Skip to main content
SpringerLink
Log in

Regular formal moduli spaces and arithmetic transfer conjectures

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We define various formal moduli spaces of p-divisible groups which are regular, and morphisms between them. We formulate arithmetic transfer conjectures, which are variants of the arithmetic fundamental lemma conjecture of the third author in the presence of ramification. These conjectures include the AT conjecture of our previous joint work. We prove these conjectures in low-dimensional cases.

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

Similar content being viewed by others

Notes

  1. Recall that, since \(F/F_0\) is unramified, we have \(\breve{F}=\breve{F}_0\) as in the Introduction.

  2. Recall that essentially proper means that each irreducible component of the reduced underlying scheme is proper over \({{\mathrm{Spec}}}\overline{k}\).

  3. Here and below we replace the symbol W used in loc. cit. with .

  4. Here and below we interchange r and s in the notation relative to [32].

  5. Note that the statement in loc. cit. considers quasi-isogenies which are compatible with the polarizations only up to scalar, whereas here we are requiring compatibility on the nose.

  6. Strictly speaking, loc. cit. shows these these implications hold in the case of signature opposite to ours, but up to isomorphism the local model is the same, cf. [24, p. 19 fn. 5]. The same remark applies to essentially all of the subsequent references we make to the local models in [3, 20], [21, §2.6], and [32].

  7. Exceptional in three ways: in this case the space is defined over \({{\mathrm{Spf}}}O_{\breve{F}_0}\) instead of \({{\mathrm{Spf}}}O_{\breve{F}}\); it is regular; and the corresponding parahoric subgroup of \(\text {GU}(h)(F_0)\) is not a maximal parahoric subgroup, but an Iwahori subgroup, cf. [21, Rem. 2.35].

  8. Note that later on, for technical convenience, we will rescale the polarization \(\lambda _\mathbb {X} \), cf. Example 9.4.

  9. Note that the quantity \(\eta \) in loc. cit. should be a square root of \(-\epsilon ^{-1}\), rather than a square root of \(\epsilon ^{-1}\).

  10. Here the condition \(VL' \subset ^{\le 1} VL' + \pi L'\) is manifestly equivalent to the wedge condition in the moduli problem for \(\mathcal {P} _n\), via the canonical isomorphism \({{\mathrm{Lie\,}}}X' \cong L'/VL'\). For the purposes of the proof, we only need to know that the moduli problem implies this condition on \(L'\); we leave it as an exercise to show that, for \(\overline{k}\)-points, condition (9.2) imposes nothing further.

  11. Here conditions (6) and (7) make sense as written whenever S is an \(O_F\)-scheme, and when \(r = s\) they descend to conditions on \(M_I^\text {naive} \) over \({{\mathrm{Spec}}}O_E = {{\mathrm{Spec}}}O_{F_0}\).

  12. Strictly speaking loc. cit. requires I to have the property that if n is even and \(m - 1 \in I\), then \(m \in I\), but here we drop this requirement. Note that on the other hand, Proposition 9.12 below allows one to view sets I not satisfying this property as being redundant in some sense.

  13. When \((r,s) = (1,1)\), these schemes are defined over \({{\mathrm{Spec}}}O_{F_0}\), and here we implicitly replace them with their base change to \({{\mathrm{Spec}}}O_F\).

  14. After extension of scalars, this becomes an Iwahori subgroup.

  15. Note that in loc. cit. n is odd, which in the case of our \(\widetilde{\eta }\) implies that \(\widetilde{\eta }(\gamma _1^{-1}\gamma _2)^{n-1} = 1\); however the proof for arbitrary n is analogous. The last statement in (iv) is immediate from the specific choice of \(\phi _\mathrm {corr}'\) given in (5.13) in loc. cit.

References

  1. Ahsendorf, T., Cheng, C., Zink, Th: \(\cal{O}\)-displays and \(\pi \)-divisible formal \(\cal{O}\)-modules. J. Algebra 457, 129–193 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. ARGOS seminar on intersections of modular correspondences, Astérisque 312 (2007)

  3. Arzdorf, K.: On local models with special parahoric level structure. Mich. Math. J. 58(3), 683–710 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Drinfeld, V.G.: Coverings of \(p\)-adic symmetric domains. Funkc. Anal. i Prilož. 10(2), 29–40 (1976). (Russian)

    MathSciNet  Google Scholar 

  5. Fargues, L.: Cohomologie des espaces de modules de groupes \(p\)-divisibles et correspondances de Langlands locales. Astérisque 291, 1–199 (2004)

    MathSciNet  MATH  Google Scholar 

  6. Görtz, U.: On the flatness of models of certain Shimura varieties of PEL-type. Math. Ann. 321(3), 689–727 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  7. Görtz, U., He, X.: Basic loci in Shimura varieties of Coxeter type. Camb. J. Math. 3(3), 323–353 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gordon, J.: Transfer to characteristic zero, appendix to [39]

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Jacquet, H., Rallis, S.: On the Gross–Prasad conjecture for unitary groups. In: On Certain \(L\)-Functions, Clay Math. Proc., 13, Amer. Math. Soc., Providence, RI, pp. 205–264 (2011)

  11. Kudla, S., Rapoport, M.: Special cycles on unitary Shimura varieties I. Unramified local theory. Invent. Math. 184(3), 629–682 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kudla, S., Rapoport, M.: An alternative description of the Drinfeld \(p\)-adic half-plane. Ann. Inst. Fourier (Grenoble) 64(3), 1203–1228 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kudla, S., Rapoport, M.: Special cycles on the \(\Gamma _0(p^n)\)-moduli curve (unpublished)

  14. Lau, E.: Smoothness of the truncated display functor. J. Am. Math. Soc. 26(1), 129–165 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mihatsch, A.: An arithmetic transfer identity. Manuscr. Math. 150(1), 1–19 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mihatsch, A.: On the arithmetic fundamental lemma conjecture through Lie algebras, to appear in Math. Z.

  17. Mihatsch, A.: Relative unitary RZ-spaces and the Arithmetic Fundamental Lemma, preprint (2016). arXiv:1611.06520 [math.AG]

  18. Pappas, G.: On the arithmetic moduli schemes of PEL Shimura varieties. J. Algebraic Geom. 9(3), 577–605 (2000)

    MathSciNet  MATH  Google Scholar 

  19. Pappas, G., Rapoport, M.: Twisted loop groups and their affine flag varieties. Adv. Math. 219(1), 118–198 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Pappas, G., Rapoport, M.: Local models in the ramified case. III. Unitary groups. J. Inst. Math. Jussieu 8(3), 507–564 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Pappas, G., Rapoport, M., Smithling, B.: Local models of Shimura varieties, I. Geometry and combinatorics. In: Handbook of Moduli. Vol. III, Adv. Lect. Math. (ALM), vol. 26, pp. 135–217. Int. Press, Somerville, MA (2013)

  22. Pappas, G., Zhu, X.: Local models of Shimura varieties and a conjecture of Kottwitz. Invent. Math. 194(1), 147–254 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Rallis, S., Schiffmann, G.: Multiplicity one conjectures, preprint (2007). arXiv:0705.2168 [math.NT]

  24. Rapoport, M., Smithling, B., Zhang, W.: On the arithmetic transfer conjecture for exotic smooth formal moduli spaces, to appear in Duke Math. J. (accepted)

  25. Rapoport, M., Smithling, B., Zhang, W.: Arithmetic diagonal cycles on unitary Shimura varieties (in preparation)

  26. Rapoport, M., Terstiege, U., Wilson, S.: The supersingular locus of the Shimura variety for \({\rm GU}(1, n - 1)\) over a ramified prime. Math. Z. 276(3–4), 1165–1188 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. Rapoport, M., Terstiege, U., Zhang, W.: On the arithmetic fundamental lemma in the minuscule case. Compos. Math. 149(10), 1631–1666 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Rapoport, M., Zink, Th: Period Spaces for \(p\)-Divisible Groups, Annals of Mathematics Studies, vol. 141. Princeton University Press, Princeton (1996)

    MATH  Google Scholar 

  29. Roy, A.: Cancellation of quadratic forms over commutative rings. J. Algebra 10, 286–298 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  30. Sankaran, S.: Unitary cycles on Shimura curves and the Shimura lift I. Doc. Math. 18, 1403–1464 (2013)

    MathSciNet  MATH  Google Scholar 

  31. Sankaran, S.: e-mail to Rapoport (11 April 2015)

  32. Smithling, B.: On the moduli description of local models for ramified unitary groups. Int. Math. Res. Not. 2015(24), 13493–13532 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  33. Vollaard, I.: Endomorphisms of quasi-canonical lifts, in [2], pp. 105–112

  34. Vollaard, I.: The supersingular locus of the Shimura variety for \(GU(1, s)\). Can. J. Math. 62, 668–720 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. Vollaard, I., Wedhorn, T.: The supersingular locus of the Shimura variety for \({\rm GU}(1, n- 1)\) II. Invent. Math. 184(3), 591–627 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wewers, S.: Canonical and quasi-canonical liftings, in [2], pp. 67–86

  37. Wu, H.: The supersingular locus of unitary Shimura varieties with exotic good reduction, preprint (2016). arXiv:1609.08775 [math.AG]

  38. Yu, S.: in preparation

  39. Yun, Z.: The fundamental lemma of Jacquet–Rallis in positive characteristics. Duke Math. J. 156(2), 167–228 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  40. Yun, Z.: An arithmetic fundamental lemma for function fields (in preparation)

  41. Zhang, W.: On arithmetic fundamental lemmas. Invent. Math. 188(1), 197–252 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  42. Zhang, W.: On the smooth transfer conjecture of Jacquet–Rallis for \(n=3\). Ramanujan J. 29(1–3), 225–256 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  43. Zhang, W.: Fourier transform and the global Gan–Gross–Prasad conjecture for unitary groups. Ann. Math. (2) 180(3), 971–1049 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We are grateful to U. Görtz, X. He, and S. Yu for helpful discussions. We also acknowledge the hospitality of the ESI (Vienna) and the MFO (Oberwolfach), where part of this work was carried out. We finally thank the referee for his/her remarks on the text. M.R. is supported by a Grant from the Deutsche Forschungsgemeinschaft through the Grant SFB/TR 45. B.S. is supported by a Simons Foundation Grant #359425 and an NSA Grant H98230-16-1-0024. W.Z. is supported by NSF Grants DMS #1301848 and #1601144, and by a Sloan research fellowship.

Author information

Authors and Affiliations

  1. Mathematisches Institut der Universität Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    M. Rapoport

  2. Department of Mathematics, Johns Hopkins University, 3400 N. Charles St., Baltimore, MD, 21218, USA

    B. Smithling

  3. Department of Mathematics, Columbia University, 2990 Broadway, New York, NY, 10027, USA

    W. Zhang

Authors
  1. M. Rapoport
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. Smithling
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. W. Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. Smithling.

Additional information

Communicated by Toby Gee.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rapoport, M., Smithling, B. & Zhang, W. Regular formal moduli spaces and arithmetic transfer conjectures. Math. Ann. 370, 1079–1175 (2018). https://doi.org/10.1007/s00208-017-1526-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-017-1526-2

Search

Navigation