Abstract
Thanks to the work of Wiles [Wi], completed by Taylor-Wiles [TW] and extended by Diamond [Di], we now know that all elliptic curves over the rationals (having good or semi-stable reduction at 3 and 5) are modular. This breakthrough has far-reaching consequences for the arithmetic of elliptic curves. As Mazur wrote in [Ma3], “It has been abundantly clear for years that one has a much more tenacious hold on the arithmetic of an elliptic curve E/Q if one supposes that it is […] parametrized [by a modular curve].” This expository article explores some of the implications of Wiles’ theorem for the theory of elliptic curves, with particular emphasis on the Birch and Swinnerton-Dyer conjecture, now the main outstanding problem in the field.
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Darmon, H. (1997). Wiles’ Theorem and the Arithmetic of 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_21
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DOI: https://doi.org/10.1007/978-1-4612-1974-3_21
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