Duke Mathematical Journal

On the arithmetic transfer conjecture for exotic smooth formal moduli spaces

M. Rapoport, B. Smithling, and W. Zhang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In the relative trace formula approach to the arithmetic Gan–Gross–Prasad conjecture, we formulate a local conjecture (arithmetic transfer) in the case of an exotic smooth formal moduli space of p-divisible groups, associated to a unitary group relative to a ramified quadratic extension of a p-adic field. We prove our conjecture in the case of a unitary group in three variables.

Article information

Source
Duke Math. J. Volume 166, Number 12 (2017), 2183-2336.

Dates
Received: 31 March 2015
Revised: 3 November 2016
First available in Project Euclid: 9 June 2017

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1496995226

Digital Object Identifier
doi:10.1215/00127094-2017-0003

Subjects
Primary: 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35]
Secondary: 14G17: Positive characteristic ground fields 22E50: Representations of Lie and linear algebraic groups over local fields [See also 20G05]

Keywords
arithmetic Gan–Gross–Prasad conjecture arithmetic fundamental lemma Rapoport–Zink space special cycles

Citation

Rapoport, M.; Smithling, B.; Zhang, W. On the arithmetic transfer conjecture for exotic smooth formal moduli spaces. Duke Math. J. 166 (2017), no. 12, 2183--2336. doi:10.1215/00127094-2017-0003. https://projecteuclid.org/euclid.dmj/1496995226


Export citation

References

  • [1] K. Arzdorf, On local models with special parahoric level structure, Michgan Math. J. 58 (2009), 683–710.
  • [2] B. Conrad, “Gross–Zagier revisited,” in Heegner Points and Rankin $L$-Series, Math. Sci. Res. Inst. Publ. 49, Cambridge Univ. Press, Cambridge, 2004, 67–163.
  • [3] L. Fargues, “Cohomologie des espaces de modules de groupes $p$-divisibles et correspondances de Langlands locales,” in Variétés de Shimura, espaces de Rapoport-Zink et correspondances de Langlands locales, Astérisque 291, Soc. Math. France, Paris, 2004, 1–199.
  • [4] W. T. Gan, B. Gross, and D. Prasad, “Symplectic local root numbers, central critical $L$-values, and restriction problems in the representation theory of classical groups,” in Sur les conjectures de Gross et Prasad, I, Astérisque 346, Soc. Math. France, Paris, 2012, 1–109.
  • [5] U. Görtz and M. Rapoport, eds., ARGOS Seminar on Intersections of Modular Correspondences, Astérisque 312, Soc. Math. France, Paris, 2007.
  • [6] B. Gross and D. Zagier, Heegner points and derivatives of $L$-series, Invent. Math. 84 (1986), 225–320.
  • [7] R. Jacobowitz, Hermitian forms over local fields, Amer. J. Math. 84 (1962), 441–465.
  • [8] H. Jacquet and S. Rallis, “On the Gross-Prasad conjecture for unitary groups,” in On Certain $L$-Functions, Clay Math. Proc. 13, Amer. Math. Soc., Providence, 2011, 205–264.
  • [9] S. Kudla and M. Rapoport, Special cycles on unitary Shimura varieties, I: Unramified local theory, Invent. Math. 184 (2011), 629–682.
  • [10] S. Kudla and M. Rapoport, An alternative description of the Drinfeld $p$-adic half-plane, Ann. Inst. Fourier (Grenoble) 64 (2014), 1203–1228.
  • [11] S. Kudla and M. Rapoport, Special cycles on unitary Shimura varieties, II: Global theory, J. Reine Angew. Math. 697 (2014), 91–157.
  • [12] A. Mihatsch, On the arithmetic fundamental lemma through Lie algebras, Math. Z., published online 7 January 2017.
  • [13] G. Pappas, On the arithmetic moduli schemes of PEL Shimura varieties, J. Algebraic Geom. 9 (2000), 577–605.
  • [14] G. Pappas and M. Rapoport, Twisted loop groups and their affine flag varieties, Adv. Math. 219 (2008), 118–198.
  • [15] G. Pappas and M. Rapoport, Local models in the ramified case, III: Unitary groups, J. Inst. Math. Jussieu 8 (2009), 507–564.
  • [16] G. Pappas, M. Rapoport, and B. Smithling, “Local models of Shimura varieties, I: Geometry and combinatorics,” in Handbook of Moduli, Vol. III, Adv. Lect. Math. (ALM) 26, Int. Press, Somerville, Mass., 2013, 135–217.
  • [17] S. Rallis and G. Schiffmann, Multiplicity one conjectures, preprint, arXiv:0705.2168v1 [math.RT].
  • [18] M. Rapoport, “Deformations of isogenies of formal groups,” in ARGOS Seminar on Intersections of Modular Correspondences, Astérisque 312, Soc. Math. France, Paris, 2007, 139–169.
  • [19] M. Rapoport, B. Smithling, and W. Zhang, Regular formal moduli spaces and arithmetic transfer conjectures, Math. Ann., published online 10 May 2017.
  • [20] M. Rapoport, B. Smithling, and W. Zhang, Arithmetic diagonal cycles on unitary Shimura varieties, in preparation.
  • [21] M. Rapoport, U. Terstiege, and S. Wilson, The supersingular locus of the Shimura variety for $\mathrm{GU}(1,n-1)$ over a ramified prime, Math. Z. 276 (2014), 1165–1188.
  • [22] M. Rapoport, U. Terstiege, and W. Zhang, On the arithmetic fundamental lemma in the minuscule case, Compos. Math. 149 (2013), 1631–1666.
  • [23] M. Rapoport and E. Viehmann, Towards a theory of local Shimura varieties, Münster J. Math. 7 (2014), 273–326.
  • [24] M. Rapoport and T. Zink, Period Spaces for $p$-Divisible Groups, Ann. Math. Stud. 141, Princeton Univ. Press, Princeton, N.J., 1996.
  • [25] B. Smithling, On the moduli description of local models for ramified unitary groups, Int. Math. Res. Not. IMRN 2015, no. 24, 13493–13532.
  • [26] J. T. Tate, Jr., Fourier analysis in number fields and Hecke’s zeta functions, Ph.D. dissertation, Princeton Univ., Princeton, 1950.
  • [27] U. Terstiege, Intersections of arithmetic Hirzebruch-Zagier cycles, Math. Ann. 349 (2011), 161–213.
  • [28] I. Vollaard, “Endomorphisms of quasi-canonical lifts,” in ARGOS Seminar on Intersections of Modular Correspondences, Astérisque 312, Soc. Math. France, Paris, 2007, 105–112.
  • [29] I. Vollaard, The supersingular locus of the Shimura variety for $\mathrm{GU}(1,s)$, Canad. J. Math. 62 (2010), 668–720.
  • [30] I. Vollaard and T. Wedhorn, The supersingular locus of the Shimura variety of $\mathrm{GU}(1,n-1)$, II, Invent. Math. 184 (2011), 591–627.
  • [31] S. Wewers, “Canonical and quasi-canonical liftings,” in ARGOS Seminar on Intersections of Modular Correspondences, Astérisque 312, Soc. Math. France, Paris, 2007, 67–86.
  • [32] Z. Yun, The fundamental lemma of Jacquet–Rallis in positive characteristics, with an appendix by J. Gordon, Duke Math. J. 156 (2011), 167–219.
  • [33] Z. Yun, An arithmetic fundamental lemma for function fields, in preparation.
  • [34] W. Zhang, On arithmetic fundamental lemmas, Invent. Math. 188 (2012), 197–252.
  • [35] W. Zhang, On the smooth transfer conjecture of Jacquet–Rallis for $n=3$, Ramanujan J. 29 (2012), 225–256.
  • [36] W. Zhang, Fourier transform and the global Gan–Gross–Prasad conjecture for unitary groups, Ann. of Math. (2) 180 (2014), 971–1049.
  • [37] W. Zhang, Relative trace formula and arithmetic Gross–Prasad conjecture, preprint, 2009.