KnowraModularity theoremLinked fromLinked fromThe 8 pages that link to Modularity theorem, each with the reason it gives.All 8Broader topic 2Related 5Narrower topic 1Fermat's Last TheoremRelated: Its applicable case contradicts the nonmodularity implied by a hypothetical Fermat counterexample.Elliptic curveRelated: It connects elliptic-curve arithmetic to the theory of modular forms.Modular formRelated: It links modular forms to elliptic curves and enabled the proof of Fermat’s Last Theorem.Birch and Swinnerton-Dyer conjectureRelated: Modularity makes the curve’s L-function analytic and enables rigorous study at 1.Langlands programBroader topic: It realizes a major arithmetic-to-automorphic correspondence predicted by Langlands ideas.Andrew WilesNarrower topic: Wiles proved enough of this broader result to establish Fermat’s Last Theorem.Robert LanglandsBroader topic: Its proof established a striking instance of the kind of correspondence Langlands envisioned.Ribet's theoremRelated: Together with Ribet's theorem, it completes the modularity argument used against a Fermat counterexample.