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The square-free sieve and the rank of elliptic curves


Authors: F. Gouvêa and B. Mazur
Journal: J. Amer. Math. Soc. 4 (1991), 1-23
MSC: Primary 11G40; Secondary 11G05, 11N35
DOI: https://doi.org/10.1090/S0894-0347-1991-1080648-7
MathSciNet review: 1080648
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  • [BM] A. Brumer and O. McGuinness, The behavior of the Mordell-Weil group of elliptic curves, Bull. Amer. Math. Soc. (NS) 23 (1990), 375-382. MR 1044170 (91b:11076)
  • [BFH] D. Bump, S. Friedberg, and J. Hoffstein, Nonvanishing theorems for $ L$-functions of modular forms and their derivatives, Invent. Math. 102 (1990), 543-618. MR 1074487 (92a:11058)
  • [E] O. Erdös, The sum $ \Sigma d\left\{ {f(k)} \right\}$, J. London Math. Soc. 27 (1952), 7-15. MR 0044565 (13:438f)
  • [Go] D. Goldfeld, Conjectures on elliptic curves over quadratic fields, Number Theory (Proc. Conf. in Carbondale, 1979) (M. B. Nathanson, ed.), Lecture Notes in Math., vol. 751, Springer-Verlag, Berlin, New York, and Heidelberg, 1979, pp. 108-118. MR 564926 (81i:12014)
  • [Gou] F. Gouvêa, The square-free sieve over number fields, preprint, 1990. MR 1200814 (93m:11045)
  • [Gr] G. Greaves, Power-free values of binary forms, preprint, Cardiff, 1990. MR 1150469 (92m:11098)
  • [H] C. Hooley, Applications of sieve methods to the theory of numbers, Cambridge Univ. Press 70 (1976). MR 0404173 (53:7976)
  • [HW] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., Oxford Univ. Press, 1979. MR 568909 (81i:10002)
  • [I] J.-I. Igusa, Lectures on forms of higher degree, Tata Inst. of Fund. Research, 1978. MR 546292 (80m:10020)
  • [K1] V. A. Kolyvagin, Euler systems, The Grothendieck Festschrift, vol. 11, a collection of articles written in honor of the 60th birthday of Alexander Grothendieck (P. Cartier et al., eds.), Progr. Math. 87 (1990), 435-483. MR 1106906 (92g:11109)
  • [K2] -, Finiteness of $ E(Q)$ and $ \mathcyr{SH}_{E/Q}$ for a subclass of Weil curves, Math USSR-Izv. 32 (3) (1989), 523-541. (English Transl.) MR 954295 (89m:11056)
  • [Ko] N. Koblitz, Introduction to elliptic curves and modular forms, Graduate Texts in Math., vol. 97, Springer-Verlag, 1984. MR 766911 (86c:11040)
  • [M] B. Mazur, Rational isogenies of prime degree, Invent. Math. 44 (1978), 129-162. MR 482230 (80h:14022)
  • [MM] M. R. Murty and K. V. Murty, Mean values of derivatives of modular $ L$-series, preprint, 1988.
  • [Se] J.-P. Serre, Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259-331. MR 0387283 (52:8126)
  • [Sh] G. Shimura, On modular forms of half-integral weight, Ann. of Math. 97 (1973), 440-481. MR 0332663 (48:10989)
  • [III] I. Shafarevitch, Algebraic number fields, Proc. Internat. Congress, Stockholm, 1962, pp. 163-176; Amer. Math. Soc. Transl. Ser. 2, vol. 31, pp. 25-39.
  • [T] J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math. 72 (1983), 323-334. MR 700775 (85d:11046)
  • [W] J.-L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl. 60 (1981), 375-484. MR 646366 (83h:10061)
  • [ZK] D. Zagier and G. Kramarz, Numerical investigations related to the $ L$-series of certain elliptic curves, J. Indian Math. Soc. 52 (1987), 51-60. MR 989230 (90d:11072)

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DOI: https://doi.org/10.1090/S0894-0347-1991-1080648-7
Article copyright: © Copyright 1991 American Mathematical Society

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