Speculations about the topology of rational points: an up-date

Columbia university number theory seminar - New-York, 1992, Astérisque, no. 228 (1995), 17 p.

@incollection{AST_1995__228__165_0,
     author = {Mazur, B.},
     title = {Speculations about the topology of rational points: an up-date},
     booktitle = {Columbia university number theory seminar - New-York, 1992},
     author = {Collectif},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {228},
     year = {1995},
     zbl = {0851.14009},
     mrnumber = {1330932},
     language = {en},
     url = {www.numdam.org/item/AST_1995__228__165_0/}
}

Mazur, B. Speculations about the topology of rational points: an up-date, dans Columbia university number theory seminar - New-York, 1992, Astérisque, no. 228 (1995), 17 p. http://www.numdam.org/item/AST_1995__228__165_0/

Bibliographie

1. [A-J] Arenstorf, R., Johnston, D.: Uniform distribution of integral points on $3$ -dimensional spheres via modular forms. Journal of Number Theory 11 (1979) 218-238 | Article | MR 535393 | Zbl 0413.10037

2. [B-M] Batyrev, V., Manin, Y.: Sur le nombre des points rationnels de hauteur bornée des variétés algébriques. Math. Ann. 286 (1990) 27-43 | Article | EuDML 164626 | MR 1032922 | Zbl 0679.14008

3. [B 1] Birch, B.: Homogeneous forms of odd degree in a large number of variables. Mathematika 4 (1957) 102-105 | Article | MR 97359 | Zbl 0081.04501

4. [B 2] Birch, B.: Forms in many variables. Proc. Roy. Soc. Ser. A 265 (1962) 245-263 | Article | MR 150129 | Zbl 0103.03102

5. [Bu] Burnol, J.-F.: Remarques sur la stabilité en arithmétique. International Math. Research Notices (1992) no. 6 in Duke Math. J. 66 (1992) 117-127 | MR 1167116 | Zbl 0764.14006

6. [C-S] Call, G., Silverman, J.: Canonical heights on varieties with morphisms. Preprint (1993) | Numdam | MR 1255693 | Zbl 0826.14015

7. [Ca-N] Cassou-Noguès, P.: Valeurs aux entiers négatifs des séries de Dirichlet associées à un polynôme II, Amer. J. of Math. 106 (1985) 255-299 | Article | MR 737776 | Zbl 0561.10020

8. [CT 1] Colliot-Thélène, J.-L.: Arithmétique des variétés rationnelles et problèmes birationnels. Proc. Int. Cong. Math. Berkeley Ca. (1986) 641-653 | MR 934267 | Zbl 0698.14060

9. Colliot-Thélène, J.-L.: Surfaces rationnelles fibrées en coniques de degré $4$ . Séminaire de Théorie des Nombres, Paris 1988-1989. Birkhäuser, Boston-Basel-Berlin (1990) 43-55 | MR 1104699 | Zbl 0731.14033

10. [CT-Sa 1] Colliot-Thélène, J.-L., Sansuc, J.-J.: Sur le principle de Hasse et l'approximation faible, et sur une hypothèse de Schinzel. Acta Arithmetica XLI.1 (1982) 33-53 | Article | EuDML 205830 | MR 667708 | Zbl 0414.10009

11. [CT-Sa 2] Colliot-Thélène, J.-L., Sansuc, J.-J.: La descente sur les variétés rationnelles II. Duke Math. J. 54 (1987) 375-492 | Article | MR 899402 | Zbl 0659.14028

12. [CT-S-SD] Colliot-Thélène, J.-L., Sansuc, J.-L., Swinnerton-Dyer, P.: Intersections of two quadrics and Châtelet surfaces, II, J. für die reine und angew. Math. 374 (1987) 72-168 | EuDML 152896 | MR 876222 | Zbl 0622.14030

13. [CT-SD] Colliot-Thélène, J.-L., Swinnerton-Dyer, P.: Hasse principle and weak approximation for pencils of Severi-Brauer and similar varieties. Preprint (1992) | MR 1285781 | Zbl 0805.14010

14. [C-V] Conn, W., Vaserstein, L.: On sums of three integral cubes. Preprint (1992) | MR 1284068 | Zbl 0808.11075

15. [C-T] Coray, D., Tsfasman, M.: Arithmetic on singular Del Pezzo surfaces, Proc. London Math. Soc. 57 (1988) 25-87 | Article | MR 940430 | Zbl 0653.14018

16. [D-W] Datskowsky, B., Wright, D.: The adelic zeta-function associated with the space of binary cubic forms II: Local theory. Journal für die reine und der angew. Math. 386 (1988) 116-138 | MR 839123

17. [D-L] Davenport, H., Lewis, D.: Homogeneous additive equations. Proc. Roy. Soc. Ser. A 274 (1963) 443-460 | Article | MR 153655 | Zbl 0118.28002

18. [D] Duke, W.: Hyperbolic distribution problems and half-integral weight Maass forms. Inv. Math. 92 (1988) 73-90 | Article | EuDML 143559 | MR 931205 | Zbl 0628.10029

19. [D-R-S] Duke, W., Rudnick, Z., Sarnak, P.: Density of integer points on affine homogeneous varieties. Preprint (1991) | MR 1230289 | Zbl 0798.11024

20. [D-S-P] Duke, W., Schulze-Pillot, R.: Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids. Inv. Math. 99 (1990) 49-57 | Article | EuDML 143750 | MR 1029390 | Zbl 0692.10020

21. [F] Faddeev, D.: Invariants of divisor classes for the curves $x^{k} (1 - x) = y^{ℓ}$ in an $ℓ$ -adic cyclotomic field. Tr. Mat. Inst. Steklova 64 (1961) 284-293 | MR 132740 | Zbl 0126.07001

22. [F-M & T] Franke, J., Manin, Y., Tschinkel, Y.: Rational points of bounded height on Fano varieties. Inv. Math. 95 (1989) 421-435 | Article | EuDML 143659 | MR 974910 | Zbl 0674.14012

23. [F-G] Fomenko, O., Golubeva, E.: Application of spherical functions to a certain problem in the theory of quadratic forms. J. Sov. Math. 38 (1987) 2054-2060 | Article | MR 787411 | Zbl 0624.10014

24. [G-G] Gordon, D., Grant, D.: Computing the Mordell-Weil rank of Jacobians of curves of genus two. Preprint (1991) | MR 1094558 | Zbl 0790.14028

25. [G-R] Gross, B., Rohrlich, D.: Some results on the Mordell-Weil group of the jacobian of the Fermat curve. Inv. Math. 44 (1978) 201-224 | Article | EuDML 142528 | MR 491708 | Zbl 0369.14011

26. [H-B] Heath-Brown, D. R.: The density of zeros of forms for which weak approximation fails. Mathematics of Computation 59 (1992) 613-623 | Article | MR 1146835 | Zbl 0778.11017

27. [Hi] Hironaka, H.: Triangulations of algebraic sets. Proc. Sympos. Pure Math. 29 A.M.S. Providence, R.I. (1975) 165-185 | Article | MR 374131 | Zbl 0332.14001

28. [Ho] Hooley, C.: On nonary cubic forms: I. Journal für die reine und der angew. Math. 386 (1988) 32-98 | EuDML 153017 | MR 936992 | Zbl 0641.10019

Hooley, C.: On nonary cubic forms: II. Journal für die reine und der angew. Math. 415 (1991) 95-165 | EuDML 153309 | MR 1096903 | Zbl 0719.11017

29. [Ig] Igusa, J.-I.: Forms of higher degree, Tata Institute of Fundamental Research, Bombay (1978) | MR 546292 | Zbl 0417.10015

30. [Iw] Iwaniec, H.: Fourier coefficients of modular forms of half integral weight. Inv. Math. 87 (1987) 385-401 | Article | EuDML 143426 | MR 870736 | Zbl 0606.10017

31. [K] Kolyvagin, V. A.: Euler systems, pp. 435-483 in: The Grothendieck Festschrift II, Progress in Mathematics 87, Birkhäuser, Boston-Basel-Berlin (1990) | MR 1106906 | Zbl 0742.14017

32. [K-R] Koblitz, N., Rohrlich, D.: Simple factors in the jacobian of a Fermat curve. Can. J. Math. 30 (1978) 1183-1205 | Article | MR 511556 | Zbl 0399.14023

33. [K-W] Kuwata, M., Wang, L.: Topology of rational points on isotrivial elliptic surfaces. To appear: in Int. Math. Research Notices, in Duke Math. J. (1993) | MR 1214702 | Zbl 0804.14008

34. [Licht] Lichtin, B.: On the moderate growth of generalized Dirichlet series for hypoelliptic polynomials. Compositio Math. 80 (1991) 337-354 | EuDML 90128 | Numdam | MR 1134260 | Zbl 0759.11027

35. [Linn] Linnik, Y.: Ergodic properties of algebraic fields. Springer-Verlag 1968 | Article | MR 238801 | Zbl 0162.06801

36. [Lo] Lojasiewicz, S.: Triangulation of semi-analytic sets. Ann. Scuo. Norm. Pisa 18 (1964) 449-474 | EuDML 83333 | Numdam | MR 173265 | Zbl 0128.17101

37. [Mal] Malyshev, A.: On representations of integers by positive quadratic forms. Trudy V.A. Steklov Math. Inst. AN SSSR 65 (1962) 1-212 | MR 142511 | Zbl 0135.09801

38. [Mn 1] Manin, Y.: Cubic Forms, algebra, geometry, arithmetic (2nd Ed.) North-Holland (1986) | MR 833513 | Zbl 0582.14010

39. [Ms 2] Manin, Y.: Algebraic Varieties With Many Rational Points, Lecture Notes, MIT (1990)

40. [Mn 3] Manin, Y.: Notes on the arithmetic of Fano threefolds, Composito Mathematica 85 (1993) 37-55 | EuDML 90190 | Numdam | MR 1199203 | Zbl 0780.14022

41. [M-Tsch] Manin, Y., Tschinkel, Y.: Points of bounded height on Del Pezzo surfaces. Preprint no. 655. Univ. of Utrecht (1991) | Numdam | MR 1214451 | Zbl 0782.14033

42. [Mz 1] Mazur, B.: The topology of rational points. Journ. Exp. Math. 1 (1992) 35-45 | EuDML 231027 | MR 1181085 | Zbl 0784.14012

43. [Mz 2] Mazur, B.: Questions of decidability and undecidability in Number Theory. To appear: Journal of Symbolic Logic | MR 1276620 | Zbl 0814.11059

44. [Mc] Mccallum, W.: The arithmetic of Fermat curves. Math. Ann. 294 (1992) 503-511 | Article | EuDML 165014 | MR 1188134 | Zbl 0766.14013

45. [Pe] Peyre, E.: Hauteurs et nombres de Tamagawa sur les variétés de Fano, preprint | Zbl 0901.14025

46. [P] Pommerenke, C.: Über die Gleichverteilung von Gitterpunkten auf $m$ -dimensionalen Ellipsoiden. Acta Arith. 5 (1957) 227-257 | Article | EuDML 206405 | MR 122788 | Zbl 0089.26802

47. [R] Rohrlich, D.: Variation of the root number in families of elliptic curves. Preprint (1992) | Numdam | MR 1219633 | Zbl 0791.11026

48. [Sa] Sargos, P.: Croissance de certaines séries de Dirichlet et applications. Journal für die reine und der angew. Math. 367 (1986) 139-154 | EuDML 152828 | MR 839128 | Zbl 0575.10035

49. [Sch] Schmidt, W.: The density of integer points on homogeneous varieties. Acta Mathematica 154 (1985) 243-296 | Article | MR 781588 | Zbl 0561.10010

50. [Se] Serre, J.-P.: Lectures on the Mordell-Weil Theorem. Aspects of Math. 15 Vieweg (1989) | MR 1002324 | Zbl 0676.14005

51. [Si 1] Silverman, J.: Rational points on $K 3$ surfaces: a new canonical height. Inv. Math. 105 (1991) 347-373 | Article | EuDML 143916 | MR 1115546 | Zbl 0754.14023

52. [Si 2] Silverman, J.: Counting integer and rational points on varieties. Preprint (1991); in this volume. | MR 1330936 | Zbl 0834.11029

53. [S-D] Swinnerton-Dyer, P.: Counting rational points on cubic surfaces. Preprint 1993 | MR 1272709 | Zbl 0844.14008

54. [Th] Thunder, J.: An asymptotic estimate for heights of algebraic subspaces. PhD Thesis, Univ. of Colorado (1990) | MR 2638744

55. [Tsch] Tschinkel, Y.: Arithmetic of algebraic surfaces. M.I.T. PhD thesis (1992) | MR 2716382

56. [We] Wehler, J.: $K$ - $3$ surfaces with Picard number $2$ . Arch. Math. 50 (1988) 73-82 | Article | MR 925498 | Zbl 0602.14038

57. [Wr] Wright, D.: The adelic zeta-function associated with the space of binary cubic forms II: Global theory. Global theory Math. Ann. 270 (1985) 503-534 | EuDML 182976 | MR 776169 | Zbl 0533.10020

58. [W-Y] Wright, D., Yukie, A.: Prehomogeneous vector spaces and field extensions. Inv. Math. 110 (1992) 283-314 | Article | EuDML 144052 | MR 1185585 | Zbl 0803.12004