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
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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