Rational points and fundamental groups: applications of the p -adic cohomology

Antoine Chambert-loir

Séminaire Bourbaki (2002-2003)

  • Volume: 45, page 125-146
  • ISSN: 0303-1179

Abstract

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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.

How to cite

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Chambert-loir, Antoine. "Points rationnels et groupes fondamentaux : applications de la cohomologie $p$-adique." Séminaire Bourbaki 45 (2002-2003): 125-146. <http://eudml.org/doc/252128>.

@article{Chambert2002-2003,
abstract = {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.},
author = {Chambert-loir, Antoine},
journal = {Séminaire Bourbaki},
keywords = {Fano varieties; chain rationaly connected varieties; rational points; fundamental group; rigid cohomology; slopes},
language = {fre},
pages = {125-146},
publisher = {Association des amis de Nicolas Bourbaki, Société mathématique de France},
title = {Points rationnels et groupes fondamentaux : applications de la cohomologie $p$-adique},
url = {http://eudml.org/doc/252128},
volume = {45},
year = {2002-2003},
}

TY - JOUR
AU - Chambert-loir, Antoine
TI - Points rationnels et groupes fondamentaux : applications de la cohomologie $p$-adique
JO - Séminaire Bourbaki
PY - 2002-2003
PB - Association des amis de Nicolas Bourbaki, Société mathématique de France
VL - 45
SP - 125
EP - 146
AB - 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.
LA - fre
KW - Fano varieties; chain rationaly connected varieties; rational points; fundamental group; rigid cohomology; slopes
UR - http://eudml.org/doc/252128
ER -

References

top
  1. [1] J. Ax – “Zeroes of polynomials over finite fields”, Amer. J. Math.86 (1964), p. 255–261. Zbl0121.02003MR160775
  2. [2] P. Berthelot – Cohomologie cristalline des schémas de caractéristique p g t ; 0 , Lecture Notes in Math., vol. 407, Springer Verlag, 1974. Zbl0298.14012MR384804
  3. [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. MR865810
  4. [4] —, “Cohomologie rigide et cohomologie rigide à supports propres. Première partie”, Prépublication, IRMAR, Université Rennes 1, 1996. 
  5. [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. Zbl0908.14006MR1692313
  6. [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. Zbl0908.14005MR1440308
  7. [7] P. Berthelot & A. Ogus – Notes on crystalline cohomology, Math. Notes, vol. 21, Princeton Univ. Press, 1978. Zbl0383.14010MR491705
  8. [8] S. Bloch – Lectures on algebraic cycles, Duke University Mathematics Series, IV, Duke University Mathematics Department, Durham, N.C., 1980. Zbl1201.14006MR558224
  9. [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. Zbl0508.14004MR605343
  10. [10] S. Bloch, H. Esnault & M. Levine – “Decomposition of the diagonal and eigenvalues of Frobenius for Fano hypersurfaces”, 2003, arXiv:math.AG/0302109. Zbl1087.14017MR2115665
  11. [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. Zbl0337.14006MR435073
  12. [12] E. Bombieri – “On exponential sums in finite fields. II”, Invent. Math. 47 (1978), no. 1, p. 29–39. Zbl0396.14001MR506272
  13. [13] F. Campana – “Remarques sur les groupes de Kähler nilpotents”, Ann. Sci. École Norm. Sup.28 (1995), p. 307–316. Zbl0829.32006MR1326670
  14. [14] A. Chambert-Loir – “À propos du groupe fondamental des variétés rationnellement connexes par chaînes”, 2003, arXiv:math.AG/0303051. MR2932435
  15. [15] B. Chiarellotto – “Weights in rigid cohomology applications to unipotent F -isocrystals”, Ann. Sci. École Norm. Sup. 31 (1998), no. 5, p. 683–715. Zbl0933.14008MR1643966
  16. [16] B. Chiarellotto & B. Le Stum – “ F -isocristaux unipotents”, Compositio Math. 116 (1999), no. 1, p. 81–110. Zbl0936.14017MR1669440
  17. [17] —, “Pentes en cohomologie rigide et F -isocristaux unipotents”, Manuscripta Math. 100 (1999), no. 4, p. 455–468. Zbl0980.14016MR1734795
  18. [18] J.-L. Colliot-Thélène & D.A. Madore – “Surfaces de Del Pezzo sans point rationnel sur un corps de dimension cohomologique un”, 2003. Zbl1056.14030MR2036596
  19. [19] R.M. Crew – “Etale p -covers in characteristic p ”, Compositio Math. 52 (1984), no. 1, p. 31–45. Zbl0558.14009MR742696
  20. [20] O. Debarre – Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001. Zbl0978.14001MR1841091
  21. [21] —, “Variétés rationnellement connexes”, in Séminaire Bourbaki, Vol. 2001/02, Astérisque, Soc. Math. France, Paris, 2004, exposé 905, p. 243–266. 
  22. [22] P. Deligne – “Théorie de Hodge II”, Publ. Math. Inst. Hautes Études Sci.40 (1972), p. 5–57. Zbl0219.14007MR498551
  23. [23] —, “La conjecture de Weil, I”, Publ. Math. Inst. Hautes Études Sci.43 (1974), p. 273–307. Zbl0287.14001MR340258
  24. [24] B. Dwork – “On the rationality of the zeta function of an algebraic variety”, Amer. J. Math.82 (1960), p. 631–648. Zbl0173.48501MR140494
  25. [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. Zbl0591.14010MR735130
  26. [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. Zbl1092.14010MR1943746
  27. [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. Zbl0789.14015MR1225991
  28. [28] —, “Fonctions L associées aux F -isocristaux surconvergents. II. Zéros et pôles unités”, Invent. Math. 127 (1997), no. 1, p. 1–31. Zbl0911.14011MR1423023
  29. [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. Zbl0632.14016MR902593
  30. [30] T. Graber, J. Harris & J. Starr – “Families of rationally connected varieties”, J. Amer. Math. Soc. 16 (2003), no. 1, p. 57–67. Zbl1092.14063MR1937199
  31. [31] E. Grosse-Klönne – “Finiteness of de Rham cohomology in rigid analysis”, Duke Math. J. 113 (2002), no. 1, p. 57–91. Zbl1057.14023MR1905392
  32. [32] A. Grothendieck – “Crystals and the de Rham cohomology of schemes”, in Dix exposés sur la cohomologie des schémas [34], p. 306–358. Zbl0215.37102MR269663
  33. [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. [34] A. Grothendieck et al. – Dix exposés sur la cohomologie des schémas, Adv. Stud. in pure Math., North-Holland, 1968. Zbl0192.57801
  35. [35] R. Hartshorne – “On the De Rham cohomology of algebraic varieties”, Publ. Math. Inst. Hautes Études Sci.45 (1975), p. 5–99. Zbl0326.14004MR432647
  36. [36] L. Illusie – “Complexe de de Rham–Witt et cohomologie cristalline”, Ann. Sci. École Norm. Sup.12 (1979), p. 501–661. Zbl0436.14007MR565469
  37. [37] A.J. de Jong – “Smoothness, semistability and alterations”, Publ. Math. Inst. Hautes Études Sci.83 (1996), p. 51–93. Zbl0916.14005
  38. [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. Zbl1063.14025
  39. [39] B. Kahn – “Number of points of function fields over finite fields”, 2002, arXiv:math.NT/0210202. 
  40. [40] N.M. Katz – “On a theorem of Ax”, Amer. J. Math.93 (1971), p. 485–499. Zbl0237.12012MR288099
  41. [41] —, “Le niveau de la cohomologie des intersections complètes”, in Groupes de monodromie en géométrie algébrique [33], SGA 7, p. 363–399. 
  42. [42] —, “Slope filtration of F-crystals”, in Journées de Géométrie algébrique de Rennes, Astérisque, vol. 63, Soc. Math. France, Paris, 1979, p. 113–164. Zbl0426.14007MR563463
  43. [43] N.M. Katz & W. Messing – “Some consequences of the Riemann hypothesis for varieties over finite fields”, Invent. Math.23 (1974), p. 73–77. Zbl0275.14011MR332791
  44. [44] K.S. Kedlaya – “Counting points on hyperelliptic curves using Monsky-Washnitzer cohomology”, J. Ramanujan Math. Soc. 16 (2001), no. 4, p. 323–338. Zbl1066.14024MR1877805
  45. [45] —, “Finiteness of rigid cohomology with coefficients”, 2002, arXiv:math.AG/0208027. Zbl1133.14019
  46. [46] —, “Fourier transforms and p -adic Weil II”, 2002, arXiv:math.NT/0210149. 
  47. [47] M. Kim – “A vanishing theorem for Fano varieties in positive characteristic”, 2002, arXiv:math.AG/0201183. 
  48. [48] S. Kleiman – “Algebraic cycles and the Weil conjectures”, in Dix exposés sur la cohomologie des schémas [34], p. 359–386. Zbl0198.25902MR292838
  49. [49] J. Kollár – “Shafarevich maps and plurigenera of algebraic varieties”, Invent. Math.113 (1993), p. 177–216. Zbl0819.14006MR1223229
  50. [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. Zbl0970.14014MR1752011
  51. [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). Zbl1067.11078MR1916921
  52. [52] Yu.I. Manin – “The theory of commutative formal groups over fields of finite characteristic”, Russian Math. Surveys18 (1963), p. 1–80. Zbl0128.15603MR157972
  53. [53] —, “Notes on the arithmetic of Fano threefolds”, Compositio Math. 85 (1993), no. 1, p. 37–55. Zbl0780.14022MR1199203
  54. [54] B. Mazur – “Frobenius and the Hodge filtration”, Bull. Amer. Math. Soc.78 (1972), p. 653–667. Zbl0258.14006MR330169
  55. [55] —, “Frobenius and the Hodge filtration (estimates)”, Ann. of Math.98 (1973), p. 58–95. Zbl0261.14005MR321932
  56. [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. Zbl0926.14007MR1473068
  57. [57] J.S. Milne – “Zero cycles on algebraic varieties in nonzero characteristic : Roĭtman’s theorem”, Compositio Math. 47 (1982), no. 3, p. 271–287. Zbl0506.14006MR681610
  58. [58] P. Monsky & G. Washnitzer – “Formal cohomology I”, Ann. of Math.88 (1968), p. 181–217. Zbl0162.52504MR248141
  59. [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. Zbl0824.11017MR1314464
  60. [60] N. Nygaard – “On the fundamental group of a unirational 3 -fold”, Invent. Math. 44 (1978), no. 1, p. 75–86. Zbl0427.14014MR491731
  61. [61] D. Petrequin – “Classes de Chern et classes de cycles en cohomologie rigide”, Bull. Soc. Math. France131 (2003), p. 59–121. Zbl1083.14505MR1975806
  62. [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. Zbl0504.14006MR577137
  63. [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. Zbl1009.11051MR1801221
  64. [64] J.-P. Serre – “On the fundamental group of a unirational variety”, J. London Math. Soc.34 (1959), p. 481–484. Zbl0097.36301MR109155
  65. [65] N.I. Shepherd-Barron – “Fano threefolds in positive characteristic”, Compositio Math. 105 (1997), no. 3, p. 237–265. Zbl0889.14021MR1440723
  66. [66] T. Shioda – “An example of unirational surface in characteristic p ”, Math. Ann.211 (1974), p. 233–236. Zbl0276.14018MR374149
  67. [67] N. Tsuzuki – “Cohomological descent of rigid cohomology for proper coverings”, Invent. Math. 151 (2003), no. 1, p. 101–133. Zbl1085.14019MR1943743
  68. [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. Zbl0821.11062MR1215208
  69. [69] E. Warning – “Bemerkung zur vorstehenden Arbeit von Herr Chevalley”, Abh. Math. Sem. Univ. Hamburg11 (1936), p. 76–83. Zbl0011.14601JFM61.1043.02
  70. [70] A. Weil – “Number of solutions of equations in finite fields”, Bull. Amer. Math. Soc.55 (1949), p. 397–508. Zbl0032.39402MR29393

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