An odd irreducible representation ρ : Gal(ℚ̄/ℚ) → GL₂(𝔽_ℓ) is modular. Serre conjectured that every such continuous homomorphism arises from a cuspidal eigenform whose weight and level are read off from the local ramification of ρ. The same Galois module that appears in the étale cohomology of an algebraic variety, or in the torsion of a modular Jacobian J_{0,ns}^+(q), also governs the Hecke eigenvalues of a holomorphic function on the upper half-plane. Formulated in a 1973 letter to Tate and given its precise shape in 1987, the statement turns questions about the absolute Galois group into questions about automorphic forms. Once proved by Khare and Wintenberger, it made Fermat’s equation a special case of a reciprocity law that identifies arithmetic monodromy with modular cohomology.

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