Hostname: page-component-cb9f654ff-d5ftd Total loading time: 0 Render date: 2025-08-09T15:20:40.336Z Has data issue: false hasContentIssue false
  • English
  • Français

Arithmetic diagonal cycles on unitary Shimura varieties

Part of: Arithmetic problems. Diophantine geometry Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  27 October 2020

M. Rapoport ,
B. Smithling  and
W. Zhang
M. Rapoport
Affiliation:
Mathematisches Institut der Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany rapoport@math.uni-bonn.de Department of Mathematics, University of Maryland, College Park, MD 20742, USA
B. Smithling
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA corgcorgcorg@gmail.com
W. Zhang
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA weizhang@mit.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We define variants of PEL type of the Shimura varieties that appear in the context of the arithmetic Gan–Gross–Prasad (AGGP) conjecture. We formulate for them a version of the AGGP conjecture. We also construct (global and semi-global) integral models of these Shimura varieties and formulate for them conjectures on arithmetic intersection numbers. We prove some of these conjectures in low dimension.

Keywords

arithmetic Gan–Gross–Prasad conjectureunitary Shimura varietiesarithmetic diagonal cyclesrelative trace formulaGross–Zagier formulaspecial values of L-functions

MSC classification

Primary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 14G35: Modular and Shimura varieties 14G40: Arithmetic varieties and schemes; Arakelov theory; heights

Information

Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 9 , September 2020 , pp. 1745 - 1824
Check for updates
Copyright
Copyright © The Author(s) 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Arancibia, N., Moeglin, C. and Renard, D., Paquets d'Arthur des groupes classiques et unitaires, Ann. Fac. Sci. Toulouse Math. (6) 27 (2018), 1023–1105.10.5802/afst.1590CrossRefGoogle Scholar
Arthur, J., Unipotent automorphic representations: conjectures, Astérisque 171–172 (1989), 13–71.Google Scholar
Arthur, J., The endoscopic classification of representations. Orthogonal and symplectic groups, American Mathematical Society Colloquium Publications, vol. 61 (American Mathematical Society, Providence, RI, 2013).Google Scholar
Beilinson, A., Height pairing between algebraic cycles, in Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985), Contemporary Mathematics, vol. 67 (American Mathematical Society, Providence, RI, 1987), 1–24.Google Scholar
Bloch, S., Height pairings for algebraic cycles, J. Pure Appl. Algebra 34 (1984), 119–145.10.1016/0022-4049(84)90032-XCrossRefGoogle Scholar
Borel, A. and Wallach, N., Continuous cohomology, discrete subgroups, and representations of reductive groups, second edition, Mathematical Surveys and Monographs, vol. 67 (American Mathematical Society, Providence, RI, 2000).Google Scholar
Bruinier, J., Howard, B., Kudla, S., Rapoport, M. and Yang, T., Modularity of generating series of divisors on unitary Shimura varieties, Astérisque 421 (2020), 7–125.Google Scholar
Chai, C.-L., Conrad, B. and Oort, F., Complex multiplication and lifting problems, Mathematical Surveys and Monographs, vol. 195 (American Mathematical Society, Providence, RI, 2014).Google Scholar
Chenevier, G., email to the authors, 1 Oct. 2016.Google Scholar
Cho, S., The basic locus of the unitary Shimura variety with parahoric level structure, and special cycles, Preprint (2018), arXiv:1807.09997.Google Scholar
Deligne, P., Travaux de Shimura, in Séminaire Bourbaki, 23ème année (1970/71), Exp. No. 389, Lecture Notes in Mathematics, vol. 244 (Springer, Berlin, 1971), 123–165.10.1007/BFb0058700CrossRefGoogle Scholar
Faltings, G., Almost étale extensions, Astérisque 279 (2002), 185–270.Google Scholar
Fontaine, J.-M., Le corps des périodes $p$-adiques, with an appendix by P. Colmez, Astérisque 223 (1994), 59–111.Google Scholar
Gan, W. T., Gross, B. and Prasad, D., Symplectic local root numbers, central critical $L$-values, and restriction problems in the representation theory of classical groups, Astérisque 346 (2012), 1–109.Google Scholar
Gillet, H., Arithmetic intersection theory on Deligne–Mumford stacks, in Motives and algebraic cycles, Fields Institute Communications, vol. 56 (American Mathematical Society, Providence, RI, 2009), 93–109.Google Scholar
Gillet, H. and Soulé, C., Arithmetic intersection theory, Publ. Math. Inst. Hautes Etudes Sci. 72 (1990), 93–174.Google Scholar
Görtz, U., On the flatness of models of certain Shimura varieties of PEL-type, Math. Ann. 321 (2001), 689–727.CrossRefGoogle Scholar
Gross, B. and Zagier, D., Heegner points and derivatives of $L$-series, Invent. Math. 84 (1986), 225–320.10.1007/BF01388809CrossRefGoogle Scholar
Harris, M. and Taylor, R., The geometry and cohomology of some simple Shimura varieties, with an appendix by V. Berkovich, Annals of Mathematics Studies, vol. 151 (Princeton University Press, Princeton, NJ, 2001).Google Scholar
Howard, B., Complex multiplication cycles and Kudla–Rapoport divisors, Ann. of Math. (2) 176 (2012), 1097–1171.10.4007/annals.2012.176.2.9CrossRefGoogle Scholar
Howard, B., Complex multiplication cycles and Kudla–Rapoport divisors, II, Amer. J. Math. 137 (2015), 639–698.10.1353/ajm.2015.0021CrossRefGoogle Scholar
Jacquet, H., Piatetskii-Shapiro, I. I. and Shalika, J. A., Rankin–Selberg convolutions, Amer. J. Math. 105 (1983), 367–464.Google Scholar
Jacquet, H. and Rallis, S., On the Gross-Prasad conjecture for unitary groups, in On certain L-functions, Clay Mathematics Proceedings, vol. 13 (American Mathematical Society, Providence, RI, 2011), 205–264.Google Scholar
Jannsen, U., Mixed motives and algebraic K-theory, with appendices by S. Bloch and C. Schoen, Lecture Notes in Mathematics, vol. 1400 (Springer, Berlin, 1990).10.1007/BFb0085080CrossRefGoogle Scholar
Kaletha, T., Minguez, A., Shin, S. W. and White, P.-J., Endoscopic classification of representations: inner forms of unitary groups, Preprint (2014), arXiv:1409.3731.Google Scholar
Katz, N. and Mazur, B., Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, vol. 108 (Princeton University Press, Princeton, NJ, 1985).Google Scholar
Kisin, M. and Pappas, G., Integral models of Shimura varieties with parahoric level structure, Publ. Math. Inst. Hautes Études Sci. 128 (2018), 121–218.10.1007/s10240-018-0100-0CrossRefGoogle Scholar
Kottwitz, R., Shimura varieties and λ-adic representations, in Automorphic forms, Shimura varieties, and L-functions, vol. I (Ann Arbor, MI, 1988), Perspectives in Mathematics, vol. 10 (Academic Press, Boston, MA, 1990), 161–209.Google Scholar
Kottwitz, R., Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5 (1992), 373–444.Google Scholar
Kudla, S. and Rapoport, M., Special cycles on unitary Shimura varieties I. Unramified local theory, Invent. Math. 184 (2011), 629–682.10.1007/s00222-010-0298-zCrossRefGoogle Scholar
Kudla, S. and Rapoport, M., Special cycles on unitary Shimura varieties II. Global theory, J. Reine Angew. Math. 697 (2014), 91–157.Google Scholar
Kudla, S. and Rapoport, M., New cases of $p$-adic uniformization, Astérisque 370 (2015), 207–241.Google Scholar
Kudla, S., Rapoport, M. and Zink, Th., On the p-adic uniformization of unitary Shimura curves, Preprint (2020), arXiv:2007.05211Google Scholar
Künnemann, K., Projective regular models for abelian varieties, semistable reduction, and the height pairing, Duke Math. J. 95 (1998), 161–212.CrossRefGoogle Scholar
Künnemann, K., Height pairings for algebraic cycles on abelian varieties, Ann. Sci. Éc. Norm. Supér. 34 (2001), 503–523.Google Scholar
Mihatsch, A., Relative unitary RZ-spaces and the arithmetic fundamental lemma, J. Inst. Math. Jussieu, to appear. PhD thesis, Bonn (2016), arXiv:1611.06520.Google Scholar
Mihatsch, A., On the arithmetic fundamental lemma conjecture through Lie algebras, Math. Z. 287 (2017), 181–197.Google Scholar
Mok, C. P., Endoscopic classification of representations of quasi-split unitary groups, Mem. Amer. Math. Soc. 235 (2015).Google Scholar
Morel, S. and Suh, J., The standard sign conjecture on algebraic cycles: The case of Shimura varieties, J. Reine Angew. Math. 748 (2019), 139–151.CrossRefGoogle Scholar
Pappas, G. and Rapoport, M., Local models in the ramified case. III. Unitary groups, J. Inst. Math. Jussieu 8 (2009), 507–564.CrossRefGoogle Scholar
Pappas, G., Rapoport, M. and Smithling, B., Local models of Shimura varieties, I. Geometry and combinatorics, in Handbook of moduli, vol. III, Advanced Lectures in Mathematics, vol. 26 (International Press, Somerville, MA, 2013), 135–217.Google Scholar
Rapoport, M., Smithling, B. and Zhang, W., On the arithmetic transfer conjecture for exotic smooth formal moduli spaces, Duke Math. J. 166 (2017), 2183–2336.10.1215/00127094-2017-0003CrossRefGoogle Scholar
Rapoport, M., Smithling, B. and Zhang, W., Regular formal moduli spaces and arithmetic transfer conjectures, Math. Ann. 370 (2018), 1079–1175.10.1007/s00208-017-1526-2CrossRefGoogle Scholar
Rapoport, M., Smithling, B. and Zhang, W., On Shimura varieties for unitary groups, Pure Appl. Math. Q., to appear. Preprint (2019), arXiv:1906.12346.Google Scholar
Rapoport, M., Terstiege, U. and Wilson, S., The supersingular locus of the Shimura variety for $\textrm {GU}(1, n - 1)$ over a ramified prime, Math. Z. 276 (2014), 1165–1188.10.1007/s00209-013-1240-zCrossRefGoogle Scholar
Rapoport, M., Terstiege, U. and Zhang, W., On the arithmetic fundamental lemma in the minuscule case, Compos. Math. 149 (2013), 1631–1666.10.1112/S0010437X13007239CrossRefGoogle Scholar
Rapoport, M. and Zink, Th., Period spaces for p-divisible groups, Annals of Mathematics Studies, vol. 141 (Princeton University Press, Princeton, NJ, 1996).10.1515/9781400882601CrossRefGoogle Scholar
Rapoport, M. and Zink, Th., On the Drinfeld moduli problem of $p$-divisible groups, Camb. J. Math. 5 (2017), 229–279.10.4310/CJM.2017.v5.n2.a2CrossRefGoogle Scholar
Smithling, B., On the moduli description of local models for ramified unitary groups, Int. Math. Res. Not. IMRN 2015 (2015), 13493–13532.Google Scholar
Tsuji, T., $p$-adic étale cohomology and crystalline cohomology in the semi-stable reduction case, Invent. Math. 137 (1999), 233–411.CrossRefGoogle Scholar
Yuan, X., Zhang, S.-W. and Zhang, W., The Gross–Zagier formula on Shimura curves, Annals of Mathematics Studies, vol. 184 (Princeton University Press, Princeton, NJ, 2013).10.1515/9781400845644CrossRefGoogle Scholar
Yun, Z., The fundamental lemma of Jacquet–Rallis in positive characteristics, Duke Math. J. 156 (2011), 167–228.10.1215/00127094-2010-210CrossRefGoogle Scholar
Zhang, W., Relative trace formula and arithmetic Gross–Prasad conjecture, Unpublished manuscript (2009).Google Scholar
Zhang, W., On arithmetic fundamental lemmas, Invent. Math. 188 (2012), 197–252.10.1007/s00222-011-0348-1CrossRefGoogle Scholar
Zhang, W., Gross-Zagier formula and arithmetic fundamental lemma, in Fifth International Congress of Chinese Mathematicians. Part 1, 2, AMS/IP Studies in Advanced Mathematics, vol. 51 (American Mathematical Society, Providence, RI, 2012), 447–459.10.1090/amsip/051.1/32CrossRefGoogle Scholar
Zhang, W., Fourier transform and the global Gan–Gross–Prasad conjecture for unitary groups, Ann. of Math. (2) 180 (2014), 971–1049.10.4007/annals.2014.180.3.4CrossRefGoogle Scholar
Zhang, W., Automorphic periods and the central value of Rankin-Selberg $L$-function, J. Amer. Math. Soc. 27 (2014), 541–612.Google Scholar
Zhang, W., Periods, cycles, and $L$-functions: a relative trace formula approach, in Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures (World Scientific, Hackensack, NJ, 2018), 487–521.Google Scholar
Zink, T., Über die schlechte Reduktion einiger Shimuramannigfaltigkeiten, Compos. Math. 45 (1982), 15–107.Google Scholar
Zydor, M., Les formules des traces relatives de Jacquet-Rallis grossières, J. Reine Angew. Math. 762 (2020), 195–259.Google Scholar