nLab Ribet's theorem




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.


Let ff be a weight 22 newform of level Γ 0(qN)\Gamma_{0}(q N), with qq not dividing NN. Let ρ¯ f,p\overline{\rho}_{f,p} be the associated absolutely irreducible 2-dimensional mod pp Galois representation which is unramified if qpq\neq p and finite flat if q=pq=p.

Then there exists another weight 22 newform gg, of level Γ 0(N)\Gamma_{0}(N), with associated mod pp Galois representation ρ¯ g,p\overline{\rho}_{g,p}, such that

ρ¯ f,pρ¯ g,p.\overline{\rho}_{f,p}\simeq\overline{\rho}_{g,p}.


Last revised on November 23, 2022 at 21:48:15. See the history of this page for a list of all contributions to it.