Worked solution: Wiles's modularity proof via the Taylor–Wiles method (1994)

An elliptic curve $E$ over $\mathbb{Q}$ is the set of rational solutions $(x, y)$ to a cubic equation $y^2 = x^3 + a_2 x^2 + a_4 x + a_6$ with $a_2, a_4, a_6 \in \mathbb{Q}$ whose right-hand cubic has three distinct complex roots, together with a distinguished point at infinity $\mathcal{O}$ that acts as the identity element for the chord-and-tangent addition law (Qiu et al. 2025, Section 3.3.1, Definition 3.1).

Suppose for contradiction that $a^p + b^p = c^p$ is a counterexample to FLT with prime $p \ge 5$ and nonzero pairwise coprime integers $a, b, c$. Following earlier constructions of Yves Hellegouarch (1969), Gerhard Frey (1985/1986) attached to this hypothetical solution the elliptic curve $E_{a,b,c}: y^2 = x(x - a^p)(x + b^p)$ (Wiles 1995, Introduction, p. 443 and p. 448; Qiu et al. 2025, Section 3.1.2). Because $a, b, c$ are nonzero and $a^p + b^p = c^p \ne 0$, the three roots $r_1 = 0$, $r_2 = a^p$, and $r_3 = -b^p$ are distinct, so $E_{a,b,c}$ is a genuine non-singular elliptic curve over $\mathbb{Q}$.

Frey's insight was that the equation $a^p + b^p = c^p$ is now encoded inside the geometry of $E_{a,b,c}$: the differences between the three roots are $a^p - 0 = a^p$, $0 - (-b^p) = b^p$, and $a^p - (-b^p) = a^p + b^p = c^p$, every single one of which is a perfect $p$-th power. The next step measures how unnatural that is.