Skip to main content

Arithmetic intersection on GSpin Rapoport–Zink spaces

  • Chao Li (a1) and Yihang Zhu (a2)

We prove an explicit formula for the arithmetic intersection number of diagonal cycles on GSpin Rapoport–Zink spaces in the minuscule case. This is a local problem arising from the arithmetic Gan–Gross–Prasad conjecture for orthogonal Shimura varieties. Our formula can be viewed as an orthogonal counterpart of the arithmetic–geometric side of the arithmetic fundamental lemma proved by Rapoport–Terstiege–Zhang in the minuscule case.

Hide All
[BP17] Bueltel, O. and Pappas, G., $(G,\unicode[STIX]{x1D707})$ -displays and Rapoport–Zink spaces, Preprint (2017),arXiv:1702.00291.
[DL76] Deligne, P. and Lusztig, G., Representations of reductive groups over finite fields , Ann. of Math. (2) 103 (1976), 103161.
[GGP12] Gan, W. T., Gross, B. H. and Prasad, D., Symplectic local root numbers, central critical L values, and restriction problems in the representation theory of classical groups , Astérisque 346 (2012), 1109; sur les conjectures de Gross et Prasad. I.
[GK92] Gross, B. H. and Kudla, S. S., Heights and the central critical values of triple product L-functions , Compos. Math. 81 (1992), 143209.
[GS95] Gross, B. H. and Schoen, C., The modified diagonal cycle on the triple product of a pointed curve , Ann. Inst. Fourier (Grenoble) 45 (1995), 649679.
[GZ86] Gross, B. H. and Zagier, D. B., Heegner points and derivatives of L-series , Invent. Math. 84 (1986), 225320.
[Har95] Harris, J., Algebraic geometry. A first course, Graduate Texts in Mathematics, vol. 33 (Springer, New York, 1995); corrected reprint of the 1992 original.
[HP14] Howard, B. and Pappas, G., On the supersingular locus of the GU(2, 2) Shimura variety , Algebra Number Theory 8 (2014), 16591699.
[HP17] Howard, B. and Pappas, G., Rapoport–Zink spaces for spinor groups , Compos. Math. 153 (2017), 10501118.
[Ive72] Iversen, B., A fixed point formula for action of tori on algebraic varieties , Invent. Math. 16 (1972), 229236.
[Kim13] Kim, W., Rapoport–Zink spaces of Hodge type, Preprint (2013), arXiv:1308.5537.
[Kis10] Kisin, M., Integral models for Shimura varieties of abelian type , J. Amer. Math. Soc. 23 (2010), 9671012.
[Kri16] Krishna, R. M., Relative trace formula for SO2 $\times$ SO3 and the Waldspurger formula, ProQuest LLC, Ann Arbor, MI, PhD thesis, Columbia University (2016).
[LZ17] Li, C. and Zhu, Y., Remarks on the arithmetic fundamental lemma , Algebra Number Theory 11 (2017), 24252445.
[Lus76/77] Lusztig, G., Coxeter orbits and eigenspaces of Frobenius , Invent. Math. 38 (1976–1977), 101159.
[Lus11] Lusztig, G., From conjugacy classes in the Weyl group to unipotent classes , Represent. Theory 15 (2011), 494530.
[Mad16] Madapusi Pera, K., Integral canonical models for spin Shimura varieties , Compos. Math. 152 (2016), 769824.
[RTZ13] Rapoport, M., Terstiege, U. and Zhang, W., On the arithmetic fundamental lemma in the minuscule case , Compos. Math. 149 (2013), 16311666.
[RZ96] Rapoport, M. and Zink, T., Period spaces for p-divisible groups, Annals of Mathematics Studies, vol. 141 (Princeton University Press, Princeton, NJ, 1996).
[YZZ12] Yuan, X., Zhang, S.-W. and Zhang, W., Triple product L-series and Gross–Kudla–Schoen cycles, Preprint (2012),∼wzhang/math/online/triple.pdf.
[YZZ13] 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).
[Zha12] Zhang, W., On arithmetic fundamental lemmas , Invent. Math. 188 (2012), 197252.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
  • URL: /core/journals/compositio-mathematica
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

MSC classification


Full text views

Total number of HTML views: 2
Total number of PDF views: 42 *
Loading metrics...

Abstract views

Total abstract views: 82 *
Loading metrics...

* Views captured on Cambridge Core between 16th May 2018 - 25th June 2018. This data will be updated every 24 hours.