Points de hauteur bornée, topologie adélique et mesures de Tamagawa

 Access to full text

 Full (PDF)

1. [Ar] S.J. Arakelov, Theory of intersections on the arithmetic surface. Proceedings of the international congress of mathematicians, Vol. 1 (Vancouver, 1974), Canad. Math. Congress, Montréal, 1975, pp. 405-408. Zbl0351.14003MR466150
2. [Art] E. Artin, Über eine neue Art von L-Reihen. Abh. Math. Semin. Univ. Hamburg3 (1924), 89-108. JFM49.0123.01
3. [Ba] V.V. Batyrev, The cone of effective divisors of threefolds. Proceedings of the International Conference on Algebra, Part 3 (Novosibirsk, 1989), Contemp. Math., vol. 131, Part 3, Amer. Math. Soc., Providence, 1992, pp. 337-352. Zbl0781.14027MR1175891
4. [BM] V.V. Batyrev, Y.I. Manin, Sur le nombre des points rationnels de hauteur bornée des variétés algébriques. Math. Ann.286 (1990), 27-43. Zbl0679.14008MR1032922
5. [BT1] V.V. Batyrev, Y. Tschinkel, Rational points of bounded height on compactifications of anisotropic tori. Internat. Math. Res. Notices12 (1995), 591-635. Zbl0890.14008MR1369408
6. [BT2] _, Rational points on some Fano cubic bundles. C. R. Acad. Sci. Paris Sér. I Math.323 (1996), 41-46. Zbl0879.14007MR1401626
7. [BT3] ____, Height zeta functions of toric varieties. J. Math. Sci.82 (1996), n° 1, 3220-3239. Zbl0915.14013MR1423638
8. [BT4] ____, Manin's conjecture for toric varieties. J. Algebraic Geom.7 (1998), n° 1, 15-53. Zbl0946.14009MR1620682
9. [BT5] _____, Tamagawa numbers of polarized algebraic varieties. Nombre et répartition de points de hauteur bornée, Astérisque, vol. 251, SMF, Paris, 1998, pp. 299-340. Zbl0926.11045MR1679843
10. [Bir] B.J. Birch, Forms in many variables. Proc. Roy. Soc. London265A (1962), 245-263. Zbl0103.03102MR150129
11. [BR] M. Borovoi, Z. Rudnick, Hardy-Littlewood varieties and semi-simple groups. Invent. math.119 (1995), 37-66. Zbl0917.11025MR1309971
12. [BGS] J.-B. Bost, H. Gillet, C. Soulé, Heights of projective varieties and positive Green forms. J. Amer. Math. Soc.7 (1994), 903-1027. Zbl0973.14013MR1260106
13. [Bo] D. Bourqui, Fonction zêta des hauteurs des surfaces de Hirzebruch dans le cas fonctionnel. J. Number Theory94 (2002), 343-358. Zbl1006.11032MR1916278
14. [Bre1] R. De La Bretèche.Compter des points d'une variété torique, J. Number theory87 (2001), 315-331. Zbl1020.11045MR1824152
15. [Bre2] _____, Nombre de points de hauteur bornée sur les surfaces de Del Pezzo de degré 5. Duke Math. J.113 (2002), n° 3, 421-464. Zbl1054.14025MR1909606
16. [BV] M. Brion, M. Vergne, Arrangement of hyperplanes. I: rational functions and Jeffrey-Kirwan residue. Ann. Sci. École Norm. Sup. (4) 32 (1999), 715-741. Zbl0945.32003MR1710758
17. [Bro] N. Broberg, Rational points on cubic surfaces. Rational points on algebraic varieties, Progress in Math., vol. 199, Birkhaüser, Basel, 2001, pp. 13-35. Zbl1080.14517
18. [CLT1] A. Chambert-Loir, Y. Tschinkel, Points of bounded height on equivariant compactifications of vector groups, I. Compositio Math.124 (2000), n° 1, 65-93. Zbl0963.11033MR1797654
19. [CLT2] ____, Points of bounded height on equivariant compactifications of vector groups, II. J. Number Theory85 (2000), n° 2, 172-188. Zbl0963.11034MR1802710
20. [CLT3] ____, On the distribution of points of bounded height on equivariant compactifications of vector groups. Invent. Math.148 (2002), n° 2, 421-452. Zbl1067.11036MR1906155
21. [CTS1] J.-L. Colliot-Thélène, J.-J. Sansuc, Torseurs sous des groupes de type multiplicatif; applications à l'étude des points rationnels de certaines variétés algébriques. C. R. Acad. Sci. Paris Sér. I Math.282 (1976), 1113-1116. Zbl0337.14014MR414556
22. [CTS2] _____, La descente sur les variétés rationnelles. Journées de géométrie algébrique d'Angers (1979) (A. Beauville, ed.), Sijthoff & Noordhoff, Alphen aan den Rijn, 1980, pp. 223-237. MR605344
23. [CTS3] ____, La descente sur les variétés rationnelles, II. Duke Math. J.54 (1987), n° 2, 375-492. Zbl0659.14028MR899402
24. [Co] D. Cox, The homogeneous coordinate ring of a toric variety. J. Algebraic Geom.4 (1995), 17-50. Zbl0846.14032MR1299003
25. [De1] P. Deligne, La conjecture de Weil I.. Publ. Math. I.H.E.S.43 (1973), 273-307. Zbl0287.14001MR340258
26. [Delz] T. Delzant, Hamiltoniens périodiques et images convexes de l'application moment. Bull. Soc. Math. France116 (1988), 315-339. Zbl0676.58029MR984900
27. [Dr] P.K.J. Draxl, L-Funktionen algebraischer Tori. J. Number Theory3 (1971), 444-467. Zbl0231.12018MR291099
28. [FMT] J. Franke, Y.I. Manin, Y. Tschinkel, Rational points of bounded height on Fano varieties. Invent. Math.95 (1989), 421-435. Zbl0674.14012MR974910
29. [HW] G.H. Hardy, E.M. Wright, An introduction to the theory of numbers, 3rd ed.. Oxford, Clarendon Press, 1954. Zbl0058.03301MR67125
30. [HB1] D.R. Heath-Brown, The density of zeros of forms for which weak approximation fails. Math. Comp.59 (1992), 613-623. Zbl0778.11017MR1146835
31. [HB2] _____, Counting rational points on cubic surfaces. Nombre et répartition de points de hauteur bornée, Astérisque, vol. 251, SMF, Paris, 1998, pp. 13-30. Zbl0926.11046MR1679837
32. [Lan] R.P. Langlands, On the functional equations satisfied by Eisenstein series. Lecture Notes in Math., vol. 544, Springer-Verlag, Berlin, Heidelberg and New York, 1976. Zbl0332.10018MR579181
33. [Ma] Y.I. Manin, Notes on the arithmetic of Fano threefolds. Compositio Math85 (1993), 37-55. Zbl0780.14022MR1199203
34. [MP] A.S. Merkurjev, I.A. Panin, K-theory of algebraic tori and toric varieties. K-Theory12 (1997), n° 2, 101-143. Zbl0882.19002MR1469139
35. [Mi] J.S. Milne, Étale cohomology. Princeton Math. Series, vol. 33, Princeton University Press, 1980. Zbl0433.14012MR559531
36. [Né] A. Néron, L'arithmétique sur les variétés algébriques [d'après A. Weil]. Séminaire Bourbaki 4-ème année, 1951/52, n° 66. 
37. [Ono] T. Ono, On the Tamagawa number of algebraic tori. Ann. of Math. (2) 78 (1963), n° 1, 47-73. Zbl0122.39101MR156851
38. [Pe1] E. Peyre, Hauteurs et mesures de Tamagawa sur les variétés de Fano. Duke Math. J.79 (1995), n° 1, 101-218. Zbl0901.14025MR1340296
39. [Pe2] ____, Terme principal de la fonction zéta des hauteurs et torseurs universels. Nombre et répartition de points de hauteur bornée, Astérisque, vol. 251, SMF, Paris, 1998, pp. 259-298. Zbl0966.14016MR1679842
40. [Pe3] _____, Torseurs universels et méthode du cercle. Rational points on algebraic varieties, Progress in Math., vol. 199, Birkhaüser, Basel, 2001, pp. 221-274. Zbl1092.14506MR1875176
41. [Pe4] _____, Points de hauteur bornée sur les variétés de drapeaux en caractéristique finie. Prépublication (2001). 
42. [PT1] E. Peyre, Y. Tschinkel, Tamagawa numbers of diagonal cubic surfaces, numerical evidence. Math. Comp.70 (2000), n° 233, 367-387. Zbl0961.14012MR1681100
43. [PT2] _____, Tamagawa numbers of diagonal cubic surfaces of higher rank. Rational points on algebraic varieties, Progress in Math., vol. 199, Birkhaüser, Basel, 2001, pp. 275-305. Zbl1079.11034MR1875177
44. [Sa] P. Salberger, Tamagawa measures on universal torsors and points of bounded height on Fano varieties. Nombre et répartition de points de hauteur bornée, Astérisque, vol. 251, SMF, Paris, 1998, pp. 91-258. Zbl0959.14007MR1679841
45. [Sc] S.H. Schanuel, Heights in number fields. Bull. Soc. Math. France107 (1979), 433-449. Zbl0428.12009MR557080
46. [Sch] W. Schmidt, Asymptotic formulae for points lattices of bounded determinant and subspaces of bounded height. Duke Math. J.35 (1968), 327-339. Zbl0172.06304MR224562
47. [Se1] J.-P. Serre, Valeurs propres des endomorphismes de Frobenius (d'après P. Deligne). Séminaire Bourbaki 26-ème année, 1973/74, n° 446. Zbl0331.14013
48. [Se2] ____, Lectures on the Mordell- Weil theorem. Aspects of Mathematics, vol. E15, Vieweg, Braunschweig, Wiesbaden, 1989. MR1757192
49. [Sk] A.N. Skorobogatov, On a theorem of Enriques-Swinnerton-Dyer. Ann. Fac. Sci. Toulouse Math. (6) 2 (1993), n° 3, 429-440. Zbl0811.14020MR1260765
50. [SSD] J.B. Slater, H.P.F. Swinnerton-Dyer, Counting points on cubic surfaces, I. Nombre et répartition de points de hauteur bornée, Astérisque, vol. 251, SMF, Paris, 1998, pp. 1-12. Zbl0969.11027MR1679836
51. [ST1] M. Strauch, Y. Tschinkel, Height zeta functions of twisted products. Math. Res. Lett.4 (1997), 273-282. Zbl0918.11036MR1453059
52. [ST2] _____, Height zeta functions of toric bundles over flag varieties. Selecta Math.(N.S.) 5 (1999), n° 3, 325-396. Zbl1160.14302MR1723811
53. [SD] H.P.F. Swinnerton-Dyer, Counting rational points on cubic surfaces. Classification of algebraic varieties (L'Aquila, 1992) (C. Ciliberto, E. L. Livorni, A. J. Sommese, eds.), Contemp. Math., vol. 162, AMS, Providence, 1994, pp. 371-379. Zbl0844.14008MR1272709
54. [Va] R.C. Vaughan, The Hardy-Littlewood method. Cambridge Tracts in Mathematics, vol. 80, Cambridge University Press, Cambridge- New York, 1981. Zbl0455.10034MR628618
55. [We1] A. Weil, Arithmetic on algebraic varieties. Ann. of Math. (2) 53 (1951), 412-444. Zbl0043.27002MR42169
56. [We2] ____, Adèles and algebraic groups. Progress in Mathematics, vol. 23, Birkhaüser, Boston, Basel, Stuttgart, 1982. MR670072

1. Jean-Louis Colliot-Thélène, Venapally Suresh, Quelques questions d’approximation faible pour les tores algébriques
2. Alex Gorodnik, François Maucourant, Hee Oh, Manin’s and Peyre’s conjectures on rational points and adelic mixing
3. Régis de la Bretèche, Fonctions zêta des hauteurs