Abstract
The main purpose of this chapter is to show how arithmetical properties of elliptic curves E defined over global fields K and corresponding Galois representations are often related to interesting diophantine questions, amongst which the most prominent is without doubt Fermat’s Last Theorem, which has now become Wiles’ theorem.
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References
D. Abramovich, Subvarieties of Abelian Varieties and Jacobians of Curves, Thesis Harvard 1991.
J.B. Bost, Périodes et isogénies des variétés abeliennes sur les corps de nombres, Sém. Bourbaki 795 (1994–95).
H. Darmon, Serre’s conjecture, Sem. on Fermat’s Last Theorem, CMS Conf. Proc. AMS, to appear.
P. Deligne, Démonstration des conjectures de Tate et de Shafarevich (d’après G. Faltings),Séminaire Bourbaki 616, Astérisque 121–122 (1985), 25–41.
N.D. Elkies, ABC implies Mordell, Int. Math. Res. Not. 7, 99–109.
G. Faltings, The general case of S. Lang’s conjecture, Princeton University, 1992.
G. Frey, Links between solutions of A - B = C and elliptic curves, Number Theory (Ulm 1987), Lecture Notes in Math. 1380, Springer, 1989, pp. 31–62.
G. Frey, Curves with infinitely many points of fixed degree, Israel J. of Math. 85, 79–83.
G. Frey, On elliptic curves with isomorphic torsion structures and correspond-ing curves of genus 2, Conference on Elliptic Curves and Modular Forms (Hong Kong), Intern. Press, 1995, pp. 79–98.
R. Hartshorne, Generalized divisors and Gorenstein curves and a theorem of Noether, Journ. Math. Kyoto 26 (1986), 375–385.
Y. HellegouarchPoints d’ordre 2ph sur les courbes elliptiques, Acta Arith. 26(1975), 253–263.
E. Kani, Letter to Mazur, Sept. 1995.
E. Kani and W. Schanz, Diagonal quotient surfaces (to appear).
L. Mai and M.R. MurtyThe Phragmén-Lindelöf theorem and modular elliptic curves, Contemp. Math. 166 (1994), 335–340.
B. Mazur, Modular curves and the Eisenstein ideal, Publ. Math. IHES 47 (1977), 33–186.
B. Mazur, Rational isogenies of prime degree, Invent. Math. 44 (1978), 129–162.
A.N. ParshinThe Bogomolov-Miyaoka-Yau inequality for arithmetical sur-faces and its applications, Sém. Théorie Nombres Paris 1986/87, Progr. Math. 75 (1989), 299–312.
K. RibetOn modular representations of G(Q I Q) arising from modular forms, Journ. Math. 100 (1990), 431–476.
K. Ribet, On the equation a 9 .2 ~ +b’+c 0 = 0, Acta Arithmetica (to appear).
J.P. Serre, Sur les représentations modulaires de degré 2 de G(Q Q)Duke Math. J. 54 (1987), 179–230.
J.H. Silverman, The arithmetic of elliptic curves, Springer 1986
A. Wiles, Modular elliptic curves and Fermat’s Last Theorem, Ann. of Math.142 (1995), 443–551.
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Frey, G. (1997). On Ternary Equations of Fermat Type and Relations with Elliptic Curves. In: Cornell, G., Silverman, J.H., Stevens, G. (eds) Modular Forms and Fermat’s Last Theorem. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1974-3_20
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DOI: https://doi.org/10.1007/978-1-4612-1974-3_20
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