On the Ramanujan conjecture for automorphic forms over function fields I. Geometry

Sawin, Will; Templier, Nicolas

doi:10.1090/jams/968

On the Ramanujan conjecture for automorphic forms over function fields I. Geometry
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by Will Sawin and Nicolas Templier

J. Amer. Math. Soc. 34 (2021), 653-746

DOI: https://doi.org/10.1090/jams/968

Published electronically: April 16, 2021

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Abstract:

Let $G$ be a split semisimple group over a function field. We prove the temperedness at unramified places of automorphic representations of $G$, subject to a local assumption at one place, stronger than supercuspidality, and assuming the existence of cyclic base change with good properties. Our method relies on the geometry of $\operatorname {Bun}_G$. It is independent of the work of Lafforgue on the global Langlands correspondence.

References

Jeffrey D. Adler, Refined anisotropic $K$-types and supercuspidal representations, Pacific J. Math. 185 (1998), no. 1, 1–32. MR 1653184, DOI 10.2140/pjm.1998.185.1
James Arthur, Unipotent automorphic representations: conjectures, Astérisque 171-172 (1989), 13–71. Orbites unipotentes et représentations, II. MR 1021499
Pierre Baumann and Simon Riche, Notes on the geometric Satake equivalence, Relative aspects in representation theory, Langlands functoriality and automorphic forms, Lecture Notes in Math., vol. 2221, Springer, Cham, 2018, pp. 1–134. MR 3839695
A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR 751966
Enrico Bombieri and Nicholas M. Katz, A note on lower bounds for Frobenius traces, Enseign. Math. (2) 56 (2010), no. 3-4, 203–227. MR 2778790, DOI 10.4171/LEM/56-3-1
F. Bruhat and J. Tits, Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée, Inst. Hautes Études Sci. Publ. Math. 60 (1984), 197–376 (French). MR 756316
Colin J. Bushnell, Induced representations of locally profinite groups, J. Algebra 134 (1990), no. 1, 104–114. MR 1068417, DOI 10.1016/0021-8693(90)90213-8
Colin J. Bushnell and Guy Henniart, The local Langlands conjecture for $\rm GL(2)$, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 335, Springer-Verlag, Berlin, 2006. MR 2234120, DOI 10.1007/3-540-31511-X
W. Casselman, Introduction to the theory of admissible representations of $p$-adic reductive groups, http://www.math.ubc.ca/~cass/research/pdf/p-adic-book.pdf.
L. Clozel, Spectral theory of automorphic forms, Automorphic forms and applications, IAS/Park City Math. Ser., vol. 12, Amer. Math. Soc., Providence, RI, 2007, pp. 43–93. MR 2331344, DOI 10.1090/pcms/012/03
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
Brian Conrad, Ofer Gabber, and Gopal Prasad, Pseudo-reductive groups, 2nd ed., New Mathematical Monographs, vol. 26, Cambridge University Press, Cambridge, 2015. MR 3362817, DOI 10.1017/CBO9781316092439
M. Cowling, U. Haagerup, and R. Howe, Almost $L^2$ matrix coefficients, J. Reine Angew. Math. 387 (1988), 97–110. MR 946351
C. Cunningham and D. Roe, Commutative character sheaves and geometric types for supercuspidal representations, Preprint arXiv:1605.08820v3.
Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307 (French). MR 340258, DOI 10.1007/BF02684373
Pierre Deligne, Cohomologie étale, Séminaire de Géométrie Algébrique du Bois-Marie (SGA 4$\tfrac 12$), Lecture Notes in Mathematics, vol. 569, Springer, 1977.
Pierre Deligne and Yuval Z. Flicker, Counting local systems with principal unipotent local monodromy, Ann. of Math. (2) 178 (2013), no. 3, 921–982. MR 3092473, DOI 10.4007/annals.2013.178.3.3
V. G. Drinfel′d, Elliptic modules. II, Mat. Sb. (N.S.) 102(144) (1977), no. 2, 182–194, 325 (Russian). MR 0439758
V. G. Drinfel′d, The number of two-dimensional irreducible representations of the fundamental group of a curve over a finite field, Funktsional. Anal. i Prilozhen. 15 (1981), no. 4, 75–76 (Russian). MR 639205
V. G. Drinfel′d, Proof of the Petersson conjecture for $\textrm {GL}(2)$ over a global field of characteristic $p$, Funktsional. Anal. i Prilozhen. 22 (1988), no. 1, 34–54, 96 (Russian); English transl., Funct. Anal. Appl. 22 (1988), no. 1, 28–43. MR 936697, DOI 10.1007/BF01077720
Vladimir Drinfeld and Jonathan Wang, On a strange invariant bilinear form on the space of automorphic forms, Selecta Math. (N.S.) 22 (2016), no. 4, 1825–1880. MR 3573948, DOI 10.1007/s00029-016-0262-x
Bernard Dwork, On the rationality of the zeta function of an algebraic variety, Amer. J. Math. 82 (1960), 631–648. MR 140494, DOI 10.2307/2372974
Yuval Z. Flicker, Counting rank two local systems with at most one, unipotent, monodromy, Amer. J. Math. 137 (2015), no. 3, 739–763. MR 3357120, DOI 10.1353/ajm.2015.0016
E. Frenkel, D. Gaitsgory, and K. Vilonen, On the geometric Langlands conjecture, J. Amer. Math. Soc. 15 (2002), no. 2, 367–417. MR 1887638, DOI 10.1090/S0894-0347-01-00388-5
Wee Teck Gan and Nadya Gurevich, CAP representations of $G_2$ and the spin $L$-function of $\textrm {PGSp}_6$, Israel J. Math. 170 (2009), 1–52. MR 2506316, DOI 10.1007/s11856-009-0018-9
Radhika Ganapathy and Sandeep Varma, On the local Langlands correspondence for split classical groups over local function fields, J. Inst. Math. Jussieu 16 (2017), no. 5, 987–1074. MR 3709003, DOI 10.1017/S147474801500033X
A. Genestier and V. Lafforgue, Chtoucas restreints pour les groupes reductifs et parametrisation de Langlands locale, Preprint arXiv:1709.00978.
Benedict H. Gross, On the Satake isomorphism, Galois representations in arithmetic algebraic geometry (Durham, 1996) London Math. Soc. Lecture Note Ser., vol. 254, Cambridge Univ. Press, Cambridge, 1998, pp. 223–237. MR 1696481, DOI 10.1017/CBO9780511662010.006
Shamgar Gurevich and Ronny Hadani, The geometric Weil representation, Selecta Math. (N.S.) 13 (2007), no. 3, 465–481. MR 2383602, DOI 10.1007/s00029-007-0047-3
G. Harder, Chevalley groups over function fields and automorphic forms, Ann. of Math. (2) 100 (1974), 249–306. MR 563090, DOI 10.2307/1971073
Harish-Chandra, Harmonic analysis on reductive $p$-adic groups, Lecture Notes in Mathematics, Vol. 162, Springer-Verlag, Berlin-New York, 1970. Notes by G. van Dijk. MR 0414797, DOI 10.1007/BFb0061269
Jochen Heinloth, Bao-Châu Ngô, and Zhiwei Yun, Kloosterman sheaves for reductive groups, Ann. of Math. (2) 177 (2013), no. 1, 241–310. MR 2999041, DOI 10.4007/annals.2013.177.1.5
Guy Henniart and Bertrand Lemaire, Changement de base et induction automorphe pour $\textrm {GL}_n$ en caractéristique non nulle, Mém. Soc. Math. Fr. (N.S.) 124 (2011), vi+190 (French, with English and French summaries). MR 2841332
R. Howe and I. I. Piatetski-Shapiro, A counterexample to the “generalized Ramanujan conjecture” for (quasi-) split groups, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 315–322. MR 546605
Nicholas M. Katz, Moments, monodromy, and perversity: a Diophantine perspective, Annals of Mathematics Studies, vol. 159, Princeton University Press, Princeton, NJ, 2005. MR 2183396
N. M. Katz, A note on RH for curves and hypersurfaces over finite fields, unpublished note https://web.math.princeton.edu/~nmk/baby16.pdf, 2014.
Reinhardt Kiehl and Rainer Weissauer, Weil conjectures, perverse sheaves and $l$’adic Fourier transform, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 42, Springer-Verlag, Berlin, 2001. MR 1855066, DOI 10.1007/978-3-662-04576-3
Robert E. Kottwitz, Rational conjugacy classes in reductive groups, Duke Math. J. 49 (1982), no. 4, 785–806. MR 683003
Robert E. Kottwitz, Base change for unit elements of Hecke algebras, Compositio Math. 60 (1986), no. 2, 237–250. MR 868140
Jean-Pierre Labesse, Cohomologie, stabilisation et changement de base, Astérisque 257 (1999), vi+161 (French, with English and French summaries). Appendix A by Laurent Clozel and Labesse, and Appendix B by Lawrence Breen. MR 1695940
Laurent Lafforgue, Chtoucas de Drinfeld et correspondance de Langlands, Invent. Math. 147 (2002), no. 1, 1–241 (French, with English and French summaries). MR 1875184, DOI 10.1007/s002220100174
Vincent Lafforgue, Estimées pour les valuations $p$-adiques des valeurs propres des opérateurs de Hecke, Bull. Soc. Math. France 139 (2011), no. 4, 455–477 (French, with English and French summaries). MR 2869300, DOI 10.24033/bsmf.2614
Vincent Lafforgue, Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale, J. Amer. Math. Soc. 31 (2018), no. 3, 719–891 (French). MR 3787407, DOI 10.1090/jams/897
T. Lam and N. Templier, The mirror conjecture for minuscule flag varieties, Preprint arXiv:1705.00758.
Erasmus Landvogt, A compactification of the Bruhat-Tits building, Lecture Notes in Mathematics, vol. 1619, Springer-Verlag, Berlin, 1996. MR 1441308, DOI 10.1007/BFb0094594
Yves Laszlo and Martin Olsson, The six operations for sheaves on Artin stacks. II. Adic coefficients, Publ. Math. Inst. Hautes Études Sci. 107 (2008), 169–210. MR 2434693, DOI 10.1007/s10240-008-0012-5
Yves Laszlo and Martin Olsson, Perverse $t$-structure on Artin stacks, Math. Z. 261 (2009), no. 4, 737–748. MR 2480756, DOI 10.1007/s00209-008-0348-z
Gérard Laumon, Cohomology of Drinfeld modular varieties. Part I, Cambridge Studies in Advanced Mathematics, vol. 41, Cambridge University Press, Cambridge, 1996. Geometry, counting of points and local harmonic analysis. MR 1381898
Luis Alberto Lomelí, Functoriality for the classical groups over function fields, Int. Math. Res. Not. IMRN 22 (2009), 4271–4335. MR 2552304, DOI 10.1093/imrn/rnp089
George Lusztig, Character sheaves. I, Adv. in Math. 56 (1985), no. 3, 193–237. MR 792706, DOI 10.1016/0001-8708(85)90034-9
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
Mark Reeder and Jiu-Kang Yu, Epipelagic representations and invariant theory, J. Amer. Math. Soc. 27 (2014), no. 2, 437–477. MR 3164986, DOI 10.1090/S0894-0347-2013-00780-8
T. Richarz and X. Zhu, Construction of the full Langlands dual group via the geometric Satake correspondence, appendix to The geometric Satake correspondence for ramified groups, Ann. Sci. Ec. Norm. Super. 48 (2015), no. 4, 444–449.
W. Sawin and N. Templier, On the Ramanujan conjecture for automorphic forms over function fields. II. Base change, In preparation.
Simon Schieder, Picard-Lefschetz oscillators for the Drinfeld-Lafforgue-Vinberg degeneration for $\rm SL_2$, Duke Math. J. 167 (2018), no. 5, 835–921. MR 3782063, DOI 10.1215/00127094-2017-0044
T. A. Springer, Linear algebraic groups, 2nd ed., Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2009. MR 2458469
Shenghao Sun, Decomposition theorem for perverse sheaves on Artin stacks over finite fields, Duke Math. J. 161 (2012), no. 12, 2297–2310. MR 2972459, DOI 10.1215/00127094-1723657
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
Nguyen Quoc Thang, On Galois cohomology and weak approximation of connected reductive groups over fields of positive characteristic, Proc. Japan Acad. Ser. A Math. Sci. 87 (2011), no. 10, 203–208. MR 2863415
J. Wang, The moduli stack of $G$-bundles, Preprint arXiv:1104.4828.
Jonathan Wang, On an invariant bilinear form on the space of automorphic forms via asymptotics, Duke Math. J. 167 (2018), no. 16, 2965–3057. MR 3870080, DOI 10.1215/00127094-2018-0025
Hongjie Yu, Le nombre des systèmes locaux $\ell$-adiques sur une courbe, C. R. Math. Acad. Sci. Paris 356 (2018), no. 11-12, 1085–1089 (French, with English and French summaries). MR 3907574, DOI 10.1016/j.crma.2018.10.007
Zhiwei Yun, Rigidity in automorphic representations and local systems, Current developments in mathematics 2013, Int. Press, Somerville, MA, 2014, pp. 73–168. MR 3307715
Zhiwei Yun, Motives with exceptional Galois groups and the inverse Galois problem, Invent. Math. 196 (2014), no. 2, 267–337. MR 3193750, DOI 10.1007/s00222-013-0469-9
Zhiwei Yun, Epipelagic representations and rigid local systems, Selecta Math. (N.S.) 22 (2016), no. 3, 1195–1243. MR 3518549, DOI 10.1007/s00029-015-0204-z
Xinwen Zhu, An introduction to affine Grassmannians and the geometric Satake equivalence, Geometry of moduli spaces and representation theory, IAS/Park City Math. Ser., vol. 24, Amer. Math. Soc., Providence, RI, 2017, pp. 59–154. MR 3752460