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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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On the passage from local to global in number theory
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by B. Mazur PDF
Bull. Amer. Math. Soc. 29 (1993), 14-50 Request permission
References
  • Michael Artin, Algebraic spaces, Yale Mathematical Monographs, vol. 3, Yale University Press, New Haven, Conn.-London, 1971. A James K. Whittemore Lecture in Mathematics given at Yale University, 1969. MR 0407012
  • M. Artin, Néron models, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 213–230. MR 861977, DOI 10.1007/978-1-4613-8655-1_{8}
  • Massimo Bertolini and Henri Darmon, Kolyvagin’s descent and Mordell-Weil groups over ring class fields, J. Reine Angew. Math. 412 (1990), 63–74. MR 1079001, DOI 10.1515/crll.1990.412.63
  • Armand Borel, Introduction aux groupes arithmétiques, Publications de l’Institut de Mathématique de l’Université de Strasbourg, XV. Actualités Scientifiques et Industrielles, No. 1341, Hermann, Paris, 1969 (French). MR 0244260
  • A. Borel and J.-P. Serre, Théorèmes de finitude en cohomologie galoisienne, Comment. Math. Helv. 39 (1964), 111–164 (French). MR 181643, DOI 10.1007/BF02566948
    • S. Bosch, W. Lütkebohmert, and M. Raynaud, Néron models, Ergeb. Math. Grenzgeb., bd. 21, Springer-Verlag, New York, 1990.
  • Lawrence Breen, Bitorseurs et cohomologie non abélienne, The Grothendieck Festschrift, Vol. I, Progr. Math., vol. 86, Birkhäuser Boston, Boston, MA, 1990, pp. 401–476 (French). MR 1086889
  • J. W. S. Cassels, Rational quadratic forms, London Mathematical Society Monographs, vol. 13, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1978. MR 522835
  • J. W. S. Cassels, Lectures on elliptic curves, London Mathematical Society Student Texts, vol. 24, Cambridge University Press, Cambridge, 1991. MR 1144763, DOI 10.1017/CBO9781139172530
  • V. I. Chernousov, The Hasse principle for groups of type $E_8$, Dokl. Akad. Nauk SSSR 306 (1989), no. 5, 1059–1063 (Russian); English transl., Soviet Math. Dokl. 39 (1989), no. 3, 592–596. MR 1014762
  • John Coates, Elliptic curves with complex multiplication and Iwasawa theory, Bull. London Math. Soc. 23 (1991), no. 4, 321–350. MR 1125859, DOI 10.1112/blms/23.4.321
  • Paul J. Cohen, Decision procedures for real and $p$-adic fields, Comm. Pure Appl. Math. 22 (1969), 131–151. MR 244025, DOI 10.1002/cpa.3160220202
  • Jean-Louis Colliot-Thélène, Les grands thèmes de François Châtelet, Enseign. Math. (2) 34 (1988), no. 3-4, 387–405 (French). MR 979649
    • J.-L. Colliot-Thélène and P. Swinnerton-Dyer, Hasse principle and weak approximation for pencils of Severi-Brauer and similar varieties, preprint, 1992.
  • Henri Darmon, Euler systems and refined conjectures of Birch Swinnerton-Dyer type, $p$-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991) Contemp. Math., vol. 165, Amer. Math. Soc., Providence, RI, 1994, pp. 265–276. MR 1279613, DOI 10.1090/conm/165/01604
  • Henri Darmon, A refined conjecture of Mazur-Tate type for Heegner points, Invent. Math. 110 (1992), no. 1, 123–146. MR 1181819, DOI 10.1007/BF01231327
  • W. Duke, Hyperbolic distribution problems and half-integral weight Maass forms, Invent. Math. 92 (1988), no. 1, 73–90. MR 931205, DOI 10.1007/BF01393993
  • William Duke and Rainer Schulze-Pillot, Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids, Invent. Math. 99 (1990), no. 1, 49–57. MR 1029390, DOI 10.1007/BF01234411
  • Matthias Flach, A finiteness theorem for the symmetric square of an elliptic curve, Invent. Math. 109 (1992), no. 2, 307–327. MR 1172693, DOI 10.1007/BF01232029
  • Jean Giraud, Cohomologie non abélienne, Die Grundlehren der mathematischen Wissenschaften, Band 179, Springer-Verlag, Berlin-New York, 1971 (French). MR 0344253, DOI 10.1007/978-3-662-62103-5
    • E. Golubeva and O. Fomenko, Application of spherical functions to a certain problem in the theory of quadratic forms, J. Soviet Math. 38 (1987), 2054-2060.
  • Benedict H. Gross, Kolyvagin’s work on modular elliptic curves, $L$-functions and arithmetic (Durham, 1989) London Math. Soc. Lecture Note Ser., vol. 153, Cambridge Univ. Press, Cambridge, 1991, pp. 235–256. MR 1110395, DOI 10.1017/CBO9780511526053.009
    • A. Grothendieck, Techniques de descente et théorèmes d’existence en géométrie algébrique, Sém. Bourbaki, vol. 12 (59/60), Benjamin Inc., New York and Amsterdam, 1966.
  • Alexander Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux $(SGA$ $2)$, North-Holland Publishing Co., Amsterdam; Masson & Cie, Editeur, Paris, 1968 (French). Augmenté d’un exposé par Michèle Raynaud; Séminaire de Géométrie Algébrique du Bois-Marie, 1962; Advanced Studies in Pure Mathematics, Vol. 2. MR 0476737
  • Groupes de monodromie en géométrie algébrique. I, Lecture Notes in Mathematics, Vol. 288, Springer-Verlag, Berlin-New York, 1972 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 I); Dirigé par A. Grothendieck. Avec la collaboration de M. Raynaud et D. S. Rim. MR 0354656
    • G. Harder, Über die Galoiskohomologie halbeinfacher Matrizengruppen, Math. Z. 90 (1965), 404-428; 92 (1966), 396-415.
  • Joe Harris, Algebraic geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. A first course. MR 1182558, DOI 10.1007/978-1-4757-2189-8
  • Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
    • H. Hasse, Über die Äquivalenz quadratischer Formen im Körper der rationalen Zahlen, J. Reine Angew. Math. 152 (1923), 205-224.
  • Henryk Iwaniec, Fourier coefficients of modular forms of half-integral weight, Invent. Math. 87 (1987), no. 2, 385–401. MR 870736, DOI 10.1007/BF01389423
    • C. Jordan, Mémoire sur l’equivalence des formes, J. École Polytech XLVIII (1880), 112-150.
  • Shoshichi Kobayashi and Takushiro Ochiai, Mappings into compact manifolds with negative first Chern class, J. Math. Soc. Japan 23 (1971), 137–148. MR 288316, DOI 10.2969/jmsj/02310137
  • V. A. Kolyvagin, Finiteness of $E(\textbf {Q})$ and SH$(E,\textbf {Q})$ for a subclass of Weil curves, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 3, 522–540, 670–671 (Russian); English transl., Math. USSR-Izv. 32 (1989), no. 3, 523–541. MR 954295, DOI 10.1070/IM1989v032n03ABEH000779
    • —, On the Mordell-Weil group and the Shafarevich-Tate group of Weil elliptic curves, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), 1154-1180.
  • V. A. Kolyvagin, Euler systems, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 435–483. MR 1106906
  • V. A. Kolyvagin and D. Yu. Logachëv, Finiteness of the Shafarevich-Tate group and the group of rational points for some modular abelian varieties, Algebra i Analiz 1 (1989), no. 5, 171–196 (Russian); English transl., Leningrad Math. J. 1 (1990), no. 5, 1229–1253. MR 1036843
  • M. Kneser, Lectures on Galois cohomology of classical groups, Tata Institute of Fundamental Research Lectures on Mathematics, No. 47, Tata Institute of Fundamental Research, Bombay, 1969. With an appendix by T. A. Springer; Notes by P. Jothilingam. MR 0340440
  • Donald Knutson, Algebraic spaces, Lecture Notes in Mathematics, Vol. 203, Springer-Verlag, Berlin-New York, 1971. MR 0302647, DOI 10.1007/BFb0059750
  • Barry Mazur, Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18 (1972), 183–266. MR 444670, DOI 10.1007/BF01389815
  • B. Mazur, Number theory as gadfly, Amer. Math. Monthly 98 (1991), no. 7, 593–610. MR 1121312, DOI 10.2307/2324924
    • Y. Matijasevic, Enumerable sets are diophantine, Dokl. Akad. Nauk SSSR 191 (1970); English transl., Soviet Math. Dokl. 11 (1970), 354-358. J. Milne, The conjectures of Birch and Swinnerton-Dyer for constant abelian varieties over function fields, Thesis, Harvard Univ., 1967. —, The Tate-Shafarevich group of a constant abelian variety, Invent. Math. 5 (1968), 63-84.
  • J. S. Milne, On a conjecture of Artin and Tate, Ann. of Math. (2) 102 (1975), no. 3, 517–533. MR 414558, DOI 10.2307/1971042
  • J. S. Milne, Arithmetic duality theorems, Perspectives in Mathematics, vol. 1, Academic Press, Inc., Boston, MA, 1986. MR 881804
  • L. J. Mordell, Diophantine equations, Pure and Applied Mathematics, Vol. 30, Academic Press, London-New York, 1969. MR 0249355
  • Jan Nekovář, Kolyvagin’s method for Chow groups of Kuga-Sato varieties, Invent. Math. 107 (1992), no. 1, 99–125. MR 1135466, DOI 10.1007/BF01231883
    • —, p-Adic methods in arithmetic, Center for Pure and Appl. Math., Univ. of California, Berkeley, preprint, 1992.
  • André Néron, Modèles minimaux des variétés abéliennes sur les corps locaux et globaux, Inst. Hautes Études Sci. Publ. Math. 21 (1964), 128 (French). MR 179172, DOI 10.1007/bf02684271
    • V. Nikulin, On factor groups of the automorphism groups of hyperbolic forms modulo subgroups generated by 2-reflections, Soviet Math. Dokl. 20 (1979), 1156-1158. —, Quotient-groups of the automorphisms of hyperbolic forms by subgroups generated by 2-reflections, algebro-geometric applications, Current Problems in Math., vol. 18, Akad. Nauk SSSR, Usesoyuz. Inst. Nauchn. Tekhn. Informatsii, Moscow, 1981, pp. 3-114; English transl., J. Soviet Math. 22 (1983), no. 4. Y. Nisnevich, Etale cohomology and arithmetic of semi-simple groups, Ph.D. thesis, Harvard Univ., 1982. —, On certain arithmetic and cohomological invariants of semi-simple groups, Johns Hopkins Univ., preprint, 1989.
  • Takashi Ono, Arithmetic of orthogonal groups, J. Math. Soc. Japan 7 (1955), 79–91. MR 69823, DOI 10.2969/jmsj/00710079
  • Bernadette Perrin-Riou, Travaux de Kolyvagin et Rubin, Astérisque 189-190 (1990), Exp. No. 717, 69–106 (French). Séminaire Bourbaki, Vol. 1989/90. MR 1099872
    • I. Piatetski-Shapiro and I. Shafarevich, A Torelli theorem for algebraic surfaces of type K3, Math. USSR-Izv. 5 (1971), 547-588.
  • V. P. Platonov, Arithmetic theory of algebraic groups, Uspekhi Mat. Nauk 37 (1982), no. 3(225), 3–54, 224 (Russian). MR 659426
  • C. P. Ramanujam, A note on automorphism groups of algebraic varieties, Math. Ann. 156 (1964), 25–33. MR 166198, DOI 10.1007/BF01359978
  • Karl Rubin, Tate-Shafarevich groups and $L$-functions of elliptic curves with complex multiplication, Invent. Math. 89 (1987), no. 3, 527–559. MR 903383, DOI 10.1007/BF01388984
  • Karl Rubin, On the main conjecture of Iwasawa theory for imaginary quadratic fields, Invent. Math. 93 (1988), no. 3, 701–713. MR 952288, DOI 10.1007/BF01410205
    • —, Kolyvagin’s system of Gauss sums, Arithmetic Algebraic Geometry (van der Geer, Oort, and Steenbrink, eds.), Progr. Math., vol. 89, Birkhäuser, Boston, MA, 1991, pp. 309-324.
  • Serge Lang, Cyclotomic fields I and II, 2nd ed., Graduate Texts in Mathematics, vol. 121, Springer-Verlag, New York, 1990. With an appendix by Karl Rubin. MR 1029028, DOI 10.1007/978-1-4612-0987-4
  • Karl Rubin, The “main conjectures” of Iwasawa theory for imaginary quadratic fields, Invent. Math. 103 (1991), no. 1, 25–68. MR 1079839, DOI 10.1007/BF01239508
  • Karl Rubin, The one-variable main conjecture for elliptic curves with complex multiplication, $L$-functions and arithmetic (Durham, 1989) London Math. Soc. Lecture Note Ser., vol. 153, Cambridge Univ. Press, Cambridge, 1991, pp. 353–371. MR 1110401, DOI 10.1017/CBO9780511526053.015
  • Karl Rubin, Stark units and Kolyvagin’s “Euler systems”, J. Reine Angew. Math. 425 (1992), 141–154. MR 1151317, DOI 10.1515/crll.1992.425.141
  • Karl Rubin, The work of Kolyvagin on the arithmetic of elliptic curves, Arithmetic of complex manifolds (Erlangen, 1988) Lecture Notes in Math., vol. 1399, Springer, Berlin, 1989, pp. 128–136. MR 1034261, DOI 10.1007/BFb0095973
  • Jean-Pierre Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier (Grenoble) 6 (1955/56), 1–42 (French). MR 82175, DOI 10.5802/aif.59
  • Jean-Pierre Serre, Cohomologie galoisienne, Lecture Notes in Mathematics, No. 5, Springer-Verlag, Berlin-New York, 1965 (French). With a contribution by Jean-Louis Verdier; Troisième édition, 1965. MR 0201444, DOI 10.1007/978-3-662-21576-0
  • Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237, DOI 10.1007/978-1-4757-5673-9
  • Carl Ludwig Siegel, Equivalence of quadratic forms, Amer. J. Math. 63 (1941), 658–680. MR 5506, DOI 10.2307/2371381
  • N. M. Stephens, The diophantine equation $X^{3}+Y^{3}=DZ^{3}$ and the conjectures of Birch and Swinnerton-Dyer, J. Reine Angew. Math. 231 (1968), 121–162. MR 229651, DOI 10.1515/crll.1968.231.121
  • John T. Tate, The arithmetic of elliptic curves, Invent. Math. 23 (1974), 179–206. MR 419359, DOI 10.1007/BF01389745
  • Francisco Thaine, On the ideal class groups of real abelian number fields, Ann. of Math. (2) 128 (1988), no. 1, 1–18. MR 951505, DOI 10.2307/1971460
    • A. Weil, Sur l’analogie entre les corps de nombres algébriques et les corps de fonctions algébriques (1939), Oeuvres Sci. I, Springer-Verlag, New York, 1980, pp. 236-240.
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  • © Copyright 1993 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 29 (1993), 14-50
  • MSC (2000): Primary 11G35; Secondary 11-02, 11D25, 11G05
  • DOI: https://doi.org/10.1090/S0273-0979-1993-00414-2
  • MathSciNet review: 1202293