Glasnik Matematicki, Vol. 50, No. 2 (2015), 333-348.

ON THE NUMBER OF N-ISOGENIES OF ELLIPTIC CURVES OVER NUMBER FIELDS

Miljen Mikić and Filip Najman

Kumičićeva 20, 51000 Rijeka, Croatia
e-mail: miljen.mikic@gmail.com

Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10000 Zagreb, Croatia
e-mail: fnajman@math.hr


Abstract.   We find the number of elliptic curves with a cyclic isogeny of degree n over various number fields by studying the modular curves X0(n). We show that for n=14,15,20,21,49 there exist infinitely many quartic fields K such that # Y0(n)(Q)≠ # Y0(n)(K)< ∞ . In the case n=27 we prove that there are infinitely many sextic fields such that # Y0(n)(Q)≠ # Y0(n)(K)< ∞.

2010 Mathematics Subject Classification.   11G05, 11G18, 11R16, 14H52.

Key words and phrases.   Elliptic curves, Mordell-Weil group, isogenies.


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DOI: 10.3336/gm.50.2.05


References:

  1. I. Connell, Elliptic Curve Handbook, http://www.math.mcgill.ca/connell/public/ECH1/.

  2. S. Dasgupta and J. Voight, Heegner points and Sylvester's conjecture, Arithmetic Geometry, Clay Math. Proc., vol. 8, Amer. Math. Soc., Providence, 2009, 91-102.
    MathSciNet    

  3. M. A. Kenku, The modular curves X0(65) and X0(91) and rational isogeny, Math. Proc. Cambridge Philos. Soc. 87 (1980), 15-20.
    MathSciNet     CrossRef

  4. M. A. Kenku, The modular curve X0(169) and rational isogeny, J. London Math. Soc. (2) 22 (1980), 239-244.
    MathSciNet     CrossRef

  5. M. A. Kenku, On the modular curves X0(125), X1(25) and X1(49), J. London Math. Soc. (2) 23 (1981), 415-427.
    MathSciNet     CrossRef

  6. B. Mazur, Rational isogenies of prime degree, Invent. Math. 44 (1978), 129-162.
    MathSciNet     CrossRef

  7. F. Najman, Exceptional elliptic curves over quartic fields, Int. J. Number Theory, 8 (2012), 1231-1246.
    MathSciNet     CrossRef

  8. F. Najman, On the number of elliptic curves with prescribed isogeny or torsion group over number fields of prime degree, Glasg. Math. J. 57 (2015), 465-473.
    MathSciNet     CrossRef

  9. E. S. Selmer, The diophantine equation ax3+by3+cz3=0, Acta Math. 85 (1951), 203-362.
    MathSciNet     CrossRef

  10. J. H. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, New York, 1986.
    MathSciNet    

  11. Y. Yang, Defining equations of modular curves, Adv. Math. 204 (2006), 481-508.
    MathSciNet     CrossRef

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