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Siódmy problem milenijny : hipoteza Bircha i Swinnertona-Dyera

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  • Instytut Matematyki, Uniwersytet Warszawski, ul. Banacha 2, 02-097 Warszawa, Polska, bro@mimuw.edu.pl
Bibliografia
  • [1] C. Batut, K. Belabas, D. Bernardi, H. Cohen, M. Olivier, Pakiet PARI/GP. ftp://megrez.math.u-bordeaux.fr/pub/pari/.
  • [2] B. Birch, H. P. F. Swinnerton-Dyer, Notes on elliptic curves, I, J. Reine Angew. Math. 212 (1963) 7-25.
  • [3] —, Notes on elliptic curves, II, J. Reine Angew. Math., 218 (1965) 79-108.
  • [4] R. Bölling, Die Ordnung der Schafarewitsch-Tate Gruppe kann beliebig gross werden, Math. Nachr., 67 (1975) 157-179.
  • [5] Z. I. Borewicz, I. R. Szafarewicz, Teoria Liczb (ros.), Moskwa, Nauka, 1964.
  • [6] C. Breuil, B. Conrad, F. Diamond, R. Taylor, On the modularity of elliptic curves: Wild 3-adic exercises, J. Amer. Math. Soc., 14 (2001), no. 4, 843-939.
  • [7] A. Вrumer, O. McGuinness, The behaviour of the Mordell-Weil group of elliptic curves, Bull. Amer. Math. Soc., 23 (1990) 375-382.
  • [8] J. P. Buhler, B. H. Gross, D. B. Zagier, On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of rank 3, Math. Comp., 44 (1985), no. 170, 473-481.
  • [9] J. W. S. Cassels, Arithmetic on curves of genus 1, (VI). The Tate-Šafarevič group can be arbitrarily large, J. Reine Angew. Math., 214/5 (1964) 65-70.
  • [10] —, Arithmetic on curves of genus 1, (VIII). On conjectures of Birch and Swinnerton-Dyer, J. Reine Angew. Math., 217 (1965) 180-199.
  • [11] —, Diophantine equations with special reference to elliptic curves, J. London Math. Soc., 41 (1966) 193-291.
  • [12] J. E. Cremona, The analytic order of III for modular elliptic curves, J. Théor. Nombres, Bordeaux, 5 (1993), no. 1, 179-184.
  • [13] —, Algorithms for modular elliptic curves, 2. wyd., Cambridge Univ. Press, Cambridge, 1997 http://www.maths.nott.ac.uk/personal/jec/packages.html.
  • [14] J. E. Cremona, B. Mazur, Visualizing elements of the Shafarevich-Tate group, Exper. Math., 9 (2000), no. 1, 13-28.
  • [15] A. Dąbrowski, M. Wieczorek, Arithmetic on certain families of elliptic curves, Bull. Austral. Math. Soc., 61 (2000) 319-327.
  • [16] F. Keqin, Non-congruent numbers, odd graphs and the Birch-Swinnerton-Dyer conjecture, Acta Arith., 75 (1996), no. 1, 71-83.
  • [17] S. Fermigier, Une courbe elliptique définie sur Q de rang ≥ 22, Acta Arith., 82 (1997), no. 4, 359-363.
  • [18] D. Goldfeld, L. Szpiro, Bounds for the order of the Tate-Shafarevich group, Compositio Math., 97 (1995) 71-87.
  • [19] R. Greenberg, On the Birch and Swinnerton-Dyer conjecture, Invent. Math., 72 (1983) 241-265.
  • [20] В. H. Gross, Kolyvagin’s work on elliptic curves, in: L-functions and Arithmetic (J. Coates, M. J. Taylor, eds.), London Math. Soc. Lecture Note Series, vol. 153, Cambridge Univ. Press, Cambridge, 1991, 235-256.
  • [21] В. H. Gross, D. B. Zagier, Heegner points and derivatives of L-series, Invent. Math., 84 (1986), no. 2, 225-320.
  • [22] J. Kaczorowski, Czwarty problem milenijny: Hipoteza Riemanna, Wiadom. Mat., 38 (2002), 91-120.
  • [23] A. W. Knapp, Elliptic Curves, Math. Notes, vol. 40, Princeton University Press, Princeton, 1992.
  • [24] N. Кoblitz, Introduction to elliptic curves and modular forms, Graduate Texts in Mathematics, vol. 97, Springer-Verlag, New York, 1984.
  • [25] V. А. Коlуvagin, Finiteness of E (Q) and III (E, Q) for a subclass of Weil curves, (ros.), Izv. Akad. Nauk SSSR, Ser. Mat., 52 (1988), no. 3, 523-540.
  • [26] —, The Mordell-Weil and Shafarevich-Tate groups for Weil elliptic curves, (ros.), Izv. Akad. Nauk SSSR, Ser. Mat., 52 (1988), no. 6, 1154-1180.
  • [27] —, On the Mordell-Weil group and the Shafarevich-Tate group of modular elliptic curve, Proc. Intern. Congr. Math., Kyoto 1988. I (1990), 429-436.
  • [28] S. Lang, Algebra, wyd. 2, PWN, Warszawa, 1984.
  • [29] Delang Li, Ye Tian, On the Birch-Swinnerton-Dyer conjecture, Acta Math. Sinica, English Series, 16 (2000), no. 2, 229-236.
  • [30] Е. Lutz, Sur l’équation y2 = x3 – Ax - B dans les corps p-adiques, J. Reine Angew. Math., 177 (1937) 238-247.
  • [31] W. G. MсСallum, Kolyvagin’s work on Shafarevich-Tate groups, in: L-functions and Arithmetic (J. Coates, M. J. Taylor, eds), London Math. Soc. Lecture Note Series, vol. 153, Cambridge Univ. Press, Cambridge, 1991, 295-316.
  • [32] Yu. I. Manin, Cyclotomic fields and modular curves (ros.), Uspehi Mat. Nauk, 26 (1971), no. 6, (162), 7-71.
  • [33] В. Mazur, Modular curves and the Eisenstein ideal, Publ. Math. IHES, 47 (1977) 33-186.
  • [34] —, Rational isogenies of prime degree, Invent. Math., 44 (1978) 129-162.
  • [35] J. S. Milne, The Tate-Šafarevič group of a constant abelian variety, Invent. Math., 6 (1968) 91-105.
  • [36] T. Nagell, Solution de quelques problèmes dans la théorie arithmétique des cubiques planes du premier genre, Wid. Akad. Skrifter, Oslo, 1 (1935), no. 1, 1-25.
  • [37] A. Nitaj, Invariants des courbes de Frey-Hellegouarch et grandes groupes de Tate-Shafarevich, Acta Arith., 93 (2000) 303-327.
  • [38] K. A. Ribet, On modular representations of Gal (Ǭ/Q) arising from modular forms, Invent. Math., 100 (1990), 431-476.
  • [39] H. E. Rose, On some elliptic curves with large Sha, Exper. Math., 9 (2000), no. 1, 85-89.
  • [40] K. Rubin, Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer, Invent. Math., 64 (1981), 455-470.
  • [41] —, Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication, Invent. Math., 89 (1987), no. 3, 527-559.
  • [42] —, The „main conjecture” of Iwasawa theory for imaginary quadratic fields, Invent. Math., 103 (1991), 25-68.
  • [43] —, Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer, in: J. Coates, R. Greenberg, K. A. Ribet, K. Rubin, Arithmetic Theory of Elliptic Curves (Cetraro, Italy 1997), Lecture Notes in Math. 1716 (C. Viola, eds.), Springer-Verlag, Berlin, 1999, 167-234.
  • [44] K. Rubin, A. Silverberg, Ranks of elliptic curves, Bull. Amer. Math. Soc., 39 (2002) 455-474.
  • [45] J.-P. Serre, Lectures on the Mordell-Weil theorem, Friedr. Vieweg & Sohn, Braunschweig, 1989.
  • [46] A. Silverberg, Open questions in arithmetic algebraic geometry, in: Arithmetic Algebraic Geometry (B. Conrad, K. Rubin, eds.), IAS/Park City Mathematics Series, vol. 9, Providence, RI, AM S, 2001, 83-142.
  • [47] J. H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, Vol. 108, Springer-Verlag, New York, 1986.
  • [48] J. H. Silverman, Advanced topics in the arithmetic of elliptic curves, Springer-Verlag, New York, 1994.
  • [49] J. H. Silverman, J. Tate, Rational points on elliptic curves, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992.
  • [50] N. M. Stephens, The diophantine equation X3 + Y3 = DZ3 and the conjectures of Birch and Swinnerton-Dyer, J. Reine Angew. Math., 231 (1968) 121-162.
  • [51] J. Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analogue, Séminaire Bourbaki (1965/66), no. 306, 415-440.
  • [52] —, The arithmetic of elliptic curves, Invent. Math., 23 (1974) 179-206.
  • [53] R. Taylor, A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. Math., 141 (1995), no. 3, 553-572.
  • [54] J. Tunnell, A classical Diophantine problem and modular form of weight 3/2, Invent. Math., 72 (1983), 323-334.
  • [55] В. M. M. de Wegeг, А + В = С and big III's, Quart. J. Math. Oxford, 49 (1998) 105-128.
  • [56] A. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. Math., 141 (1995), no. 3, 443-551.
  • [57] —, The Birch and Swinnerton-Dyer conjecture http://www.claymath.org/prizeproblems/birchsd.htm.
  • [58] S. Zhang, A note on Sha of some elliptic curves, Adv. Math. (China), 26 (1997), no. 6, 551-555.
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Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-BUS2-0003-0021
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