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Weil representation and arithmetic fundamental lemma. (English) Zbl 1486.11059

The theorem of Gross and Zagier relates the Néron-Tate heights of Heegner points on modular curves to the central derivative of certain \(L\)-functions. The arithmetic Gan-Gross-Prasad conjecture is a generalization of this theorem to higher-dimensional Shimura varieties. The author has proposed an approach to establish the conjecture using an “arithmetic” relative trace formula. The paper is a major breakthrough in this approach. The paper proves the fundamental lemma for the relative trace formula, with the assumption that the local field is \({\mathbb Q}_p\) where \(p\) is a large enough prime. The argument should work for all local fields with large residue characteristic, in the general case there are just some claims that are expected to be true but need to be checked.
A relative trace formula is a distribution which can be decomposed into a sum of orbital integrals. The fundamental lemma are identities for the orbital integrals over a local field, when the orbital integrals are associated to a base function (identity element of Hecke algebra). Once such identities are established, it is strong evidence that the trace formula approach will work, and some partial results can follow with relatively small amount of extra work.
The reviewer does not claim to understand the details of the proof of the fundamental lemma, however the brilliance of the idea behind it is clear. Previously Jacquet introduced an idea to use Fourier transform in the proof of fundamental lemma. The current paper further developed the idea, and uses Weil representation (which encodes Fourier transform as Weyl group action) in the proof. This is combined with a local-global argument where the local orbital integral identity is derived from a global one, and an induction argument. The paper also used the same argument to reprove the fundamental lemma for Jacquet-Rallis relative trace formula that is used for Gan-Gross-Prasad conjecture for unitary groups. That fundamental lemma was originally proved using arithmetic geometric method by Z. Yun [Duke Math. J. 156, No. 2, 167–227 (2011; Zbl 1211.14039)] for positive characteristic case and extended to any local field with large residue characteristic using a standard argument in logic. In comparison, the proof presented in current paper can be considered an elementary proof.

MSC:

11F27 Theta series; Weil representation; theta correspondences
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
14C25 Algebraic cycles
14G35 Modular and Shimura varieties

Citations:

Zbl 1211.14039
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References:

[1] Beuzart-Plessis, R., Comparison of local spherical characters and the {I}chino-Ikeda conjecture for unitary groups (2016)
[2] Beuzart-Plessis, R., A new proof of {J}acquet-{R}allis’s fundamental lemma (2019)
[3] Bost, J.-B.; Gillet, H.; Soul\'{e}, C., Heights of projective varieties and positive {G}reen forms, J. Amer. Math. Soc.. Journal of the American Mathematical Society, 7, 903-1027 (1994) · Zbl 0973.14013 · doi:10.2307/2152736
[4] Burgos Gil, J. I.; Kramer, J.; K\"{u}hn, U., Cohomological arithmetic {C}how rings, J. Inst. Math. Jussieu. Journal of the Institute of Mathematics of Jussieu. JIMJ. Journal de l’Institut de Math\'{e}matiques de Jussieu, 6, 1-172 (2007) · Zbl 1115.14013 · doi:10.1017/S1474748007000011
[5] Bruinier, Jan Hendrik, Regularized theta lifts for orthogonal groups over totally real fields, J. Reine Angew. Math.. Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle’s Journal], 672, 177-222 (2012) · Zbl 1268.11058 · doi:10.1515/crelle.2011.163
[6] Bruinier, Jan Hendrik; Howard, B.; Kudla, S.; Rapoport, M.; Yang, T., Modularity of generating series of divisors on unitary {S}himura varieties (2017)
[7] Bruinier, Jan Hendrik; Kudla, Stephen S.; Yang, Tonghai, Special values of {G}reen functions at big {CM} points, Int. Math. Res. Not. IMRN. International Mathematics Research Notices. IMRN, 1917-1967 (2012) · Zbl 1281.11063 · doi:10.1093/imrn/rnr095
[8] Ehlen, Stephan; Sankaran, Siddarth, On two arithmetic theta lifts, Compos. Math.. Compositio Mathematica, 154, 2090-2149 (2018) · Zbl 1436.11046 · doi:10.1112/s0010437x18007327
[9] Gan, Wee Teck; Gross, Benedict H.; Prasad, Dipendra, Symplectic local root numbers, central critical {\(L\)} values, and restriction problems in the representation theory of classical groups. Sur les Conjectures de Gross et Prasad. I, Ast\'{e}risque, 346, 1-109 (2012) · Zbl 1280.22019
[10] Gillet, Henri, Arithmetic intersection theory on {D}eligne-{M}umford stacks. Motives and Algebraic Cycles, Fields Inst. Commun., 56, 93-109 (2009) · Zbl 1196.14022 · doi:10.1090/fic/056/04
[11] Gillet, Henri; Soul\'{e}, Christophe, Intersection theory using {A}dams operations, Invent. Math.. Inventiones Mathematicae, 90, 243-277 (1987) · Zbl 0632.14009 · doi:10.1007/BF01388705
[12] Gillet, Henri; Soul\'{e}, Christophe, Arithmetic intersection theory, Inst. Hautes \'{E}tudes Sci. Publ. Math.. Institut des Hautes \'{E}tudes Scientifiques. Publications Math\'{e}matiques, 93-174 (1990) · Zbl 0741.14012
[13] Gordon, J., Transfer to characteristic zero
[14] Gross, Benedict H.; Zagier, Don B., Heegner points and derivatives of {\(L\)}-series, Invent. Math.. Inventiones Mathematicae, 84, 225-320 (1986) · Zbl 0608.14019 · doi:10.1007/BF01388809
[15] Grothendieck, A., \'{E}l\'{e}ments de g\'{e}om\'{e}trie alg\'{e}brique. {I}. {L}e langage des sch\'{e}mas, Inst. Hautes \'{E}tudes Sci. Publ. Math.. Institut des Hautes \'{E}tudes Scientifiques. Publications Math\'{e}matiques, 5-228 (1960)
[16] Grothendieck, A., \'{E}l\'{e}ments de g\'{e}om\'{e}trie alg\'{e}brique. {III}. \'{E}tude cohomologique des faisceaux coh\'{e}rents. Premi\`{e}re partie, Inst. Hautes \'{E}tudes Sci. Publ. Math.. Institut des Hautes \'{E}tudes Scientifiques. Publications Math\'{e}matiques, 5-167 (1961)
[17] He, Xuhua; Li, Chao; Zhu, Yihang, Fine {D}eligne-{L}usztig varieties and arithmetic fundamental lemmas, Forum Math. Sigma. Forum of Mathematics. Sigma, 7, 47-55 (2019) · Zbl 1439.11142 · doi:10.1017/fms.2019.45
[18] Howard, Benjamin, Complex multiplication cycles and {K}udla-{R}apoport divisors, Ann. of Math. (2). Annals of Mathematics. Second Series, 176, 1097-1171 (2012) · Zbl 1327.14126 · doi:10.4007/annals.2012.176.2.9
[19] Jacquet, H.; Langlands, R. P., Automorphic Forms on {\({\rm GL}(2)\)}, Lecture Notes in Math, 114, vii+548 pp. (1970) · Zbl 0236.12010 · doi:10.1007/BFb0058988
[20] Jacquet, H.; Rallis, Stephen, On the {G}ross-{P}rasad conjecture for unitary groups. On Certain {\(L\)}-Functions, Clay Math. Proc., 13, 205-264 (2011) · Zbl 1222.22018
[21] Jacquet, H., Sur un r\'{e}sultat de {W}aldspurger, Ann. Sci. \'{E}cole Norm. Sup. (4). Annales Scientifiques de l’\'{E}cole Normale Sup\'{e}rieure. Quatri\`eme S\'{e}rie, 19, 185-229 (1986) · Zbl 0605.10015 · doi:10.24033/asens.1506
[22] Kudla, S. S., Algebraic cycles on {S}himura varieties of orthogonal type, Duke Math. J.. Duke Mathematical Journal, 86, 39-78 (1997) · Zbl 0879.11026 · doi:10.1215/S0012-7094-97-08602-6
[23] Kudla, S. S., Central derivatives of {E}isenstein series and height pairings, Ann. of Math. (2). Annals of Mathematics. Second Series, 146, 545-646 (1997) · Zbl 0990.11032 · doi:10.2307/2952456
[24] Kudla, Stephen; Rapoport, Michael, Special cycles on unitary {S}himura varieties {I}. {U}nramified local theory, Invent. Math.. Inventiones Mathematicae, 184, 629-682 (2011) · Zbl 1229.14020 · doi:10.1007/s00222-010-0298-z
[25] Kudla, Stephen; Rapoport, Michael, Special cycles on unitary {S}himura varieties {II}: {G}lobal theory, J. Reine Angew. Math.. Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle’s Journal], 697, 91-157 (2014) · Zbl 1316.11050 · doi:10.1515/crelle-2012-0121
[26] Li, Chao; Zhang, Wei, Kudla–{R}apoport cycles and derivatives of local densities (2019)
[27] Li, Chao; Zhu, Yihang, Remarks on the arithmetic fundamental lemma, Algebra Number Theory. Algebra & Number Theory, 11, 2425-2445 (2017) · Zbl 1439.11143 · doi:10.2140/ant.2017.11.2425
[28] Liu, Yifeng, Arithmetic theta lifting and {\(L\)}-derivatives for unitary groups, {I}, Algebra Number Theory. Algebra & Number Theory, 5, 849-921 (2011) · Zbl 1258.11060 · doi:10.2140/ant.2011.5.849
[29] Liu, Yifeng, Relative trace formulae toward {B}essel and {F}ourier-{J}acobi periods on unitary groups, Manuscripta Math.. Manuscripta Mathematica, 145, 1-69 (2014) · Zbl 1301.11050 · doi:10.1007/s00229-014-0666-x
[30] Liu, Yifeng, Fourier-{J}acobi cycles and arithmetic relative trace formula (with an appendix by {C. L}i and {Y. Z}hu) (2021)
[31] Mihatsch, Andreas, On the arithmetic fundamental lemma through {L}ie algebras, Math. Z.. Mathematische Zeitschrift, 287, 181-197 (2017) · Zbl 1410.11071 · doi:10.1007/s00209-016-1822-7
[32] Mihatsch, A., Relative unitary {RZ}-spaces and the arithmetic fundamental lemma, Journal of the Institute of Mathematics of Jussieu, 61 pp. pp. · Zbl 1492.11105 · doi:10.1017/S1474748020000079
[33] Mihatsch, A., Local constancy of intersection numbers (2020)
[34] Ng\^o, Bao Ch\^au, Le lemme fondamental pour les alg\`ebres de {L}ie, Publ. Math. Inst. Hautes \'{E}tudes Sci.. Publications Math\'{e}matiques. Institut de Hautes \'{E}tudes Scientifiques, 1-169 (2010) · Zbl 1200.22011 · doi:10.1007/s10240-010-0026-7
[35] Oberhettinger, F., On the derivative of {B}essel functions with respect to the order, J. Math. and Phys.. Journal of Mathematics and Physics, 37, 75-78 (1958) · Zbl 0111.06607 · doi:10.1002/sapm195837175
[36] Oda, Takayuki; Tsuzuki, Masao, Automorphic {G}reen functions associated with the secondary spherical functions, Publ. Res. Inst. Math. Sci.. Kyoto University. Research Institute for Mathematical Sciences. Publications, 39, 451-533 (2003) · Zbl 1044.11033 · doi:10.2977/prims/1145476077
[37] Rapoport, M.; Smithling, B.; Zhang, W., On the arithmetic transfer conjecture for exotic smooth formal moduli spaces, Duke Math. J.. Duke Mathematical Journal, 166, 2183-2336 (2017) · Zbl 1411.11056 · doi:10.1215/00127094-2017-0003
[38] Rapoport, M.; Smithling, B.; Zhang, W., Regular formal moduli spaces and arithmetic transfer conjectures, Math. Ann.. Mathematische Annalen, 370, 1079-1175 (2018) · Zbl 1408.14143 · doi:10.1007/s00208-017-1526-2
[39] Rapoport, M.; Smithling, B.; Zhang, W., Arithmetic diagonal cycles on unitary {S}himura varieties, Compos. Math.. Compositio Mathematica, 156, 1745-1824 (2020) · Zbl 1456.14031 · doi:10.1112/s0010437x20007289
[40] Rapoport, M.; Smithling, B.; Zhang, W., On {S}himura varieties for unitary groups, {\em Pure Appl. Math. Q.} (special issue in honor of D. Mumford), to appear (2019) · Zbl 1456.14031
[41] Rapoport, Michael; Terstiege, Ulrich; Zhang, Wei, On the arithmetic fundamental lemma in the minuscule case, Compos. Math.. Compositio Mathematica, 149, 1631-1666 (2013) · Zbl 1300.11069 · doi:10.1112/S0010437X13007239
[42] Vollaard, Inken; Wedhorn, Torsten, The supersingular locus of the {S}himura variety of {\({\rm GU}(1,n-1)\)} {II}, Invent. Math.. Inventiones Mathematicae, 184, 591-627 (2011) · Zbl 1227.14027 · doi:10.1007/s00222-010-0299-y
[43] Xue, Hang, On the global {G}an–{G}ross–{P}rasad conjecture for unitary groups: approximating smooth transfer of {J}acquet-{R}allis, J. Reine Angew. Math.. Journal f\"{u}r die Reine und Angewandte Mathematik. [Crelle’s Journal], 756, 65-100 (2019) · Zbl 1473.22015 · doi:10.1515/crelle-2017-0016
[44] Yuan, Xinyi; Zhang, Shou-Wu; Zhang, Wei, The {G}ross–{K}ohnen–{Z}agier theorem over totally real fields, Compos. Math.. Compositio Mathematica, 145, 1147-1162 (2009) · Zbl 1235.11056 · doi:10.1112/S0010437X08003734
[45] Yun, Zhiwei, The fundamental lemma of {J}acquet and {R}allis, Duke Math. J.. Duke Mathematical Journal, 156, 167-227 (2011) · Zbl 1211.14039 · doi:10.1215/00127094-2010-210
[46] Zhang, Wei, On arithmetic fundamental lemmas, Invent. Math.. Inventiones Mathematicae, 188, 197-252 (2012) · Zbl 1247.14031 · doi:10.1007/s00222-011-0348-1
[47] Zhang, Wei, Fourier transform and the global {G}an-{G}ross-{P}rasad conjecture for unitary groups, Ann. of Math. (2). Annals of Mathematics. Second Series, 180, 971-1049 (2014) · Zbl 1322.11048 · doi:10.4007/annals.2014.180.3.4
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