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

For an elliptic curve $E/\mathbb{Q}$ and a prime $q \nmid \Delta$ of good reduction, let $|E(\mathbb{F}_q)|$ be the number of solutions $(x, y) \in \mathbb{F}_q^2$ plus the point at infinity $\mathcal{O}$, and define the trace of Frobenius $a_q(E) = q + 1 - |E(\mathbb{F}_q)|$ (Qiu et al. 2025, Section 3.2.1). These integers package into the Hasse–Weil $L$-series $L(E, s) = \sum_{n=1}^{\infty} a_n(E) n^{-s}$.

On the analytic side, a cusp form $f$ of weight $k = 2$ and level $N$ for the congruence subgroup $\Gamma_0(N) = \{\binom{a\ b}{c\ d} \in \operatorname{SL}_2(\mathbb{Z}) : N \mid c\}$ is a holomorphic function on the upper half-plane satisfying $f((a\tau+b)/(c\tau+d)) = (c\tau+d)^2 f(\tau)$ for all matrices in $\Gamma_0(N)$ and vanishing at the cusps (Wiles 1995, p. 445; Qiu et al. 2025, Section 3.2.2). When $f$ is an eigenform for all Hecke operators $T_n$, say $T_n f = c(n, f) f$, its Fourier expansion $f(\tau) = \sum_{n=1}^{\infty} c(n, f) e^{2\pi i n \tau}$ carries arithmetic eigenvalues $c(n, f)$.

By the theorem of Eichler and Shimura (Wiles 1995, p. 444–445, Theorem 0.1), an elliptic curve $E/\mathbb{Q}$ of conductor $N$ is modular—meaning it is covered by the modular curve $X_0(N)$—if and only if there is a weight-$2$, level-$N$ Hecke eigenform $f$ with rational coefficients such that $c(q, f) = a_q(E)$ for every prime $q \nmid N$. This bridges the world of cubic curves with the rigid finite-dimensional spaces of modular forms.