Mazur's torsion classification

Mazur's 1977 proof translates the existence of a rational point of exact order $N$ on $E$ into the existence of a non-cuspidal rational point on the modular curve $X_1(N)$, which classifies pairs $(E, P)$ with $P$ of order $N$. The strategy studies the Jacobian $J_0(N)$ of the related modular curve $X_0(N)$ and the Eisenstein ideal $\mathcal{I}$ — the ideal in the Hecke algebra generated by $T_\ell - \ell - 1$ for primes $\ell \nmid N$ — acting on it. By analyzing the Eisenstein quotient of $J_0(N)$ and its reduction modulo auxiliary primes, Mazur shows that for $N$ outside the allowed list, $X_1(N)(\mathbb{Q})$ consists only of cusps, so no elliptic curve over $\mathbb{Q}$ can have a rational point of that exact order; explicit constructions (for instance using the curves $y^2=x^3+ax$ and $y^2=x^3+b$) exhibit examples realizing each of the $15$ permitted groups.