Geodesic
ON THE NUMBER OF ELLIPTIC CURVES WITH PRESCRIBED ISOGENY OR TORSION GROUP OVER NUMBER FIELDS OF PRIME DEGREE
Glasgow mathematical journal, Tome 57 (2015) no. 2, pp. 465-473

Voir la notice de l'article provenant de la source Cambridge University Press

Let p be a prime and K a number field of degree p. We determine the finiteness of the number of elliptic curves, up to K-isomorphism, having a prescribed property, where this property is either that the curve contains a fixed torsion group as a subgroup or that it has a cyclic isogeny of prescribed degree. 
NAJMAN, FILIP. ON THE NUMBER OF ELLIPTIC CURVES WITH PRESCRIBED ISOGENY OR TORSION GROUP OVER NUMBER FIELDS OF PRIME DEGREE. Glasgow mathematical journal, Tome 57 (2015) no. 2, pp. 465-473. doi: 10.1017/S0017089514000421
@article{10_1017_S0017089514000421,
     author = {NAJMAN, FILIP},
     title = {ON {THE} {NUMBER} {OF} {ELLIPTIC} {CURVES} {WITH} {PRESCRIBED} {ISOGENY} {OR} {TORSION} {GROUP} {OVER} {NUMBER} {FIELDS} {OF} {PRIME} {DEGREE}},
     journal = {Glasgow mathematical journal},
     pages = {465--473},
     year = {2015},
     volume = {57},
     number = {2},
     doi = {10.1017/S0017089514000421},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089514000421/}
}
TY  - JOUR
AU  - NAJMAN, FILIP
TI  - ON THE NUMBER OF ELLIPTIC CURVES WITH PRESCRIBED ISOGENY OR TORSION GROUP OVER NUMBER FIELDS OF PRIME DEGREE
JO  - Glasgow mathematical journal
PY  - 2015
SP  - 465
EP  - 473
VL  - 57
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089514000421/
DO  - 10.1017/S0017089514000421
ID  - 10_1017_S0017089514000421
ER  - 
%0 Journal Article
%A NAJMAN, FILIP
%T ON THE NUMBER OF ELLIPTIC CURVES WITH PRESCRIBED ISOGENY OR TORSION GROUP OVER NUMBER FIELDS OF PRIME DEGREE
%J Glasgow mathematical journal
%D 2015
%P 465-473
%V 57
%N 2
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089514000421/
%R 10.1017/S0017089514000421
%F 10_1017_S0017089514000421

[1] 1.Cojocaru, A. C., On the surjectivity of the Galois representations associated to non-CM elliptic curves, With an appendix by E. Kani Can. Math. Bull. 48 (2005), 16–31. Google Scholar

[2] 2.Cremona, J., Algorithms for modular elliptic curves (Cambridge University Press, Cambridge, 1992). Google Scholar

[3] 3.Diamond, F. and Im, J., Modular forms and modular curves, in Seminar on Fermat's last theorem (Kumar, Murty V., Editor) (AMS, Providence, 1997), 39–133. Google Scholar

[4] 4.Dieulefait, L., González-Jiménez, E. and Jiménez, J.Urroz, On fields of definition of torsion points of elliptic curves with complex multiplication Proc. Am. Math. Soc. 139 (2011), 1961–1969. Google Scholar

[5] 5.Faltings, G., Endlichkeitssätze für abelsche Varietäten über Zahlkörpern Invent. Math. 73 (1983), 349–366. Google Scholar | DOI

[6] 6.Jeon, D., Kim, C. H. and Park, E., On the torsion of elliptic curves over quartic number fields, J. London Math. Soc. 74 (2) (2006), 1–12. Google Scholar

[7] 7.Kamienny, S. and Najman, F., Torsion groups of elliptic curves over quadratic fields Acta Arith. 152 (2012), 291–305. Google Scholar

[8] 8.Kenku, M. A. and Momose, F., Torsion points on elliptic curves defined over quadratic fields Nagoya Math. J. 109 (1988), 125–149. Google Scholar | DOI

[9] 9.Mazur, B., Modular curves and the Eisenstein ideal Publ. Math. IHES 47 (1978), 33–186. Google Scholar

[10] 10.Mazur, B., Rational isogenies of prime degree Invent. Math. 44 (1978), 129–162. Google Scholar | DOI

[11] 11.Mazur, B. and Rubin, K., Ranks of twists of elliptic curves and Hilbert's tenth problem Invent. Math. 181 (2010), 541–575. Google Scholar

[12] 12.Najman, F., Torsion of elliptic curves over cubic fields J. Number Theory 132 (2012), 26–36. Google Scholar

[13] 13.Najman, F., Exceptional elliptic curves over quartic fields, Int. J. Number Theory, 8 (2012), 1231–1246. Google Scholar

[14] 14.Serre, J. P., Propriétés galoisiennes des points d'ordre fini des courbes elliptiques Invent. Math. 15 (1972) 259–311. Google Scholar

[15] 15.Stein, W. A.et al., Sage Mathematics Software (Version 4.7.1) (The Sage Development Team, 2011), http://wiki.sagemath.org/Publications_using_SAGE, accessed 28 October 2014. Google Scholar

[16] 16.Sutherland, A., Constructing elliptic curves with prescribed torsion over finite fields Math. Comput. 81 (2012), 1131–1147. Google Scholar | DOI

[17] 17.Washington, L., Elliptic curves. Number theory and cryptography. Discrete Mathematics and its Applications (Chapman & Hall/CRC, Boca Raton, FL, 2003). Google Scholar

[18] 18.Yang, Y., Defining equations of modular curves Adv. Math. 204 (2006), 481–508. Google Scholar

Cité par Sources :