Chapter

Modular Curves and Abelian Varieties

Volume 224 of the series Progress in Mathematics pp 241-261

Abelian Varieties over Q and Modular Forms

  • Kenneth A. RibetAffiliated withUC Mathematics Department

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Abstract

Let C be an elliptic curve over Q. Let N be the conductor of C. The Taniyama conjecture asserts that there is a non-constant map of algebraic curves X 0 (N) — C which is defined over Q. Here, X o (N) is the standard modular curve associated with the problem of classifying elliptic curves E together with cyclic subgroups of E having order N.