent. Math. 110 (1992), 263281.
8. Dasgupta, S., Darmon, H. and Pollack, R., Hilbert modular forms and the Gross–Stark conjecture, Ann. of Math. (2) 174 (2011), 439484.
9. Deligne, P. and Pappas, G., Singularités des espaces de modules de Hilbert, en les caractéristiques divisant le discriminant, Compos. Math. 90 (1994), 5979.
10. Deligne, P. and Serre, J.-P., Formes modulaires de poids 1, Ann. Sci. Éc. Norm. Supér. (4) 74 (1974), 507530.
11. Dimitrov, M., Galois representations modulo p and cohomology of Hilbert modular varieties, Ann. Sci. Éc. Norm. Supér. (4) 38 (2005), 505551.
12. Dimitrov, M., On Ihara’s lemma for Hilbert modular varieties, Compos. Math. 145 (2009), 11141146.
13. Dimitrov, M., Compactifications arithmétiques des variétés de Hilbert et formes modulaires de Hilbert pour 𝛤1(c, n), in Geometric Aspects of Dwork Theory, pp. 527554 (Walter de Gruyter, Berlin, 2004).
14. Dimitrov, M. and Tilouine, J., Variétés et formes modulaires de Hilbert arithmétiques pour 𝛤1(c, n), in Geometric Aspects of Dwork Theory, pp. 555614 (Walter de Gruyter, Berlin, 2004).
15. Edixhoven, S. J., The weight in Serre’s conjectures on modular forms, Invent. Math. 109 (1992), 563594.
16. Emerton, M., Reduzzi, D. A. and Xiao, L., Unramifiedness of Galois representations arising from Hilbert modular surfaces, Forum Math. (2017), to appear.
17. Faltings, G. and Chai, C.-L., Degeneration of Abelian Varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Volume 22 (Springer, Berlin, 1990). With an appendix by David Mumford, xii+316 pp.
18. Gee, T. and Kassaei, P., Companion forms in parallel weight 1, Compos. Math. 149 (2013), 903913.
19. Gee, T., Liu, T. and Savitt, D., The weight part of Serre’s conjecture for GL(2), Forum Math. 3(e2) (2015), 52 pages.
20. Goren, E. and Kassaei, P., Canonical subgroups over Hilbert modular varieties, J. Reine Angew. Math. 670 (2012), 163.
21. Gross, B. H., A tameness criterion for Galois representations associated to modular forms (mod p), Duke Math. J. 61 (1990), 445517.
22. Grothendieck, A., Éléments de géométrie algébrique, Publ. Math. Hautes Etudes Sci., 4, 8, 11, 17, 20, 24, 28, 32 (1961).
23. Hida, H., p-adic Automorphic Forms on Shimura Varieties, Springer Monographs in Mathematics (Springer, New York, 2004).
24. Katz, N., p-adic L-functions for CM fields, Invent. Math. 49 (1978), 199297.
25. Khare, C. and Wintenberger, J.-P., Serre’s modularity conjecture (I), Invent. Math. 178 (2009), 485504.
26. Kisin, M. and Lai, K., Overconvergent Hilbert modular forms, Amer. J. Math. 127 (2005), 735783.
27. Lan, K.-W. and Suh, J., Liftability of mod p cusp forms of parallel weights, Int. Math. Res. Not. IMRN 8 (2011), 18701879.
28. Ohta, M., Hilbert modular forms of weight 1 and Galois representations, in Automorphic Forms of Several Variables (Katata, 1983), Progress in Mathematics, Volume 46, pp. 333352 (Birkhäuser Boston, Boston, MA, 1984).
29. Pappas, G., Arithmetic models for Hilbert modular varieties, Compos. Math. 98 (1995), 4376.
30. Rogawski, J. D. and Tunnell, J. B., On Artin L-functions associated to Hilbert modular forms of weight 1, Invent. Math. 74 (1983), 142.
31. Rapoport, M., Compactification de l’espace de modules de Hilbert–Blumenthal, Compos. Math. 36 (1978), 255335.
32. Raynaud, M., Faisceaux amples sur les schémas en groupes et les espaces homogènes, Lecture Notes in Mathematics, Volume 119 (Springer, Berlin-New York, 1970),ii+218 pp.
33. Serre, J.-P., Sur les représentations modulaires de degré 2 de Gal(/ℚ), Duke Math. J. 54 (1987), 179230.
34. Shimura, G., The special values of the zeta functions associated with Hilbert modular forms, Duke Math. J. 45 (1978), 637679.
35.The Stacks Project Authors, Stacks project, http://stacks.math.columbia.edu, 2016.
36. Taylor, R., On Galois representations associated to Hilbert modular forms, Invent. Math. 98 (1989), 265280.
37. Van Hirtum, J., Explicit methods for Hilbert modular forms of weight 1, Preprint, 2017, arXiv:1710.02287.
38. Wiese, G., On Galois representations of weight one, Doc. Math. 19 (2014), 689707.
39. Wiles, A., On p-adic representations for totally real fields, Ann. of Math. (2) 123 (1986), 407456.
40. Wiles, A., On ordinary 𝜆-adic representations associated to modular forms, Invent. Math. 94 (1988), 529573.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of the Institute of Mathematics of Jussieu
  • ISSN: 1474-7480
  • EISSN: 1475-3030
  • URL: /core/journals/journal-of-the-institute-of-mathematics-of-jussieu
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 11 *
Loading metrics...

Abstract views

Total abstract views: 40 *
Loading metrics...

* Views captured on Cambridge Core between 23rd April 2018 - 21st May 2018. This data will be updated every 24 hours.