Langlands correspondence for isocrystals and the existence of crystalline companions for curves

Abe, Tomoyuki

doi:10.1090/jams/898

Langlands correspondence for isocrystals and the existence of crystalline companions for curves
HTML articles powered by AMS MathViewer

by Tomoyuki Abe;

J. Amer. Math. Soc. 31 (2018), 921-1057

DOI: https://doi.org/10.1090/jams/898

Published electronically: May 22, 2018

PDF | Request permission

Abstract:

In this paper, we show the Langlands correspondence for isocrystals on curves, which asserts the existence of crystalline companions in the case of curves. For the proof we generalize the theory of arithmetic $\mathscr {D}$-modules to algebraic stacks whose diagonal morphisms are finite. Finally, combining with methods of Deligne and Drinfeld, we show the existence of an “$\ell$-adic companion” for any isocrystal on a smooth scheme of any dimension under the assumption of a Bertini-type conjecture.

References

Tomoyuki Abe, Explicit calculation of Frobenius isomorphisms and Poincaré duality in the theory of arithmetic $\scr {D}$-modules, Rend. Semin. Mat. Univ. Padova 131 (2014), 89–149. MR 3217753, DOI 10.4171/RSMUP/131-7
Tomoyuki Abe, Langlands program for $p$-adic coefficients and the petits camarades conjecture, J. Reine Angew. Math. 734 (2018), 59–69. MR 3739313, DOI 10.1515/crelle-2015-0045
T. Abe and D. Caro, Theory of weights in $p$-adic cohomology, Amer. J. Math. (to appear), available at http://arxiv.org/abs/1303.0662.
T. Abe and D. Caro, On Beilinson’s equivalence for $p$-adic cohomology, Selecta Math. (to appear), available at http://arxiv.org/abs/1309.4517.
T. Abe and H. Esnault, A Lefschetz theorem for overconvergent isocrystals with Frobenius structure, Ann. ENS. (to appear), available at http://arxiv.org/abs/1607.07112.
Tomoyuki Abe and Adriano Marmora, Product formula for $p$-adic epsilon factors, J. Inst. Math. Jussieu 14 (2015), no. 2, 275–377. MR 3315058, DOI 10.1017/S1474748014000024
Kai A. Behrend, Derived $l$-adic categories for algebraic stacks, Mem. Amer. Math. Soc. 163 (2003), no. 774, viii+93. MR 1963494, DOI 10.1090/memo/0774
A. A. Beĭlinson, On the derived category of perverse sheaves, $K$-theory, arithmetic and geometry (Moscow, 1984–1986) Lecture Notes in Math., vol. 1289, Springer, Berlin, 1987, pp. 27–41. MR 923133, DOI 10.1007/BFb0078365
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
A. Beilinson and V. Drinfeld, Quantization of Hitchin’s integrable system and Hecke eigensheaves, preprint available at http://www.math.uchicago.edu/~mitya/langlands/hitchin/BD-hitchin.pdf.
P. Berthelot, Cohomologie rigide et cohomologie rigide à supports propres. Premiére partie (version provisoire 1991), Prépublication IRMR 96-03 (1996), available at https://perso.univ-rennes1.fr/pierre.berthelot/publis/Cohomologie_Rigide_I.pdf.
Pierre Berthelot, ${\scr D}$-modules arithmétiques. I. Opérateurs différentiels de niveau fini, Ann. Sci. École Norm. Sup. (4) 29 (1996), no. 2, 185–272 (French, with English summary). MR 1373933, DOI 10.24033/asens.1739
Pierre Berthelot, $\scr D$-modules arithmétiques. II. Descente par Frobenius, Mém. Soc. Math. Fr. (N.S.) 81 (2000), vi+136 (French, with English and French summaries). MR 1775613
Pierre Berthelot, Introduction à la théorie arithmétique des $\scr D$-modules, Astérisque 279 (2002), 1–80 (French, with French summary). Cohomologies $p$-adiques et applications arithmétiques, II. MR 1922828
Daniel Caro, Comparaison des foncteurs duaux des isocristaux surconvergents, Rend. Sem. Mat. Univ. Padova 114 (2005), 131–211 (2006) (French, with English summary). MR 2207865
Daniel Caro, $\scr D$-modules arithmétiques surholonomes, Ann. Sci. Éc. Norm. Supér. (4) 42 (2009), no. 1, 141–192 (French, with English and French summaries). MR 2518895, DOI 10.24033/asens.2092
Daniel Caro, $\scr D$-modules arithmétiques associés aux isocristaux surconvergents. Cas lisse, Bull. Soc. Math. France 137 (2009), no. 4, 453–543 (French, with English and French summaries). MR 2572180, DOI 10.24033/bsmf.2581
Daniel Caro, Sur la compatibilité à Frobenius de l’isomorphisme de dualité relative, Rend. Semin. Mat. Univ. Padova 122 (2009), 235–286 (French, with English summary). MR 2582839, DOI 10.4171/RSMUP/122-14
Daniel Caro, Pleine fidélité sans structure de Frobenius et isocristaux partiellement surconvergents, Math. Ann. 349 (2011), no. 4, 747–805 (French, with French summary). MR 2777033, DOI 10.1007/s00208-010-0539-x
Daniel Caro, Sur la stabilité par produit tensoriel de complexes de $\mathcal {D}$-modules arithmétiques, Manuscripta Math. 147 (2015), no. 1-2, 1–41 (French, with English summary). MR 3336937, DOI 10.1007/s00229-014-0716-4
Daniel Caro and Nobuo Tsuzuki, Overholonomicity of overconvergent $F$-isocrystals over smooth varieties, Ann. of Math. (2) 176 (2012), no. 2, 747–813. MR 2950764, DOI 10.4007/annals.2012.176.2.2
Bruno Chiarellotto, Weights in rigid cohomology applications to unipotent $F$-isocrystals, Ann. Sci. École Norm. Sup. (4) 31 (1998), no. 5, 683–715 (English, with English and French summaries). MR 1643966, DOI 10.1016/S0012-9593(98)80004-9
B. Conrad, Cohomological descent, available at http://math.stanford.edu/~conrad/papers/hypercover.pdf.
B. Conrad, Keel-Mori theorem via stacks, available at http://math.stanford.edu/~conrad/papers/coarsespace.pdf.
Brian Conrad, Max Lieblich, and Martin Olsson, Nagata compactification for algebraic spaces, J. Inst. Math. Jussieu 11 (2012), no. 4, 747–814. MR 2979821, DOI 10.1017/S1474748011000223
Richard Crew, $F$-isocrystals and their monodromy groups, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 4, 429–464. MR 1186910, DOI 10.24033/asens.1655
Pierre Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 137–252 (French). MR 601520, DOI 10.1007/BF02684780
P. Deligne, Le groupe fondamental de la droite projective moins trois points, Galois groups over $\textbf {Q}$ (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 16, Springer, New York, 1989, pp. 79–297 (French). MR 1012168, DOI 10.1007/978-1-4613-9649-9_{3}
P. Deligne and J. Milne, Tannakian categories, Lecture Notes in Math. 900, Springer, pp. 101–228 (1982).
Vladimir Drinfeld, On a conjecture of Deligne, Mosc. Math. J. 12 (2012), no. 3, 515–542, 668 (English, with English and Russian summaries). MR 3024821, DOI 10.17323/1609-4514-2012-12-3-515-542
Hélène Esnault and Moritz Kerz, A finiteness theorem for Galois representations of function fields over finite fields (after Deligne), Acta Math. Vietnam. 37 (2012), no. 4, 531–562. MR 3058662
Jean-Yves Étesse and Bernard Le Stum, Fonctions $L$ associées aux $F$-isocristaux surconvergents. I. Interprétation cohomologique, Math. Ann. 296 (1993), no. 3, 557–576 (French). MR 1225991, DOI 10.1007/BF01445120
Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik, Tensor categories, Mathematical Surveys and Monographs, vol. 205, American Mathematical Society, Providence, RI, 2015. MR 3242743, DOI 10.1090/surv/205
A. Grothendieck, Éléments de géométrie algébrique. I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. 4 (1960), 228 (French). MR 217083
A. Grothendieck, Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes de morphismes, Inst. Hautes Études Sci. Publ. Math. 8 (1961), 222 (French). MR 217084
A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I, Inst. Hautes Études Sci. Publ. Math. 20 (1964), 259 (French). MR 173675
Kazuhiro Fujiwara, Independence of $l$ for intersection cohomology (after Gabber), Algebraic geometry 2000, Azumino (Hotaka), Adv. Stud. Pure Math., vol. 36, Math. Soc. Japan, Tokyo, 2002, pp. 145–151. MR 1971515, DOI 10.2969/aspm/03610145
William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620, DOI 10.1007/978-3-662-02421-8
Pierre Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448 (French). MR 232821, DOI 10.24033/bsmf.1583
Henri Gillet, Riemann-Roch theorems for higher algebraic $K$-theory, Adv. in Math. 40 (1981), no. 3, 203–289. MR 624666, DOI 10.1016/S0001-8708(81)80006-0
Jean Giraud, Méthode de la descente, Bull. Soc. Math. France Mém. 2 (1964), viii+150 (French). MR 190142
Alexander Grothendieck, Sur quelques points d’algèbre homologique, Tohoku Math. J. (2) 9 (1957), 119–221 (French). MR 102537, DOI 10.2748/tmj/1178244839
Robin Hartshorne, Residues and duality, Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64; With an appendix by P. Deligne. MR 222093, DOI 10.1007/BFb0080482
Christine Huyghe, $\scr D^\dagger (\infty )$-affinité des schémas projectifs, Ann. Inst. Fourier (Grenoble) 48 (1998), no. 4, 913–956 (French, with English and French summaries). MR 1656002, DOI 10.5802/aif.1643
Luc Illusie, Crystalline cohomology, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 43–70. MR 1265522, DOI 10.1090/pspum/055.1/1265522
Masaki Kashiwara, $t$-structures on the derived categories of holonomic $\scr D$-modules and coherent $\scr O$-modules, Mosc. Math. J. 4 (2004), no. 4, 847–868, 981 (English, with English and Russian summaries). MR 2124169, DOI 10.17323/1609-4514-2004-4-4-847-868
Masaki Kashiwara and Pierre Schapira, Categories and sheaves, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 332, Springer-Verlag, Berlin, 2006. MR 2182076, DOI 10.1007/3-540-27950-4
Kazuya Kato and Takeshi Saito, Ramification theory for varieties over a perfect field, Ann. of Math. (2) 168 (2008), no. 1, 33–96. MR 2415398, DOI 10.4007/annals.2008.168.33
Kiran S. Kedlaya, Full faithfulness for overconvergent $F$-isocrystals, Geometric aspects of Dwork theory. Vol. I, II, Walter de Gruyter, Berlin, 2004, pp. 819–835. MR 2099088
Kiran S. Kedlaya, Fourier transforms and $p$-adic ‘Weil II’, Compos. Math. 142 (2006), no. 6, 1426–1450. MR 2278753, DOI 10.1112/S0010437X06002338
Kiran S. Kedlaya, $p$-adic cohomology, Algebraic geometry—Seattle 2005. Part 2, Proc. Sympos. Pure Math., vol. 80, Amer. Math. Soc., Providence, RI, 2009, pp. 667–684. MR 2483951, DOI 10.1090/pspum/080.2/2483951
Kiran S. Kedlaya, Semistable reduction for overconvergent $F$-isocrystals. I. Unipotence and logarithmic extensions, Compos. Math. 143 (2007), no. 5, 1164–1212. MR 2360314, DOI 10.1112/S0010437X07002886
Kiran S. Kedlaya, Semistable reduction for overconvergent $F$-isocrystals, IV: local semistable reduction at nonmonomial valuations, Compos. Math. 147 (2011), no. 2, 467–523. MR 2776611, DOI 10.1112/S0010437X10005142
K. S. Kedlaya, Notes on isocrystals, preprint, available at http://kskedlaya.org/papers/isocrystals.pdf.
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
Andrew Kresch, Cycle groups for Artin stacks, Invent. Math. 138 (1999), no. 3, 495–536. MR 1719823, DOI 10.1007/s002220050351
Laurent Lafforgue, Chtoucas de Drinfeld et conjecture de Ramanujan-Petersson, Astérisque 243 (1997), ii+329 (French). MR 1600006
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
Yves Laszlo and Martin Olsson, The six operations for sheaves on Artin stacks. I. Finite coefficients, Publ. Math. Inst. Hautes Études Sci. 107 (2008), 109–168. MR 2434692, DOI 10.1007/s10240-008-0011-6
Gérard Laumon and Laurent Moret-Bailly, Champs algébriques, 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. 39, Springer-Verlag, Berlin, 2000 (French). MR 1771927, DOI 10.1007/978-3-540-24899-6
Bernard Le Stum, Constructible $\nabla$-modules on curves, Selecta Math. (N.S.) 20 (2014), no. 2, 627–674. MR 3177929, DOI 10.1007/s00029-013-0130-x
Madhav V. Nori, Constructible sheaves, Algebra, arithmetic and geometry, Part I, II (Mumbai, 2000) Tata Inst. Fund. Res. Stud. Math., vol. 16, Tata Inst. Fund. Res., Bombay, 2002, pp. 471–491. MR 1940678
Martin C. Olsson, Logarithmic geometry and algebraic stacks, Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 5, 747–791 (English, with English and French summaries). MR 2032986, DOI 10.1016/j.ansens.2002.11.001
Martin C. Olsson, On proper coverings of Artin stacks, Adv. Math. 198 (2005), no. 1, 93–106. MR 2183251, DOI 10.1016/j.aim.2004.08.017
Martin Olsson, Sheaves on Artin stacks, J. Reine Angew. Math. 603 (2007), 55–112. MR 2312554, DOI 10.1515/CRELLE.2007.012
Atsushi Shiho, Cut-by-curves criterion for the log extendability of overconvergent isocrystals, Math. Z. 269 (2011), no. 1-2, 59–82. MR 2836060, DOI 10.1007/s00209-010-0716-3
Revêtements étales et groupe fondamental (SGA 1), Documents Mathématiques (Paris) [Mathematical Documents (Paris)], vol. 3, Société Mathématique de France, Paris, 2003 (French). Séminaire de géométrie algébrique du Bois Marie 1960–61. [Algebraic Geometry Seminar of Bois Marie 1960-61]; Directed by A. Grothendieck; With two papers by M. Raynaud; Updated and annotated reprint of the 1971 original [Lecture Notes in Math., 224, Springer, Berlin; MR0354651 (50 #7129)]. MR 2017446
Schémas en groupes. I: Propriétés générales des schémas en groupes, Lecture Notes in Mathematics, Vol. 151, 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 274458
Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Lecture Notes in Mathematics, Vol. 269, Springer-Verlag, Berlin-New York, 1972 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4); Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. MR 354652
P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, vol. 569, Springer-Verlag, Berlin, 1977 (French). Séminaire de géométrie algébrique du Bois-Marie SGA $4\frac {1}{2}$. MR 463174, DOI 10.1007/BFb0091526
Nobuo Tsuzuki, Morphisms of $F$-isocrystals and the finite monodromy theorem for unit-root $F$-isocrystals, Duke Math. J. 111 (2002), no. 3, 385–418. MR 1885826, DOI 10.1215/S0012-7094-02-11131-4
Anne Virrion, Trace et dualité relative pour les $\scr D$-modules arithmétiques, Geometric aspects of Dwork theory. Vol. I, II, Walter de Gruyter, Berlin, 2004, pp. 1039–1112 (French). MR 2099095
Anne Virrion, Dualité locale et holonomie pour les $\scr D$-modules arithmétiques, Bull. Soc. Math. France 128 (2000), no. 1, 1–68 (French, with English and French summaries). MR 1765829, DOI 10.24033/bsmf.2362

AMS Website Down

Journal of the American Mathematical Society

Langlands correspondence for isocrystals and the existence of crystalline companions for curves
HTML articles powered by AMS MathViewer

Abstract: