Serre's modularity conjecture

Remarks

Serre stated the conjecture in 1987 in a strong form. In it, $N$ is the Artin conductor of $\bar\rho$ away from $p$, and $k$ is an explicit weight $k(\bar\rho)$ read off from the restriction of $\bar\rho$ to inertia at $p$. Ribet's level-lowering theorem and later work showed that the weak form above implies the strong form. Khare and Wintenberger proved the conjecture, with a step by Kisin, in work published in 2009. Their proof is an induction on the prime $p$ and the conductor, using modularity lifting theorems and the existence of compatible families of Galois representations. Oddness is necessary: for every modular eigenform the attached representation is odd. Example: for an elliptic curve $E$ over $\Q$, the action on the $p$-torsion points $E[p]$ gives an odd $\bar\rho$, since $\det \bar\rho$ is the cyclotomic character modulo $p$. When this $\bar\rho$ is irreducible, its modularity also follows from the modularity theorem.

Going deeper

For a student

The strong form gives a route to Fermat's Last Theorem. A putative solution would give a representation of level $2$ and weight $2$, and there are no nonzero cusp forms of that level and weight. The theorem is a mod $p$ case of the Langlands program for $\mathrm{GL}_2$ over $\Q$. It also implies that the Artin conjecture holds for odd two-dimensional complex Galois representations of $\Q$.