Skip to main content
SpringerLink
Log in

Semistable models for modular curves of arbitrary level

  • Published:
Inventiones mathematicae Aims and scope

Abstract

We produce an integral model for the modular curve \(X(Np^m)\) over the ring of integers of a sufficiently ramified extension of \(\mathbf {Z}_p\) whose special fiber is a semistable curve in the sense that its only singularities are normal crossings. This is done by constructing a semistable covering (in the sense of Coleman) of the supersingular part of \(X(Np^m)\), which is a union of copies of a Lubin–Tate curve. In doing so we tie together non-abelian Lubin–Tate theory to the representation-theoretic point of view afforded by Bushnell–Kutzko types. For our analysis it was essential to work with the Lubin–Tate curve not at level \(p^m\) but rather at infinite level. We show that the infinite-level Lubin–Tate space (in arbitrary dimension, over an arbitrary nonarchimedean local field) has the structure of a perfectoid space, which is in many ways simpler than the Lubin–Tate spaces of finite level.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. In much of this paper we work with adic spaces rather than rigid spaces, because presently we will be using adic spaces which do not come from rigid spaces. There is a fully faithful functor (see [22]) from the category of rigid analytic varieties to the category of adic spaces, which identifies admissible opens with opens, and admissible open covers with open covers. A separated adic space lies in the image of this functor if it is locally topologically of finite type.

  2. Through the paper, the “height” of a formal \(\mathcal {O}_K\)-module will be understood to mean its height relative to K. The same convention holds for quasi-isogenies between formal \(\mathcal {O}_K\)-modules.

References

  1. Berkovich, V.G.: Vanishing cycles for formal schemes. II. Invent. Math. 125(2), 367–390 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bushnell, C.J., Henniart, G.: The local Langlands conjecture for \({\rm GL(2)}\). In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 335. Springer, Berlin (2006)

  3. Bushnell, C.J., Kutzko, P.C.: The admissible dual of \({\rm GL}(N)\) via compact open subgroups. In: Annals of Mathematics Studies, vol. 129. Princeton University Press, Princeton (1993)

  4. Boyer, P.: Mauvaise réduction des variétés de Drinfeld et correspondance de Langlands locale. Invent. Math. 138(3), 573–629 (1999)

    Article  MathSciNet  Google Scholar 

  5. Bouw, I.I., Wewers, S.: Stable reduction of modular curves. In: Modular Curves and Abelian Varieties. Progress in Mathematics, vol. 224, pp. 1–22. Birkhäuser, Basel (2004)

  6. Carayol, H.: Sur les représentations \(\ell \)-adiques attachees aux formes modulaires de Hilbert. C. R. Acad. Sci. Paris. 296(15), 629–632 (1983)

    MathSciNet  MATH  Google Scholar 

  7. Carayol, H.: Sur les représentations \(\ell \)-adiques associées aux formes modulaires de Hilbert. Annales scientifiques de l’É.N.S. 19(3), 409–468 (1986)

  8. Carayol, H.: Nonabelian Lubin–Tate theory. In: Automorphic Forms, Shimura Varieties, and \(L\)-Functions, vol. II (Ann Arbor, MI, 1988). Perspectives in Mathematics, vol. 11, pp. 15–39. Academic Press, Boston (1990)

  9. Coleman, R.F., McMurdy, K.: Stable reduction of \(X_0(Np^3)\). Algebra Number Theory 4(4), 357–431 (2010). With an appendix by Everett W. Howe

  10. Coleman, R.F.: Stable maps of curves. Doc. Math. (2003). No. Extra Vol., 217–225, Kazuya Kato’s fiftieth birthday

  11. Deligne, P., Rapoport, M.: Les schémas de modules de courbes elliptiques. In: Lecture Notes in Mathematics, vol. 349, pp. 143–316 (1973)

  12. Drinfel’d, V.G.: Elliptic modules. Mat. Sb. (N.S.) 94(136), 594–627, 656 (1974)

  13. Edixhoven, B.: Minimal resolution and stable reduction of \(X_0(N)\). Ann. Inst. Fourier (Grenoble) 40(1), 31–67 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fargues, L., Fontaine, J.-M.: Courbes et fibrés vectoriels en théorie de Hodge \(p\)-adique (2011) (preprint)

  15. Fargues, L., Genestier, A., Lafforgue, V.: L’isomorphisme entre les tours de Lubin–Tate et de Drinfeld. In: Progress in Mathematics, vol. 262. Birkhäuser Verlag, Basel (2008)

  16. Fontaine, J.-M., Wintenberger, J.-P.: Le “corps des normes” de certaines extensions algébriques de corps locaux. C. R. Acad. Sci. Paris Sér. A-B 288(6), A367–A370 (1979)

  17. Gross, B.H., Hopkins, M.J.: Equivariant vector bundles on the Lubin–Tate moduli space. In: Topology and Representation Theory (Evanston, IL, 1992). Contemporary Mathematics, vol. 158, pp. 23–88. American Mathematical Society, Providence (1994)

  18. Gross, B.H.: On canonical and quasicanonical liftings. Invent. Math. 84(2), 321–326 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hedayatzadeh, S.M.H.: Exterior powers of Lubin–Tate groups. J. Théor. Nombres Bordeaux 27(1), 77–148 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Harris, M., Taylor, R.: The geometry and cohomology of some simple Shimura varieties. In: Annals of Mathematics Studies, vol. 151. Princeton University Press, Princeton, NJ (2001). With an appendix by Vladimir G. Berkovich

  21. Huber, R.: A generalization of formal schemes and rigid analytic varieties. Math. Z. 217(4), 513–551 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  22. Huber, R.: Étale cohomology of rigid analytic varieties and adic spaces. In: Aspects of Mathematics, E30. Friedr. Vieweg & Sohn, Braunschweig (1996)

  23. Imai, N., Tsushima, T.: Stable models of Lubin–Tate curves with level three (2011). arXiv:1111.1893

  24. Katz, N.M.: Crystalline cohomology, Dieudonné modules, and Jacobi sums. In: Automorphic Forms, Representation Theory and Arithmetic (Bombay, 1979). Tata Institute of Fundamental Research Studies in Mathematics, vol. 10, pp. 165–246. Tata Institute of Fundamental Research, Bombay (1981)

  25. Lubin, J., Tate, J.: Formal complex multiplication in local fields. Ann. Math. 2(81), 380–387 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  26. Scholze, P.: Perfectoid spaces. Publ. Math. Inst. Hautes Études Sci. 116, 245–313 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Strauch, M.: Deformation spaces of one-dimensional formal modules and their cohomology. Adv. Math. 217(3), 889–951 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Strauch, M.: Geometrically connected components of Lubin–Tate deformation spaces with level structures. Pure Appl. Math. Q. 4(4), 1215–1232 (2008). Special Issue: In honor of Jean-Pierre Serre. Part 1

  29. Scholze, P., Weinstein, J.: Moduli of \(p\)-divisible groups. Camb. J. Math. 1(2), 145–237 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. Tsushima, T.: Stable reduction of \(X_0(p^4)\) (2011). arXiv:1109.4378

  31. Weinstein, J.: The local Jacquet–Langlands correspondence via Fourier analysis. J. Théor. Nombres Bordeaux 22(2), 483–512 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Yoshida, T.: On non-abelian Lubin–Tate theory via vanishing cycles. In: Algebraic and Arithmetic Structures of Moduli Spaces (Sapporo 2007). Advanced Studies in Pure Mathematics, vol. 58, pp. 361–402. The Mathematical Society of Japan, Tokyo (2010)

  33. Zink, T.: The display of a formal \(p\)-divisible group. In: Astérisque, vol. 278, pp. 127–248 (2002). Cohomologies \(p\)-adiques et applications arithmétiques, I

Download references

Acknowledgments

This project originated in conversations with Robert Coleman while the author was a graduate student at Berkeley. We thank Peter Scholze for his work on perfectoid spaces, thus introducing a category which contains our deformation spaces at infinite level. We thank Mitya Boyarchenko, Jay Pottharst and Peter Scholze for very helpful conversations, and Toby Gee and the referee for pointing out numerous errors. Parts of this project were completed at the Institute for Advanced Study, and supported by NSF grants DMS-0803089 and DMS-1303312.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Boston University, Boston, MA, USA

    Jared Weinstein

Authors
  1. Jared Weinstein
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jared Weinstein.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Weinstein, J. Semistable models for modular curves of arbitrary level. Invent. math. 205, 459–526 (2016). https://doi.org/10.1007/s00222-015-0641-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-015-0641-5

Search

Navigation