Ribet's theorem

Ribet's theorem shows that, under suitable conditions, a modular Galois representation arising from a modular form can also arise from a form of lower level. It supplied the level-lowering step in the proof of Fermat's Last Theorem.

Modular Galois representation: A representation of a Galois group arising from a modular form or related arithmetic object. The theorem starts with such a representation and shows it can arise at a lower level.
Jean-Pierre Serre: A French mathematician whose work shaped number theory, representation theory, and algebraic geometry. Serre's conjectures on modular Galois representations framed the setting in which level lowering became central.
Epsilon conjecture: A conjecture of Serre asserting that a Frey curve's modular form has a particular level and weight. Ribet proved the level-lowering statement needed to establish this conjecture.
Level raising: The process of producing modular forms at a higher level while preserving a related Galois representation. It moves in the opposite direction from Ribet's reduction of modular level.
Modular form: A complex analytic function with prescribed transformation and growth properties under a modular group. Ribet's theorem transfers a representation from one modular form to another of lower level.
Kenneth Ribet: An American mathematician known for major contributions to number theory and the proof of Fermat's Last Theorem. Ribet proved the level-lowering result that bears his name.
Modularity theorem: The theorem that every elliptic curve over the rationals corresponds to a modular form. Together with Ribet's theorem, it completes the modularity argument used against a Fermat counterexample.
Modularity: The property of being describable or governed by modular forms. Ribet's theorem refines modularity by specifying how low a form's level can be.
Conductor: An arithmetic invariant measuring the ramification and complexity of a representation or number field. Level lowering removes certain prime factors from the representation's associated level.
Serre's modularity conjecture: A conjecture predicting that odd irreducible two-dimensional mod-p Galois representations arise from modular forms. Ribet's theorem supplied a key level-lowering result toward Serre's predicted modularity.

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