nLab Ribet's theorem

Idea

In number theory, Ken Ribet‘s theorem is a statement about Galois representations associated with modular forms.

Jean-Pierre Serre ad Ken Ribet observed that together with assuming the Taniyama-Shimura modularity conjecture (as yet unproven at that time), Ribet’s theorem implies Fermat's last theorem. The modularity theorem was then proven by Andrew Wiles and Richard Taylor, thus often known now as Wiles' proof of Fermat's last theorem.

Statement

Let $f$ be a weight $2$ newform of level $\Gamma_{0}(q N)$, with $q$ not dividing $N$. Let $\overline{\rho}_{f,p}$ be the associated absolutely irreducible 2-dimensional mod $p$ Galois representation which is unramified if $q\neq p$ and finite flat if $q=p$.

Then there exists another weight $2$ newform $g$, of level $\Gamma_{0}(N)$, with associated mod $p$ Galois representation $\overline{\rho}_{g,p}$, such that

$$\overline{\rho}_{f,p}\simeq\overline{\rho}_{g,p}.$$

References

• Wikipedia, Ribet's theorem