The arithmetic fundamental lemma: An update

Science China Mathematics - Tập 62 - Trang 2409-2422 - 2019
Wei Zhang1
1Department of Mathematics, Massachusetts Institute of Technology, Cambridge, USA

Tóm tắt

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.

Tài liệu tham khảo

Bruinier J, Howard B, Kudla S, et al. Modularity of generating series of divisors on unitary Shimura varieties. ArXiv:1702.07812, 2017 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, 1990 Fulton W. Intersection Theory, 2nd ed. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 2. Berlin: Springer-Verlag, 1998 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: 1–109 Gillet H, Soulé C. Intersection theory using Adams operations. Invent Math, 1987, 90: 243–277 Gross B H, Zagier D. Heegner points and derivatives of L-series. Invent Math, 1986, 84: 225–320 He X, Li C, Zhu Y. Fine Deligne-Lusztig varieties and arithmetic fundamental lemmas. ArXiv:1901.02870, 2019 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, 205–264 Kudla S, Rapoport M. Special cycles on unitary Shimura varieties, I: Unramified local theory. Invent Math, 2011, 184: 629–682 Kudla S, Rapoport M. Special cycles on unitary Shimura varieties, II: Global theory. J Reine Angew Math, 2014: 697: 91–157 Li C, Zhu Y. Remarks on the arithmetic fundamental lemma. Algebra Number Theory, 2017, 11: 2425–2445 Liu Y. Fourier-Jacobi cycles and arithmetic relative trace formula. https://doi.org/gauss.math.yale.edu/~yl2269/FJcycle.pdf, 2018 Mihatsch A. On the arithmetic fundamental lemma conjecture through Lie algebras. Math Z, 2017, 287: 181–197 Mumford D, Fogarty J, Kirwan F. Geometric Invariant Theory, 3rd ed. Ergebnisse der Mathematik und ihrer Grenzgebiete (2), vol. 34. Berlin: Springer-Verlag, 1994 Rapoport M, Smithling B, Zhang W. On the arithmetic transfer conjecture for exotic smooth formal moduli spaces. Duke Math J, 2017, 166: 2183–2336 Rapoport M, Smithling B, Zhang W. Arithmetic diagonal cycles on unitary Shimura varieties. ArXiv:1710.06962, 2017 Rapoport M, Smithling B, Zhang W. Regular formal moduli spaces and arithmetic transfer conjectures. Math Ann, 2018, 370: 1079–1175 Rapoport M, Smithling B, Zhang W. On Shimura varieties for unitary groups. ArXiv:1906.12346, 2019 Rapoport M, Terstiege U, Zhang W. On the arithmetic fundamental lemma in the minuscule case. Compos Math, 2013, 149: 1631–1666 Smithling B. An Arithmetic Intersection Conjecture. Proceedings of the First Annual Meeting of the ICCM. Guangzhou: Sun Yat-Sen University, 2017 Yun Z. The fundamental lemma of Jacquet-Rallis in positive characteristics. Duke Math J, 2011, 156: 167–228 Yun Z, Zhang W. Shtukas and the Taylor expansion of L-functions. Ann of Math (2), 2017, 186: 767–911 Zhang W. On arithmetic fundamental lemmas. Invent Math, 2012, 188: 197–252 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, 447–459 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, 485–520 Zhang W. Weil representation and arithmetic fundamental lemma. http://math.mit.edu/~wz2113/math/online/AFL2019.pdf, 2019