1.2. Modular forms 7 Figure 3. Pierre de Fermat (1601-1665). would have some unexpected properties that would contradict a well- known conjecture that Taniyama, Shimura and Weil had formulated in the 1950’s. Their conjecture spelled out a strong connection be- tween elliptic curves and modular forms, which we will describe in Section 5.4. Ken Ribet proved that, indeed, such a curve would con- tradict the Taniyama-Shimura-Weil (TSW) conjecture. Finally, An- drew Wiles was able to prove the TSW conjecture in a special case that would cover the hypothetical curve E. Therefore, E cannot exist and the triple (a, b, c) cannot exist, either. The Taniyama-Shimura-Weil conjecture (Conjecture 5.4.5), i.e., the modularity theorem 5.4.6, was fully proved by Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor in their article [BCDT01]. 1.2. Modular forms Let C be the complex plane and let H be the upper half of the complex plane, i.e., H = {a+bi : a, b ∈ R, b 0}. A modular form is a function

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