Skip to main content
SpringerLink
Log in

Slopes of Modular Forms

  • Conference paper
  • First Online:
Families of Automorphic Forms and the Trace Formula

Part of the book series: Simons Symposia ((SISY))

Abstract

We survey the progress (or lack thereof!) that has been made on some questions about the p-adic slopes of modular forms that were raised by the first author in Buzzard (Astérisque 298:1–15, 2005), discuss strategies for making further progress, and examine other related questions.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    A preprint “Slopes of modular forms and the ghost conjecture” by John Bergdall and Robert Pollack gives a much more natural conjectural algorithm for the slopes, the output of which presumably coincides with Buzzard’s algorithm.

References

  1. L. Berger, C. Breuil, Sur quelques représentations potentiellement cristallines de GL2(Q p ). Astérisque 330, 155–211 (2010)

    MathSciNet  Google Scholar 

  2. K. Buzzard, F. Calegari, A counterexample to the Gouvêa-Mazur conjecture. C. R. Math. Acad. Sci. Paris 338 (10), 751–753 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. K. Buzzard, F. Calegari, Slopes of overconvergent 2-adic modular forms. Compos. Math. 141 (3), 591–604 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. L. Berger, Local constancy for the reduction \(\bmod p\) of 2-dimensional crystalline representations. Bull. Lond. Math. Soc. 44 (3), 451–459 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. K. Buzzard, T. Gee, Explicit reduction modulo p of certain two-dimensional crystalline representations. Int. Math. Res. Not. IMRN 12, 2303–2317 (2009)

    MathSciNet  MATH  Google Scholar 

  6. K. Buzzard, T. Gee, Explicit reduction modulo p of certain 2-dimensional crystalline representations, II. Bull. Lond. Math. Soc. 45 (4), 779–788 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. L. Berger, H. Li, H.J. Zhu, Construction of some families of 2-dimensional crystalline representations. Math. Ann. 329 (2), 365–377 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. C. Breuil, A. Mézard, Multiplicités modulaires et représentations de GL2(Z p ) et de \(\mathrm{Gal}(\overline{\mathbf{Q}}_{p}/\mathbf{Q}_{p})\) en l = p. Duke Math. J. 115 (2), 205–310 (2002). With an appendix by Guy Henniart

    Google Scholar 

  10. K. Buzzard, Questions about slopes of modular forms. Astérisque 298, 1–15 (2005). Automorphic forms. I

    Google Scholar 

  11. F. Calegari, Even Galois representations and the Fontaine–Mazur conjecture. II. J. Am. Math. Soc. 25 (2), 533–554 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. R.F. Coleman, B. Edixhoven, On the semi-simplicity of the U p -operator on modular forms. Math. Ann. 310 (1), 119–127 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Caraiani, M. Emerton, T. Gee, D. Geraghty, V. Paškūnas, S.W. Shin, Patching and the p-adic local Langlands correspondence. Camb. J. Math. 4 (2), 197–287 (2016). DOI: http://dx.doi.org/10.4310/CJM.2016.v4.n2.a2

    Google Scholar 

  14. G. Chenevier, M. Harris, Construction of automorphic Galois representations, II. Camb. J. Math. 1 (1), 53–73 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. G. Chenevier, Sur la densité des représentations cristallines de \(\mbox{ Gal}(\overline{\mathbb{Q}}_{p}/\mathbb{Q}_{p})\). Math. Ann. 355 (4), 1469–1525 (2013)

    Article  MathSciNet  Google Scholar 

  16. L. Clozel, M. Harris, R. Taylor, Automorphy for some l-adic lifts of automorphic mod l Galois representations. Publ. Math. Inst. Hautes Études Sci. 108, 1–181 (2008). With Appendix A, summarizing unpublished work of Russ Mann, and Appendix B by Marie-France Vignéras

    Google Scholar 

  17. R.F. Coleman, p-adic Banach spaces and families of modular forms. Invent. Math. 127 (3), 417–479 (1997)

    Google Scholar 

  18. F. Diamond, The Taylor-Wiles construction and multiplicity one. Invent. Math. 128 (2), 379–391 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  19. T. Gee, Automorphic lifts of prescribed types. Math. Ann. 350 (1), 107–144 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. F. Gouvêa, B. Mazur, Families of modular eigenforms. Math. Comput. 58 (198), 793–805 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  21. F.Q. Gouvêa, Where the slopes are? J. Ramanujan Math. Soc. 16 (1), 75–99 (2001)

    MathSciNet  MATH  Google Scholar 

  22. M. Harris, K.-W. Lan, R. Taylor, J. Thorne, On the rigid cohomology of certain Shimura varieties. Preprint (2013)

    Google Scholar 

  23. M. Harris, N. Shepherd-Barron, R. Taylor, A family of Calabi-Yau varieties and potential automorphy. Ann. Math. (2) 171 (2), 779–813 (2010)

    Google Scholar 

  24. Y. Hu, F. Tan, The Breuil–Mézard conjecture for non-scalar split residual representations. arXiv preprint. arXiv:1309.1658 (2013)

    Google Scholar 

  25. L.J.P. Kilford, On the slopes of the U 5 operator acting on overconvergent modular forms. J. Théor. Nombres Bordeaux 20 (1), 165–182 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. M. Kisin, The Fontaine-Mazur conjecture for GL2. J. Am. Math. Soc. 22 (3), 641–690 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. L.J.P. Kilford, K. McMurdy, Slopes of the U 7 operator acting on a space of overconvergent modular forms. LMS J. Comput. Math. 15, 113–139 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. C. Khare, J.-P. Wintenberger, On Serre’s conjecture for 2-dimensional mod p representations of \(\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\). Ann. Math. (2) 169 (1), 229–253 (2009)

    Google Scholar 

  29. D. Loeffler, Explicit calculations of automorphic forms for definite unitary groups. LMS J. Comput. Math. 11, 326–342 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. R. Liu, D. Wan, L. Xiao, Eigencurve over the boundary of the weight space (2014). https://arxiv.org/abs/1412.2584

  31. K. Nakamura, Zariski density of crystalline representations for any p-adic field. J. Math. Sci. Univ. Tokyo 21 (1), 79–127 (2014)

    MathSciNet  MATH  Google Scholar 

  32. V. Paškūnas, On the Breuil–Mézard conjecture. Duke Math. J. 164 (2), 297–359 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. T. Saito, Modular forms and p-adic Hodge theory. Invent. Math. 129 (3), 607–620 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  35. S.W. Shin, Automorphic Plancherel density theorem. Isr. J. Math. 192 (1), 83–120 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  36. S.W. Shin, N. Templier, Sato-Tate Theorem for Families and low-lying zeros of automorphic L-functions. With Appendix 1 by R. Kottwitz and Appendix 2 by R. Cluckers, J. Gordon and I. Halupczok. Invent. Math. 203 (1), 1–177 (2016)

    Google Scholar 

  37. R. Taylor, Automorphy for some l-adic lifts of automorphic mod l Galois representations. II. Publ. Math. Inst. Hautes Études Sci. 108, 183–239 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  38. I. Varma, Local-global compatibility for regular algebraic cuspidal automorphic representation when ℓ ≠ p (2014). arXiv:1411.2520

    Google Scholar 

  39. D. Wan, Dimension variation of classical and p-adic modular forms. Invent. Math. 133 (2), 449–463 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  40. D. Wan, L. Xiao, J. Zhang, Slopes of eigencurves over boundary disks (2014). https://arxiv.org/abs/1407.0279

Download references

Acknowledgements

Kevin Buzzard and Toby Gee were supported in part by EPSRC grant EP/L025485/1. The second author was additionally supported by a Leverhulme Prize, Marie Curie Career Integration Grant 303605, and ERC Starting Grant 306326.

Author information

Authors and Affiliations

  1. Department of Mathematics, Imperial College London, London, UK

    Kevin Buzzard & Toby Gee

Authors
  1. Kevin Buzzard
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Toby Gee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kevin Buzzard .

Editor information

Editors and Affiliations

  1. Mathematical Institute, University of Bonn, Bonn, Nordrhein-Westfalen, Germany

    Werner Müller

  2. Department of Mathematics, University of California, Berkeley, Berkeley, California, USA

    Sug Woo Shin

  3. Department of Mathematics, Cornell University, Ithaca, New York, USA

    Nicolas Templier

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Buzzard, K., Gee, T. (2016). Slopes of Modular Forms. In: Müller, W., Shin, S., Templier, N. (eds) Families of Automorphic Forms and the Trace Formula. Simons Symposia. Springer, Cham. https://doi.org/10.1007/978-3-319-41424-9_2

Download citation

Publish with us

Policies and ethics

Search

Navigation