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