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

Masanari Kida

Acta Arithmetica (1996)

  • Volume: 77, Issue: 2, page 157-171
  • ISSN: 0065-1036

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Masanari Kida. "On a characterization of Shimura's elliptic curve over ℚ(√37)." Acta Arithmetica 77.2 (1996): 157-171. <http://eudml.org/doc/206915>.

@article{MasanariKida1996,
author = {Masanari Kida},
journal = {Acta Arithmetica},
keywords = {elliptic curves over real quadratic fields; cubic diophantine equations; Nebentypus cusp forms; everywhere good reduction},
language = {eng},
number = {2},
pages = {157-171},
title = {On a characterization of Shimura's elliptic curve over ℚ(√37)},
url = {http://eudml.org/doc/206915},
volume = {77},
year = {1996},
}

TY - JOUR
AU - Masanari Kida
TI - On a characterization of Shimura's elliptic curve over ℚ(√37)
JO - Acta Arithmetica
PY - 1996
VL - 77
IS - 2
SP - 157
EP - 171
LA - eng
KW - elliptic curves over real quadratic fields; cubic diophantine equations; Nebentypus cusp forms; everywhere good reduction
UR - http://eudml.org/doc/206915
ER -

References

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  2. [2] W. Casselman, On abelian varieties with many endomorphisms and a conjecture of Shimura's, Invent. Math. 12 (1971), 225-236. Zbl0213.47303
  3. [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. Zbl0752.11022
  4. [4] J. E. Cremona, Algorithms for Modular Elliptic Curves, Cambridge University Press, 1992. Zbl0758.14042
  5. [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. Zbl0281.14010
  6. [6] O. Hemer, On the Diophantine equation y²+k=x³, doctoral dissertation, Uppsala, 1952. Zbl0049.31004
  7. [7] F. Klein und R. Fricke, Vorlesungen über die Theorie der elliptischen Modulfunktionen II, Teubner, 1892. 
  8. [8] A. P. Ogg, Diophantine equations and modular forms, Bull. Amer. Math. Soc. 81 (1975), 14-27. Zbl0316.14012
  9. [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. Zbl0623.10011
  10. [10] R. G. E. Pinch, Elliptic curves over number fields, doctoral dissertation, Oxford University, 1982. 
  11. [11] K. A. Ribet, Endomorphisms of semi-stable abelian varieties over number fields, Ann. of Math. 101 (1975), 555-562. Zbl0305.14016
  12. [12] B. Setzer, Elliptic curves with good reduction everywhere over quadratic fields and having rational j-invariant, Illinois J. Math. 25 (1981), 233-245. Zbl0471.14019
  13. [13] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Publ. Math. Soc. Japan 11, Iwanami Shoten and Princeton University Press, 1971. Zbl0221.10029
  14. [14] G. Shimura, Class fields over real quadratic fields and Hecke operators, Ann. of Math. 95 (1972), 130-190. Zbl0255.10032
  15. [15] G. Shimura, On the factor of the Jacobian variety of a modular function field, J. Math. Soc. Japan 25 (1973), 523-544. Zbl0266.14017
  16. [16] K. Shiota, On the explicit models of Shimura's elliptic curves, J. Math. Soc. Japan 38 (1986), 649-659. Zbl0608.14023
  17. [17] J. H. Silverman, The Arithmetic of Elliptic Curves, Grad. Texts in Math. 106, Springer, 1986. 
  18. [18] R. P. Steiner, On Mordell's equation y²-k=x³: A problem of Stolarsky, Math. Comp. 46 (1986), 703-714. Zbl0601.10011
  19. [19] N. Tzanakis and B. M. M. de Weger, On the practical solution of the Thue equation, J. Number Theory 31 (1989), 99-132. Zbl0657.10014
  20. [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. Zbl0773.11023
  21. [21] M. J. Vélu, Isogénies entre courbes elliptiques, C. R. Acad. Sci. Paris Sér. A 273 (1971), 238-241. Zbl0225.14014
  22. [22] M. Waldschmidt, A lower bound for linear forms in logarithms, Acta Arith. 37 (1980), 257-283. Zbl0357.10017

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