Science China Mathematics

, Volume 62, Issue 11, pp 2409–2422 | Cite as

The arithmetic fundamental lemma: An update

  • Wei ZhangEmail author


This is an expository article on the recent progress on the arithmetic fundamental lemma conjecture, based largely on Zhang (2019). Beside stating the local conjecture, we will present three global intersection problems along with some constructions of algebraic cycles.


arithmetic fundamental lemma arithmetic Gan-Gross-Prasad conjecture Gross-Zagier formula relative trace formula 


11F67 11G40 14G35 



This work was supported by the National Science Foundation of USA (Grant No. DMS #1901642). The author thanks the referees for helpful comments.


  1. 1.
    Bruinier J, Howard B, Kudla S, et al. Modularity of generating series of divisors on unitary Shimura varieties. ArXiv:1702.07812, 2017Google Scholar
  2. 2.
    Faltings G, Chai C-L. Degeneration of Abelian Varieties: With An Appendix by David Mumford. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 22. Berlin: Springer-Verlag, 1990CrossRefGoogle Scholar
  3. 3.
    Fulton W. Intersection Theory, 2nd ed. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 2. Berlin: Springer-Verlag, 1998CrossRefGoogle Scholar
  4. 4.
    Gan W T, Gross B, Prasad D. Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups. Astérisque, 2012, 346: 1–109MathSciNetzbMATHGoogle Scholar
  5. 5.
    Gillet H, Soulé C. Intersection theory using Adams operations. Invent Math, 1987, 90: 243–277MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gross B H, Zagier D. Heegner points and derivatives of L-series. Invent Math, 1986, 84: 225–320MathSciNetCrossRefGoogle Scholar
  7. 7.
    He X, Li C, Zhu Y. Fine Deligne-Lusztig varieties and arithmetic fundamental lemmas. ArXiv:1901.02870, 2019Google Scholar
  8. 8.
    Jacquet H, Rallis S. On the Gross-Prasad conjecture for unitary groups. In: On Certain L-Functions. Clay Mathematics Proceedings, vol. 13. Providence: Amer Math Soc, 2011, 205–264zbMATHGoogle Scholar
  9. 9.
    Kudla S, Rapoport M. Special cycles on unitary Shimura varieties, I: Unramified local theory. Invent Math, 2011, 184: 629–682MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kudla S, Rapoport M. Special cycles on unitary Shimura varieties, II: Global theory. J Reine Angew Math, 2014: 697: 91–157MathSciNetzbMATHGoogle Scholar
  11. 11.
    Li C, Zhu Y. Remarks on the arithmetic fundamental lemma. Algebra Number Theory, 2017, 11: 2425–2445MathSciNetCrossRefGoogle Scholar
  12. 12.
    Liu Y. Fourier-Jacobi cycles and arithmetic relative trace formula., 2018
  13. 13.
    Mihatsch A. On the arithmetic fundamental lemma conjecture through Lie algebras. Math Z, 2017, 287: 181–197MathSciNetCrossRefGoogle Scholar
  14. 14.
    Mumford D, Fogarty J, Kirwan F. Geometric Invariant Theory, 3rd ed. Ergebnisse der Mathematik und ihrer Grenzgebiete (2), vol. 34. Berlin: Springer-Verlag, 1994CrossRefGoogle Scholar
  15. 15.
    Rapoport M, Smithling B, Zhang W. On the arithmetic transfer conjecture for exotic smooth formal moduli spaces. Duke Math J, 2017, 166: 2183–2336MathSciNetCrossRefGoogle Scholar
  16. 16.
    Rapoport M, Smithling B, Zhang W. Arithmetic diagonal cycles on unitary Shimura varieties. ArXiv:1710.06962, 2017Google Scholar
  17. 17.
    Rapoport M, Smithling B, Zhang W. Regular formal moduli spaces and arithmetic transfer conjectures. Math Ann, 2018, 370: 1079–1175MathSciNetCrossRefGoogle Scholar
  18. 18.
    Rapoport M, Smithling B, Zhang W. On Shimura varieties for unitary groups. ArXiv:1906.12346, 2019Google Scholar
  19. 19.
    Rapoport M, Terstiege U, Zhang W. On the arithmetic fundamental lemma in the minuscule case. Compos Math, 2013, 149: 1631–1666MathSciNetCrossRefGoogle Scholar
  20. 20.
    Smithling B. An Arithmetic Intersection Conjecture. Proceedings of the First Annual Meeting of the ICCM. Guangzhou: Sun Yat-Sen University, 2017Google Scholar
  21. 21.
    Yun Z. The fundamental lemma of Jacquet-Rallis in positive characteristics. Duke Math J, 2011, 156: 167–228MathSciNetCrossRefGoogle Scholar
  22. 22.
    Yun Z, Zhang W. Shtukas and the Taylor expansion of L-functions. Ann of Math (2), 2017, 186: 767–911MathSciNetCrossRefGoogle Scholar
  23. 23.
    Zhang W. On arithmetic fundamental lemmas. Invent Math, 2012, 188: 197–252MathSciNetCrossRefGoogle Scholar
  24. 24.
    Zhang W. Gross-Zagier formula and arithmetic fundamental lemma. In: Fifth International Congress of Chinese Mathematicians. AMS/IP Studies in Advanced Mathematics, vol. 51. Providence: Amer Math Soc, 2012, 447–459zbMATHGoogle Scholar
  25. 25.
    Zhang W. Periods, cycles, and L-functions: A relative trace formula approach. In: Proceedings of the International Congress of Mathematicians, vol. 1. Rio de Janeiro: International Congress of Mathematicians, 2018, 485–520Google Scholar
  26. 26.
    Zhang W. Weil representation and arithmetic fundamental lemma., 2019

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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