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, pp. 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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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 | Zbl

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

[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

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

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

[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 | Zbl

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

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

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

[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 | Zbl

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

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

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

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

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

[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 | Zbl

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

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

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

[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

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

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

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

[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

[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 | Zbl

[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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

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

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

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

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

[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 | Zbl

[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 | Zbl

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

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

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

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

[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 | Zbl

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

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