Abstract
For M a positive integer, let X 0 (M) Q be the modular curve over Q classifying elliptic curves with a given cyclic subgroup of order M and let J 0(M)Q be the jacobian of X 0(M)Q. An elliptic curve E over Q is said to be modular if it is an isogeny factor (isogenics over Q) of some J 0(M)Q; the smallest M for which this happens is then called the level of E. The Shimura-Taniyama conjecture states that every elliptic curve E over Q is modular, and that the level of E equals the conductor of E. A modular elliptic curve of level M is called strong if there exists a closed immersion E ↪ J 0(M)Q. It follows from the multiplicity one principle for modular forms that such an immersion is unique up to sign.
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References
B.J. Birch and H.P.F. Swinnerton-Dyer. Elliptic curves and modular functions. Modular functions of one variable IV, 2–32. Springer Lecture Notes in Mathematics 476 (1975).
H. Carayol. Sur les représentations l-adiques associées aux formes modulaires de Hubert. Ann. scient. Éc. Norm. Sup., série 4, t. 19, 409–468 (1986).
P. Deligne and M. Rapoport. Les schémas de modules des courbes elliptiques. Modular functions of one variable II, 143–316. Springer Lecture Notes in Mathematics 349.
S.J. Edixhoven. Minimal resolution and stable reduction of X 0 (N). Preprint Utrecht November 1986, accepted for publication in the Annales de l’Institut Fourier.
S.J. Edixhoven. Stable models of modular curves and applications. Thesis Utrecht June 1989, to be published.
N.M. Katz and B. Mazur. Arithmetic moduli of elliptic curves. Annals of Mathematics Studies 108, Princeton University Press (1985).
B. Mazur. Rational isogenics of prime degree. Invent. Math. 44, 129–162 (1978).
B. Mazur. Courbes elliptiques et symboles modulaires. Séminaire Bourbaki juin 1972. Springer Lecture Notes in Mathematics 317.
B. Mazur and H.P.F. Swinnerton-Dyer. Arithmetic of Weil curves. Invent. Math. 25, 1–16 (1974).
J.-F. Mestre and J. Oesterlé. Letter to B. Gross, May 1985.
F. Oort. Commutative group schemes. Springer Lecture Notes in Mathematics 15 (1966).
M. Raynaud. Spécialisation du fondeur de Picard. Publications Mathématiques de l’I.H.E.S. 38 (1970).
M. Raynaud. Schémas en groupes de type (p,…,p). Bull. Soc. Math. France 102, 241–280 (1974).
G. Stevens. Stickelberger elements and modular parametrizations of elliptic curves. To be published.
J. Tate. Algorithm for determining the type of a singular fibre in an elliptic pencil. In Modular Functions of One Variable IV, 33–52. Springer Lecture Notes in Mathematics 476 (1975).
M. Kenku. On the number of Q-isomorphism classes of elliptic curves in each Q-isogeny class. J. of Number Theory 15, 199–202 (1982).
D. Zagier. Modular parametrizations of elliptic curves. Canad. Math. Bull. 28(3), 372–384 (1985).
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Edixhoven, B. (1991). On the Manin constants of modular elliptic curves. In: van der Geer, G., Oort, F., Steenbrink, J. (eds) Arithmetic Algebraic Geometry. Progress in Mathematics, vol 89. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0457-2_3
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DOI: https://doi.org/10.1007/978-1-4612-0457-2_3
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