Geometric stabilisation via 𝑝-adic integration

Groechenig, Michael; Wyss, Dimitri; Ziegler, Paul

doi:10.1090/jams/948

Geometric stabilisation via $p$-adic integration
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by Michael Groechenig, Dimitri Wyss and Paul Ziegler

J. Amer. Math. Soc. 33 (2020), 807-873

DOI: https://doi.org/10.1090/jams/948

Published electronically: June 15, 2020

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Abstract:

In this article we give a new proof of Ngô’s geometric stabilisation theorem, which implies the fundamental lemma. This is a statement which relates the cohomology of Hitchin fibres for a quasi-split reductive group scheme $G$ to the cohomology of Hitchin fibres for the endoscopy groups $H_{\kappa }$. Our proof avoids the decomposition and support theorem, instead the argument is based on results for $p$-adic integration on coarse moduli spaces of Deligne-Mumford stacks. Along the way we establish a description of the inertia stack of the (anisotropic) moduli stack of $G$-Higgs bundles in terms of endoscopic data, and extend duality for generic Hitchin fibres of Langlands dual group schemes to the quasi-split case.

References

Kenneth Ascher and Dori Bejleri, Moduli of weighted stable elliptic surfaces & invariance of log plurigenera, arXiv preprint arXiv:1702.06107 (2017).
Dan Abramovich, Alessio Corti, and Angelo Vistoli, Twisted bundles and admissible covers, Comm. Algebra 31 (2003), no. 8, 3547–3618. Special issue in honor of Steven L. Kleiman. MR 2007376, DOI 10.1081/AGB-120022434
Michael Artin, Alexander Grothendieck, and Jean-Louis Verdier, Théorie de topos et cohomologie étale des schemas III, Springer-Verlag, 1963/1964.
Jarod Alper, Good moduli spaces for Artin stacks, Ann. Inst. Fourier (Grenoble) 63 (2013), no. 6, 2349–2402 (English, with English and French summaries). MR 3237451
Dan Abramovich, Martin Olsson, and Angelo Vistoli, Tame stacks in positive characteristic, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 4, 1057–1091 (English, with English and French summaries). MR 2427954
Victor V. Batyrev, Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs, J. Eur. Math. Soc. (JEMS) 1 (1999), no. 1, 5–33. MR 1677693, DOI 10.1007/PL00011158
Alexander Beilinson, Spencer Bloch, and Hélène Esnault, $\epsilon$-factors for Gauss-Manin determinants, Mosc. Math. J. 2 (2002), no. 3, 477–532. Dedicated to Yuri I. Manin on the occasion of his 65th birthday. MR 1988970, DOI 10.17323/1609-4514-2002-2-3-477-532
Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, Springer-Verlag, Berlin, 1990. MR 1045822, DOI 10.1007/978-3-642-51438-8
Pierre-Henri Chaudouard and Gérard Laumon, Le lemme fondamental pondéré. I. Constructions géométriques, Compos. Math. 146 (2010), no. 6, 1416–1506 (French, with English and French summaries). MR 2735371, DOI 10.1112/S0010437X10004756
Pierre-Henri Chaudouard and Gérard Laumon, Le lemme fondamental pondéré. II. Énoncés cohomologiques, Ann. of Math. (2) 176 (2012), no. 3, 1647–1781 (French, with French summary). MR 2979859, DOI 10.4007/annals.2012.176.3.6
Brian Conrad, Keel-Mori theorem via stacks, http://math.stanford.edu/ conrad/papers/coarsespace.pdf (2005).
Brian Conrad, Reductive group schemes, Autour des schémas en groupes. Vol. I, Panor. Synthèses, vol. 42/43, Soc. Math. France, Paris, 2014, pp. 93–444 (English, with English and French summaries). MR 3362641
Tsao-Hsien Chen and Xinwen Zhu, Geometric Langlands in prime characteristic, Compos. Math. 153 (2017), no. 2, 395–452. MR 3705230, DOI 10.1112/S0010437X16008113
R. Y. Donagi and D. Gaitsgory, The gerbe of Higgs bundles, Transform. Groups 7 (2002), no. 2, 109–153. MR 1903115, DOI 10.1007/s00031-002-0008-z
Jan Denef and François Loeser, Motivic integration, quotient singularities and the McKay correspondence, Compositio Math. 131 (2002), no. 3, 267–290. MR 1905024, DOI 10.1023/A:1015565912485
R. Donagi and T. Pantev, Langlands duality for Hitchin systems, Invent. Math. 189 (2012), no. 3, 653–735. MR 2957305, DOI 10.1007/s00222-012-0373-8
Dan Edidin, Brendan Hassett, Andrew Kresch, and Angelo Vistoli, Brauer groups and quotient stacks, Amer. J. Math. 123 (2001), no. 4, 761–777. MR 1844577
Edward Frenkel and Edward Witten, Geometric endoscopy and mirror symmetry, Commun. Number Theory Phys. 2 (2008), no. 1, 113–283. MR 2417848, DOI 10.4310/CNTP.2008.v2.n1.a3
Mark Goresky, Robert Kottwitz, and Robert Macpherson, Homology of affine Springer fibers in the unramified case, Duke Math. J. 121 (2004), no. 3, 509–561. MR 2040285, DOI 10.1215/S0012-7094-04-12135-9
Michael Groechenig, Dimitri Wyss, and Paul Ziegler, Mirror symmetry for moduli spaces of Higgs bundles via p-adic integration, https://arxiv.org/pdf/1707.06417.pdf (2017).
Helmut Hasse, Zahlentheorie, Akademie-Verlag, Berlin, 1963 (German). Zweite erweiterte Auflage. MR 0153659
Tamás Hausel, Global topology of the Hitchin system, Handbook of moduli. Vol. II, Adv. Lect. Math. (ALM), vol. 25, Int. Press, Somerville, MA, 2013, pp. 29–69. MR 3184173
Tamás Hausel and Michael Thaddeus, Mirror symmetry, Langlands duality, and the Hitchin system, Invent. Math. 153 (2003), no. 1, 197–229. MR 1990670, DOI 10.1007/s00222-003-0286-7
Jun-ichi Igusa, An introduction to the theory of local zeta functions, AMS/IP Studies in Advanced Mathematics, vol. 14, American Mathematical Society, Providence, RI; International Press, Cambridge, MA, 2000. MR 1743467, DOI 10.1090/amsip/014
Kenkichi Iwasawa, On Galois groups of local fields, Trans. Amer. Math. Soc. 80 (1955), 448–469. MR 75239, DOI 10.1090/S0002-9947-1955-0075239-5
Seán Keel and Shigefumi Mori, Quotients by groupoids, Ann. of Math. (2) 145 (1997), no. 1, 193–213. MR 1432041, DOI 10.2307/2951828
Andrew Kresch, On the geometry of Deligne-Mumford stacks, Algebraic geometry—Seattle 2005. Part 1, Proc. Sympos. Pure Math., vol. 80, Amer. Math. Soc., Providence, RI, 2009, pp. 259–271. MR 2483938, DOI 10.1090/pspum/080.1/2483938
Gérard Laumon and Bao Châu Ngô, Le lemme fondamental pour les groupes unitaires, Ann. of Math. (2) 168 (2008), no. 2, 477–573 (French, with English summary). MR 2434884, DOI 10.4007/annals.2008.168.477
Dino Lorenzini, Torsion and Tamagawa numbers, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 5, 1995–2037 (2012) (English, with English and French summaries). MR 2961846, DOI 10.5802/aif.2664
Jacob Lurie, Higher algebra, Preprint, available at http://www.math.harvard.edu/~lurie.
Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
Schémas en groupes. II: Groupes de type multiplicatif, et structure des schémas en groupes généraux, Lecture Notes in Mathematics, Vol. 152, Springer-Verlag, Berlin-New York, 1970 (French). Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3); Dirigé par M. Demazure et A. Grothendieck. MR 0274459
Schémas en groupes. III: Structure des schémas en groupes réductifs, Lecture Notes in Mathematics, Vol. 153, Springer-Verlag, Berlin-New York, 1970 (French). Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3); Dirigé par M. Demazure et A. Grothendieck. MR 0274460
James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
J. S. Milne, Arithmetic duality theorems, Perspectives in Mathematics, vol. 1, Academic Press, Inc., Boston, MA, 1986. MR 881804
Schémas en groupes. II: Groupes de type multiplicatif, et structure des schémas en groupes généraux, Lecture Notes in Mathematics, Vol. 152, Springer-Verlag, Berlin-New York, 1970 (French). Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3); Dirigé par M. Demazure et A. Grothendieck. MR 0274459
Bao Châu Ngô, Fibration de Hitchin et endoscopie, Invent. Math. 164 (2006), no. 2, 399–453 (French, with English summary). MR 2218781, DOI 10.1007/s00222-005-0483-7
Bao Châu Ngô, Le lemme fondamental pour les algèbres de Lie, Publ. Math. Inst. Hautes Études Sci. 111 (2010), 1–169 (French). MR 2653248, DOI 10.1007/s10240-010-0026-7
Martin C. Olsson, $\underline \textrm {Hom}$-stacks and restriction of scalars, Duke Math. J. 134 (2006), no. 1, 139–164. MR 2239345, DOI 10.1215/S0012-7094-06-13414-2
D. Prasad, http://www.math.tifr.res.in/~dprasad/dp-lecture4.pdf.
Jean-Pierre Serre, Cohomologie galoisienne, 5th ed., Lecture Notes in Mathematics, vol. 5, Springer-Verlag, Berlin, 1994 (French). MR 1324577, DOI 10.1007/BFb0108758
Stacks Project Authors, Stacks project, http://math.columbia.edu/algebraic_geometry/stacks-git.
Shenghao Sun, $L$-series of Artin stacks over finite fields, Algebra Number Theory 6 (2012), no. 1, 47–122. MR 2950161, DOI 10.2140/ant.2012.6.47
J.-L. Waldspurger, Endoscopie et changement de caractéristique, J. Inst. Math. Jussieu 5 (2006), no. 3, 423–525 (French, with English and French summaries). MR 2241929, DOI 10.1017/S1474748006000041
J.-L. Waldspurger, L’endoscopie tordue n’est pas si tordue, Mem. Amer. Math. Soc. 194 (2008), no. 908, x+261 (French, with English summary). MR 2418405, DOI 10.1090/memo/0908
André Weil, Adeles and algebraic groups, Progress in Mathematics, vol. 23, Birkhäuser, Boston, Mass., 1982. With appendices by M. Demazure and Takashi Ono. MR 670072
Takehiko Yasuda, Motivic integration over Deligne-Mumford stacks, Adv. Math. 207 (2006), no. 2, 707–761. MR 2271984, DOI 10.1016/j.aim.2006.01.004
Takehiko Yasuda, The wild McKay correspondence and $p$-adic measures, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 12, 3709–3734. MR 3730512, DOI 10.4171/JEMS/751