×

Parabolicity conjecture of \(F\)-isocrystals. (English) Zbl 07734888

Summary: In this article we prove Crew’s parabolicity conjecture of \(F\)-isocrystals. For this purpose, we introduce and study the notion of \(\dagger\)-hull of a sub-\(F\)-isocrystal. On the way, we prove a new Lefschetz theorem for overconvergent \(F\)-isocrystals.

MSC:

14F30 \(p\)-adic cohomology, crystalline cohomology
18M25 Tannakian categories
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abe, Tomoyuki, Langlands correspondence for isocrystals and the existence of crystalline companions for curves, J. Amer. Math. Soc.. Journal of the American Mathematical Society, 31, 921-1057 (2018) · Zbl 1420.14044 · doi:10.1090/jams/898
[2] Abe, Tomoyuki; Esnault, H\'{e}l\`ene, A {L}efschetz theorem for overconvergent isocrystals with {F}robenius structure, Ann. Sci. \'{E}c. Norm. Sup\'{e}r. (4). Annales Scientifiques de l’\'{E}cole Normale Sup\'{e}rieure. Quatri\`eme S\'{e}rie, 52, 1243-1264 (2019) · Zbl 1440.14097 · doi:10.24033/asens.2408
[3] Ambrosi, Emiliano; D’Addezio, Marco, Maximal tori of monodromy groups of {\(F\)}-isocrystals and an application to abelian varieties, Algebr. Geom.. Algebraic Geometry, 9, 633-650 (2022) · Zbl 1507.14028 · doi:10.14231/ag-2022-019
[4] Pr\'epublication IRMAR; available on author’s website, Cohomologie rigide et cohomologie rigide {\`a} supports propres, premi{\`e}re partie (1996)
[5] Berthelot, Pierre; Breen, Lawrence; Messing, William, Th\'{e}orie de {D}ieudonn\'{e} Cristalline. {II}, Lecture Notes in Math., 930, x+261 pp. (1982) · Zbl 0516.14015 · doi:10.1007/BFb0093025
[6] Borel, Armand, Linear Algebraic Groups, Grad. Texts in Math., 126, xii+288 pp. (1991) · Zbl 0726.20030 · doi:10.1007/978-1-4612-0941-6
[7] Cadoret, A.; Tamagawa, A., Ghosts in families of abelian varieties with a common isogeny factor (2020)
[8] Crew, Richard, {\(F\)}-isocrystals and {\(p\)}-adic representations. Algebraic Geometry, {B}owdoin, 1985, Proc. Sympos. Pure Math., 46, 111-138 (1987) · Zbl 0639.14011 · doi:10.1090/pspum/046.2/927977
[9] Crew, Richard, {\(F\)}-isocrystals and their monodromy groups, Ann. Sci. \'{E}cole Norm. Sup. (4). Annales Scientifiques de l’\'{E}cole Normale Sup\'{e}rieure. Quatri\`eme S\'{e}rie, 25, 429-464 (1992) · Zbl 0783.14008 · doi:10.24033/asens.1655
[10] Crew, Richard, The {\(p\)}-adic monodromy of a generic abelian scheme in characteristic {\(p\)}. {\(p\)}-adic Methods in Number Theory and Algebraic Geometry, Contemp. Math., 133, 59-74 (1992) · Zbl 0785.14009 · doi:10.1090/conm/133/1183970
[11] Crew, Richard, Kloosterman sums and monodromy of a {\(p\)}-adic hypergeometric equation, Compositio Math.. Compositio Mathematica, 91, 1-36 (1994) · Zbl 0806.14018
[12] D’Addezio, Marco, The monodromy groups of lisse sheaves and overconvergent {\(F\)}-isocrystals, Selecta Math. (N.S.). Selecta Mathematica. New Series, 26, 1-41 (2020) · Zbl 1454.14058 · doi:10.1007/s00029-020-00569-3
[13] D’Addezio, Marco; Esnault, H\'{e}l\`ene, On the universal extensions in {T}annakian categories, Int. Math. Res. Not. IMRN. International Mathematics Research Notices. IMRN, 14008-14033 (2022) · Zbl 07594734 · doi:10.1093/imrn/rnab107
[14] de Jong, A. J., Homomorphisms of {B}arsotti-{T}ate groups and crystals in positive characteristic, Invent. Math.. Inventiones Mathematicae, 134, 301-333 (1998) · Zbl 0929.14029 · doi:10.1007/s002220050266
[15] de Jong, A. J.; Oort, F., Purity of the stratification by {N}ewton polygons, J. Amer. Math. Soc.. Journal of the American Mathematical Society, 13, 209-241 (2000) · Zbl 0954.14007 · doi:10.1090/S0894-0347-99-00322-7
[16] Deligne, P., Le groupe fondamental de la droite projective moins trois points. Galois Groups over {\({\bf Q}\)}, Math. Sci. Res. Inst. Publ., 16, 79-297 (1989) · Zbl 0742.14022 · doi:10.1007/978-1-4613-9649-9_3
[17] Deligne, Pierre, Finitude de l’extension de {\( \Bbb Q\)} engendr\'{e}e par des traces de {F}robenius, en caract\'{e}ristique finie, Mosc. Math. J.. Moscow Mathematical Journal, 12, 497-514 (2012) · Zbl 1260.14022 · doi:10.17323/1609-4514-2012-12-3-497-514
[18] Drinfeld, Vladimir, On a conjecture of {D}eligne, Mosc. Math. J.. Moscow Mathematical Journal, 12, 515-542 (2012) · Zbl 1271.14028 · doi:10.17323/1609-4514-2012-12-3-515-542
[19] Drinfeld, Vladimir, On the pro-semisimple completion of the fundamental group of a smooth variety over a finite field, Adv. Math.. Advances in Mathematics, 327, 708-788 (2018) · Zbl 1391.14042 · doi:10.1016/j.aim.2017.06.029
[20] Drinfeld, Vladimir; Kedlaya, Kiran S., Slopes of indecomposable {\(F\)}-isocrystals, Pure Appl. Math. Q.. Pure and Applied Mathematics Quarterly, 13, 131-192 (2017) · Zbl 1431.11069 · doi:10.4310/PAMQ.2017.v13.n1.a5
[21] {\'{E}}tesse, Jean-Yves, Descente \'{e}tale des {\(F\)}-isocristaux surconvergents et rationalit\'{e} des fonctions {\(L\)} de sch\'{e}mas ab\'{e}liens, Ann. Sci. \'{E}cole Norm. Sup. (4). Annales Scientifiques de l’\'{E}cole Normale Sup\'{e}rieure. Quatri\`eme S\'{e}rie, 35, 575-603 (2002) · Zbl 1060.14028 · doi:10.1016/S0012-9593(02)01099-6
[22] Esnault, H\'{e}l\`ene, Survey on some aspects of {L}efschetz theorems in algebraic geometry, Rev. Mat. Complut.. Revista Matem\'{a}tica Complutense, 30, 217-232 (2017) · Zbl 1390.14061 · doi:10.1007/s13163-017-0223-8
[23] Helm, David, An ordinary abelian variety with an \'{e}tale self-isogeny of {\(p\)}-power degree and no isotrivial factors, Math. Res. Lett.. Mathematical Research Letters, 29, 445-453 (2022) · Zbl 1507.14063 · doi:10.4310/MRL.2022.v29.n2.a6
[24] Jouanolou, Jean-Pierre, Th\'{e}or\`emes de {B}ertini et {A}pplications, Progr. Math., 42, ii+127 pp. (1983) · Zbl 0519.14002
[25] Katz, Nicholas M., Slope filtration of {\(F\)}-crystals. Journ\'{e}es de {G}\'{e}om\'{e}trie {A}lg\'{e}brique de {R}ennes. {V}ol. {I}, Ast\'{e}risque, 63, 113-163 (1979) · Zbl 0426.14007
[26] Katz, Nicholas M., Exponential Sums and Differential Equations, Ann. of Math. Stud., 124, xii+430 pp. (1990) · Zbl 0731.14008 · doi:10.1515/9781400882434
[27] Katz, Nicholas M., Space filling curves over finite fields, Math. Res. Lett.. Mathematical Research Letters, 6, 613-624 (1999) · Zbl 1016.11022 · doi:10.4310/MRL.1999.v6.n6.a2
[28] Kedlaya, Kiran S., Full faithfulness for overconvergent {\(F\)}-isocrystals. Geometric Aspects of {D}work {T}heory. Vol. {I}., 819-835 (2004) · Zbl 1087.14018 · doi:10.1515/9783110198133.2.819
[29] Kedlaya, Kiran S., More \'{e}tale covers of affine spaces in positive characteristic, J. Algebraic Geom.. Journal of Algebraic Geometry, 14, 187-192 (2005) · Zbl 1065.14020 · doi:10.1090/S1056-3911-04-00381-9
[30] Kedlaya, Kiran S., Finiteness of rigid cohomology with coefficients, Duke Math. J.. Duke Mathematical Journal, 134, 15-97 (2006) · Zbl 1133.14019 · doi:10.1215/S0012-7094-06-13412-9
[31] Kedlaya, Kiran S., Semistable reduction for overconvergent {\(F\)}-isocrystals. {I}. {U}nipotence and logarithmic extensions, Compos. Math., 143, 1164-1212 (2007) · Zbl 1144.14012 · doi:10.1112/S0010437X07002886
[32] Kedlaya, Kiran S., Semistable reduction for overconvergent {\(F\)}-isocrystals, {II}. {A} valuation-theoretic approach, Compos. Math., 144, 657-672 (2008) · Zbl 1153.14015 · doi:10.1112/S0010437X07003296
[33] Kedlaya, Kiran S., Semistable reduction for overconvergent {\(F\)}-isocrystals, {IV}: local semistable reduction at nonmonomial valuations, Compos. Math.. Compositio Mathematica, 147, 467-523 (2011) · Zbl 1230.14023 · doi:10.1112/S0010437X10005142
[34] Kedlaya, Kiran S., Notes on isocrystals, J. Number Theory. Journal of Number Theory, 237, 353-394 (2022) · Zbl 1502.14051 · doi:10.1016/j.jnt.2021.12.004
[35] P{\'{a}}l, Ambrus, The {\(p\)}-adic monodromy group of abelian varieties over global function fields of characteristic {\(p\)}, Doc. Math.. Documenta Mathematica, 27, 1509-1579 (2022) · Zbl 1500.92083 · doi:10.3934/dcdsb.2022017
[36] Poonen, Bjorn; Voloch, Jos\'{e} Felipe, The {B}rauer-{M}anin obstruction for subvarieties of abelian varieties over function fields, Ann. of Math. (2). Annals of Mathematics. Second Series, 171, 511-532 (2010) · Zbl 1294.11110 · doi:10.4007/annals.2010.171.511
[37] R{\"{o}}ssler, Damian, On the group of purely inseparable points of an abelian variety defined over a function field of positive characteristic, {II}, Algebra Number Theory. Algebra & Number Theory, 14, 1123-1173 (2020) · Zbl 1469.11239 · doi:10.2140/ant.2020.14.1123
[38] Saavedra Rivano, Neantro, Cat\'{e}gories tannakiennes, Lecture Notes in Math., 265, ii+418 pp. (1972) · Zbl 0241.14008
[39] Serre, Jean-Pierre, Zeta and {\(L\)} functions. Arithmetical {A}lgebraic {G}eometry, {P}roc. {C}onf. {P}urdue {U}niv., 1963, 82-92 (1965) · Zbl 0171.19602
[40] Serre, Jean-Pierre, Lectures on the {M}ordell-{W}eil Theorem, Aspects of Math., E15, x+218 pp. (1989) · Zbl 0676.14005 · doi:10.1007/978-3-663-14060-3
[41] Grothendieck, A.; Raynaud, M., Rev\^etements \'{E}tales et Groupe {F}ondamental, Lecture Notes in Math., 224, xxii+447 pp. (1971) · Zbl 0234.14002 · doi:10.1007/BFb0058656
[42] Stalder, S., Scalar Extension of Abelian and {T}annakian Categories (2008)
[43] Tate, J. T., {\(p\)}-divisible groups. Proc. {C}onf. {L}ocal {F}ields, 158-183 (1967) · Zbl 0157.27601 · doi:10.1007/978-3-642-87942-5_12
[44] Tsuzuki, Nobuo, Morphisms of {\(F\)}-isocrystals and the finite monodromy theorem for unit-root {\(F\)}-isocrystals, Duke Math. J.. Duke Mathematical Journal, 111, 385-418 (2002) · Zbl 1055.14022 · doi:10.1215/S0012-7094-02-11131-4
[45] Tsuzuki, Nobuo, Minimal slope conjecture of {\(F\)}-isocrystals, Invent. Math.. Inventiones Mathematicae, 231, 39-109 (2023) · Zbl 1520.14040 · doi:10.1007/s00222-022-01146-5
[46] Vella, David C., Another characterization of parabolic subgroups, J. Algebra. Journal of Algebra, 137, 214-232 (1991) · Zbl 0715.20030 · doi:10.1016/0021-8693(91)90091-L
[47] Voloch, Jos\'{e} Felipe, Diophantine approximation on abelian varieties in characteristic {\(p\)}, Amer. J. Math.. American Journal of Mathematics, 117, 1089-1095 (1995) · Zbl 0855.11029 · doi:10.2307/2374961
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.