Points rationnels et groupes fondamentaux : applications de la cohomologie p-adique  [ Rational points and fundamental groups: applications of the p-adic cohomology ]
Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Talk no. 914, p. 125-146

I present results due to T. Ekedahl and H. Esnault concerning smooth projective varieties adefined over a field of positive characteristic, say p, two points of which can be linked by a chain of rational curves. Examples are given by weakly unirational, or Fano varieties. Notably: 1) over a finite field, such varieties have a rational point, this generalizes the Chevalley-Warning Theorem; 2) over an algebraically closed field, the fundamental group of such varieties is finite and its order is prime to p; 3) over a finite field of cardinality q, the number of rational points of two proper smooth varieties that are birational are congruent mod. q. The proofs use the p-adic rigid cohomology defined by P. Berthelot.

Je présenterai des résultats de T. Ekedahl et H. Esnault sur les variétés projectives lisses sur un corps de caractéristique strictement positive, disons p, dont deux points peuvent être liés par une chaîne de courbes rationnelles, par exemple faiblement unirationnelles, ou de Fano. Notamment : 1) sur un corps fini, de telles variétés ont un point rationnel, résultat qui généralise le théorème de Chevalley-Warning ; 2) sur un corps algébriquement clos, de telles variétés ont un groupe fondamental fini d’ordre premier à p ; 3) sur un corps fini de cardinal q, deux variétés propres et lisses qui sont birationnelles ont même nombre de points rationnels modulo q. Les démonstrations utilisent la cohomologie rigide, p-adique, de P. Berthelot.

Classification:  14M20,  14J45,  14G15,  14G05,  14H30,  14Cxx,  14F30
Keywords: Fano varieties, chain rationaly connected varieties, rational points, fundamental group, rigid cohomology, slopes
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     author = {Chambert-Loir, Antoine},
     title = {Points rationnels et groupes fondamentaux : applications de la cohomologie $p$-adique},
     booktitle = {S\'eminaire Bourbaki : volume 2002/2003, expos\'es 909-923},
     author = {Collectif},
     series = {Ast\'erisque},
     publisher = {Association des amis de Nicolas Bourbaki, Soci\'et\'e math\'ematique de France},
     address = {Paris},
     number = {294},
     year = {2004},
     note = {talk:914},
     pages = {125-146},
     zbl = {1078.14024},
     mrnumber = {2111642},
     language = {fr},
     url = {http://www.numdam.org/item/SB_2002-2003__45__125_0}
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Chambert-loir, Antoine. Points rationnels et groupes fondamentaux : applications de la cohomologie $p$-adique, in Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Talk no. 914, pp. 125-146. http://www.numdam.org/item/SB_2002-2003__45__125_0/

[1] J. Ax - “Zeroes of polynomials over finite fields”, Amer. J. Math. 86 (1964), p. 255-261. | MR 160775 | Zbl 0121.02003

[2] P. Berthelot - Cohomologie cristalline des schémas de caractéristique p>0, Lecture Notes in Math., vol. 407, Springer Verlag, 1974. | MR 384804 | Zbl 0298.14012

[3] -, “Géométrie rigide et cohomologie des variétés algébriques de caractéristique p, in Introductions aux cohomologies p-adiques (Luminy, 1984), Mém. Soc. Math. France, vol. 23, 1986, p. 7-32. | MR 865810

[4] -, “Cohomologie rigide et cohomologie rigide à supports propres. Première partie”, Prépublication, IRMAR, Université Rennes 1, 1996.

[5] -, “Dualité de Poincaré et formule de Künneth en cohomologie rigide”, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), no. 5, p. 493-498. | MR 1692313 | Zbl 0908.14006

[6] -, “Finitude et pureté cohomologique en cohomologie rigide”, Invent. Math. 128 (1997), no. 2, p. 329-377, Avec un appendice en anglais par A.J. de Jong. | MR 1440308 | Zbl 0908.14005

[7] P. Berthelot & A. Ogus - Notes on crystalline cohomology, Math. Notes, vol. 21, Princeton Univ. Press, 1978. | MR 491705 | Zbl 0383.14010

[8] S. Bloch - Lectures on algebraic cycles, Duke University Mathematics Series, IV, Duke University Mathematics Department, Durham, N.C., 1980. | MR 558224 | Zbl 1201.14006

[9] -, “On an argument of Mumford in the theory of algebraic cycles”, in Journées de Géometrie Algébrique d'Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn, 1980, p. 217-221. | MR 605343 | Zbl 0508.14004

[10] S. Bloch, H. Esnault & M. Levine - “Decomposition of the diagonal and eigenvalues of Frobenius for Fano hypersurfaces”, 2003, arXiv:math.AG/0302109. | MR 2115665 | Zbl 1087.14017

[11] S. Bloch, A. Kas & D. I. Lieberman - “Zero cycles on surfaces with p g =0, Compositio Math. 33 (1976), no. 2, p. 135-145. | Numdam | MR 435073 | Zbl 0337.14006

[12] E. Bombieri - “On exponential sums in finite fields. II”, Invent. Math. 47 (1978), no. 1, p. 29-39. | MR 506272 | Zbl 0396.14001

[13] F. Campana - “Remarques sur les groupes de Kähler nilpotents”, Ann. Sci. École Norm. Sup. 28 (1995), p. 307-316. | Numdam | MR 1326670 | Zbl 0829.32006

[14] A. Chambert-Loir - “À propos du groupe fondamental des variétés rationnellement connexes par chaînes”, 2003, arXiv:math.AG/0303051. | MR 2932435

[15] B. Chiarellotto - “Weights in rigid cohomology applications to unipotent F-isocrystals”, Ann. Sci. École Norm. Sup. 31 (1998), no. 5, p. 683-715. | Numdam | MR 1643966 | Zbl 0933.14008

[16] B. Chiarellotto & B. Le Stum - F-isocristaux unipotents”, Compositio Math. 116 (1999), no. 1, p. 81-110. | MR 1669440 | Zbl 0936.14017

[17] -, “Pentes en cohomologie rigide et F-isocristaux unipotents”, Manuscripta Math. 100 (1999), no. 4, p. 455-468. | MR 1734795 | Zbl 0980.14016

[18] J.-L. Colliot-Thélène & D. A. Madore - “Surfaces de Del Pezzo sans point rationnel sur un corps de dimension cohomologique un”, 2003. | MR 2036596 | Zbl 1056.14030

[19] R. M. Crew - “Etale p-covers in characteristic p, Compositio Math. 52 (1984), no. 1, p. 31-45. | Numdam | MR 742696 | Zbl 0558.14009

[20] O. Debarre - Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001. | MR 1841091 | Zbl 0978.14001

[21] -, “Variétés rationnellement connexes”, in Séminaire Bourbaki, Vol. 2001/02, Astérisque, Soc. Math. France, Paris, 2004, exposé 905, p. 243-266. | Numdam

[22] P. Deligne - “Théorie de Hodge II”, Publ. Math. Inst. Hautes Études Sci. 40 (1972), p. 5-57. | Numdam | MR 498551 | Zbl 0219.14007

[23] -, “La conjecture de Weil, I”, Publ. Math. Inst. Hautes Études Sci. 43 (1974), p. 273-307. | Numdam | MR 340258 | Zbl 0287.14001

[24] B. Dwork - “On the rationality of the zeta function of an algebraic variety”, Amer. J. Math. 82 (1960), p. 631-648. | MR 140494 | Zbl 0173.48501

[25] T. Ekedahl - “Sur le groupe fondamental d'une variété unirationnelle”, C. R. Acad. Sci. Paris Sér. I Math. 297 (1983), no. 12, p. 627-629. | MR 735130 | Zbl 0591.14010

[26] H. Esnault - “Varieties over a finite field with trivial Chow group of 0-cycles have a rational point”, Invent. Math. 151 (2003), no. 1, p. 187-191, arXiv:math.AG/0207022. | MR 1943746 | Zbl 1092.14010

[27] J.-Y. Étesse & B. Le Stum - “Fonctions L associées aux F-isocristaux surconvergents. I. Interprétation cohomologique”, Math. Ann. 296 (1993), no. 3, p. 557-576. | MR 1225991 | Zbl 0789.14015

[28] -, “Fonctions L associées aux F-isocristaux surconvergents. II. Zéros et pôles unités”, Invent. Math. 127 (1997), no. 1, p. 1-31. | MR 1423023 | Zbl 0911.14011

[29] J.-M. Fontaine & W. Messing - p-adic periods and p-adic étale cohomology”, in Arithmetic geometry (Arcata, 1985), Contemporary mathematics, vol. 67, Amer. Math. Soc., 1987, p. 179-207. | MR 902593 | Zbl 0632.14016

[30] T. Graber, J. Harris & J. Starr - “Families of rationally connected varieties”, J. Amer. Math. Soc. 16 (2003), no. 1, p. 57-67. | MR 1937199 | Zbl 1092.14063

[31] E. Grosse-Klönne - “Finiteness of de Rham cohomology in rigid analysis”, Duke Math. J. 113 (2002), no. 1, p. 57-91. | MR 1905392 | Zbl 1057.14023

[32] A. Grothendieck - “Crystals and the de Rham cohomology of schemes”, in Dix exposés sur la cohomologie des schémas [34], p. 306-358. | MR 269663 | Zbl 0215.37102

[33] A. Grothendieck, P. Deligne & N. M. Katz - Groupes de monodromie en géométrie algébrique, Lecture Notes in Math., no. 288-340, Springer Verlag, 1972-73, SGA 7.

[34] A. Grothendieck et al. - Dix exposés sur la cohomologie des schémas, Adv. Stud. in pure Math., North-Holland, 1968. | Zbl 0192.57801

[35] R. Hartshorne - “On the De Rham cohomology of algebraic varieties”, Publ. Math. Inst. Hautes Études Sci. 45 (1975), p. 5-99. | Numdam | MR 432647 | Zbl 0326.14004

[36] L. Illusie - “Complexe de de Rham-Witt et cohomologie cristalline”, Ann. Sci. École Norm. Sup. 12 (1979), p. 501-661. | Numdam | MR 565469 | Zbl 0436.14007

[37] A. J. De Jong - “Smoothness, semistability and alterations”, Publ. Math. Inst. Hautes Études Sci. 83 (1996), p. 51-93. | Numdam | Zbl 0916.14005

[38] A. J. De Jong & J. Starr - “Every rationally connected variety over the function field of a curve has a rational point”, 2002, http://www-math.mit.edu/~dejong/papers/familyofcurves3.dvi. | Zbl 1063.14025

[39] B. Kahn - “Number of points of function fields over finite fields”, 2002, arXiv:math.NT/0210202.

[40] N. M. Katz - “On a theorem of Ax”, Amer. J. Math. 93 (1971), p. 485-499. | MR 288099 | Zbl 0237.12012

[41] -, “Le niveau de la cohomologie des intersections complètes”, in Groupes de monodromie en géométrie algébrique [33], SGA 7, p. 363-399.

[42] -, “Slope filtration of F-crystals”, in Journées de Géométrie algébrique de Rennes, Astérisque, vol. 63, Soc. Math. France, Paris, 1979, p. 113-164. | Numdam | MR 563463

[43] N. M. Katz & W. Messing - “Some consequences of the Riemann hypothesis for varieties over finite fields”, Invent. Math. 23 (1974), p. 73-77. | MR 332791 | Zbl 0275.14011

[44] K. S. Kedlaya - “Counting points on hyperelliptic curves using Monsky-Washnitzer cohomology”, J. Ramanujan Math. Soc. 16 (2001), no. 4, p. 323-338. | MR 1877805 | Zbl 1066.14024

[45] -, “Finiteness of rigid cohomology with coefficients”, 2002, arXiv:math.AG/0208027. | Zbl 1133.14019

[46] -, “Fourier transforms and p-adic Weil II”, 2002, arXiv:math.NT/0210149.

[47] M. Kim - “A vanishing theorem for Fano varieties in positive characteristic”, 2002, arXiv:math.AG/0201183.

[48] S. Kleiman - “Algebraic cycles and the Weil conjectures”, in Dix exposés sur la cohomologie des schémas [34], p. 359-386. | MR 292838 | Zbl 0198.25902

[49] J. Kollár - “Shafarevich maps and plurigenera of algebraic varieties”, Invent. Math. 113 (1993), p. 177-216. | MR 1223229 | Zbl 0819.14006

[50] G. Lachaud & M. Perret - “Un invariant birationnel des variétés de dimension 3 sur un corps fini”, J. Algebraic Geometry 9 (2000), no. 3, p. 451-458. | MR 1752011 | Zbl 0970.14014

[51] A. G. B. Lauder & D. Q. Wan - “Computing zeta functions of Artin-Schreier curves over finite fields”, LMS J. Comput. Math. 5 (2002), p. 34-55 (electronic). | MR 1916921 | Zbl 1067.11078

[52] Yu. I. Manin - “The theory of commutative formal groups over fields of finite characteristic”, Russian Math. Surveys 18 (1963), p. 1-80. | MR 157972 | Zbl 0128.15603

[53] -, “Notes on the arithmetic of Fano threefolds”, Compositio Math. 85 (1993), no. 1, p. 37-55. | Numdam | MR 1199203 | Zbl 0780.14022

[54] B. Mazur - “Frobenius and the Hodge filtration”, Bull. Amer. Math. Soc. 78 (1972), p. 653-667. | MR 330169 | Zbl 0258.14006

[55] -, “Frobenius and the Hodge filtration (estimates)”, Ann. of Math. 98 (1973), p. 58-95. | MR 321932 | Zbl 0261.14005

[56] Z. Mebkhout - “Sur le théorème de finitude de la cohomologie p-adique d’une variété affine non singulière”, Amer. J. Math. 119 (1997), no. 5, p. 1027-1081. | MR 1473068 | Zbl 0926.14007

[57] J. S. Milne - “Zero cycles on algebraic varieties in nonzero characteristic : Roĭtman's theorem”, Compositio Math. 47 (1982), no. 3, p. 271-287. | Numdam | MR 681610 | Zbl 0506.14006

[58] P. Monsky & G. Washnitzer - “Formal cohomology I”, Ann. of Math. 88 (1968), p. 181-217. | MR 248141 | Zbl 0162.52504

[59] O. Moreno & C. J. Moreno - “Improvements of the Chevalley-Warning and the Ax-Katz theorems”, Amer. J. Math. 117 (1995), no. 1, p. 241-244. | MR 1314464 | Zbl 0824.11017

[60] N. Nygaard - “On the fundamental group of a unirational 3-fold”, Invent. Math. 44 (1978), no. 1, p. 75-86. | MR 491731 | Zbl 0427.14014

[61] D. Petrequin - “Classes de Chern et classes de cycles en cohomologie rigide”, Bull. Soc. Math. France 131 (2003), p. 59-121. | Numdam | MR 1975806 | Zbl 1083.14505

[62] A. A. Roĭtman - “The torsion of the group of 0-cycles modulo rational equivalence”, Ann. of Math. 111 (1980), no. 3, p. 553-569. | MR 577137 | Zbl 0504.14006

[63] T. Satoh - “The canonical lift of an ordinary elliptic curve over a finite field and its point counting”, J. Ramanujan Math. Soc. 15 (2000), no. 4, p. 247-270. | MR 1801221 | Zbl 1009.11051

[64] J.-P. Serre - “On the fundamental group of a unirational variety”, J. London Math. Soc. 34 (1959), p. 481-484. | MR 109155 | Zbl 0097.36301

[65] N. I. Shepherd-Barron - “Fano threefolds in positive characteristic”, Compositio Math. 105 (1997), no. 3, p. 237-265. | MR 1440723 | Zbl 0889.14021

[66] T. Shioda - “An example of unirational surface in characteristic p, Math. Ann. 211 (1974), p. 233-236. | MR 374149 | Zbl 0276.14018

[67] N. Tsuzuki - “Cohomological descent of rigid cohomology for proper coverings”, Invent. Math. 151 (2003), no. 1, p. 101-133. | MR 1943743 | Zbl 1085.14019

[68] D. Q. Wan - “A Chevalley-Warning approach to p-adic estimates of character sums”, Proc. Amer. Math. Soc. 123 (1995), no. 1, p. 45-54. | MR 1215208 | Zbl 0821.11062

[69] E. Warning - “Bemerkung zur vorstehenden Arbeit von Herr Chevalley”, Abh. Math. Sem. Univ. Hamburg 11 (1936), p. 76-83. | JFM 61.1043.02 | Zbl 0011.14601

[70] A. Weil - “Number of solutions of equations in finite fields”, Bull. Amer. Math. Soc. 55 (1949), p. 397-508. | MR 29393 | Zbl 0032.39402