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Title

On a characterization of Shimura's elliptic curve over ℚ(√37)

Authors 1, 2

Affiliations

  1. Department of Mathematics, The Johns Hopkins University, Baltimore, Maryland 21218, U.S.A.
  2. Department of Mathematical Sciences, Yamagata University, Yamagata, 990 Japan

Bibliography

  1. B. J. Birch and W. Kuyk (eds.), Modular Functions of One Variable IV, Lecture Notes in Math. 476, Springer, 1975.
  2. W. Casselman, On abelian varieties with many endomorphisms and a conjecture of Shimura's, Invent. Math. 12 (1971), 225-236.
  3. J. E. Cremona, Modular symbols for Γ₁(N) and elliptic curves with everywhere good reduction, Math. Proc. Cambridge Philos. Soc. 111 (1992), 199-218.
  4. J. E. Cremona, Algorithms for Modular Elliptic Curves, Cambridge University Press, 1992.
  5. P. Deligne et M. Rapoport, Les schémas de modules de courbes elliptiques, in: Modular Functions of One Variable III, Lecture Notes in Math. 349, Springer, 1973, 143-316.
  6. O. Hemer, On the Diophantine equation y²+k=x³, doctoral dissertation, Uppsala, 1952.
  7. F. Klein und R. Fricke, Vorlesungen über die Theorie der elliptischen Modulfunktionen II, Teubner, 1892.
  8. A. P. Ogg, Diophantine equations and modular forms, Bull. Amer. Math. Soc. 81 (1975), 14-27.
  9. A. Pethö and B. M. M. de Weger, Products of prime powers in binary recurrence sequences. Part I, The hyperbolic case, with an application to the generalized Ramanujan-Nagell equation, Math. Comp. 47 (1986), 713-727.
  10. R. G. E. Pinch, Elliptic curves over number fields, doctoral dissertation, Oxford University, 1982.
  11. K. A. Ribet, Endomorphisms of semi-stable abelian varieties over number fields, Ann. of Math. 101 (1975), 555-562.
  12. B. Setzer, Elliptic curves with good reduction everywhere over quadratic fields and having rational j-invariant, Illinois J. Math. 25 (1981), 233-245.
  13. G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Publ. Math. Soc. Japan 11, Iwanami Shoten and Princeton University Press, 1971.
  14. G. Shimura, Class fields over real quadratic fields and Hecke operators, Ann. of Math. 95 (1972), 130-190.
  15. G. Shimura, On the factor of the Jacobian variety of a modular function field, J. Math. Soc. Japan 25 (1973), 523-544.
  16. K. Shiota, On the explicit models of Shimura's elliptic curves, J. Math. Soc. Japan 38 (1986), 649-659.
  17. J. H. Silverman, The Arithmetic of Elliptic Curves, Grad. Texts in Math. 106, Springer, 1986.
  18. R. P. Steiner, On Mordell's equation y²-k=x³: A problem of Stolarsky, Math. Comp. 46 (1986), 703-714.
  19. N. Tzanakis and B. M. M. de Weger, On the practical solution of the Thue equation, J. Number Theory 31 (1989), 99-132.
  20. N. Tzanakis and B. M. M. de Weger, How to explicitly solve a Thue-Mahler equation, Comp. Math. 84 (1992), 223-288; Corrections 89 (1993), 241-242.
  21. M. J. Vélu, Isogénies entre courbes elliptiques, C. R. Acad. Sci. Paris Sér. A 273 (1971), 238-241.
  22. M. Waldschmidt, A lower bound for linear forms in logarithms, Acta Arith. 37 (1980), 257-283.
Acta Arithmetica
1996/77/2
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Pages:
157-171
Main language of publication
English
Received
1995-06-23
Accepted
1996-02-25
Published
1996
Exact and natural sciences