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The story is now well-known—in June 1993, Andrew Wiles announced, at a conference in Cambridge, a proof that every semi-stable elliptic curve over the rationals was modular. As many of the people in the audience were already aware, this result, together with another deep theorem proved by Ken Ribet in the 1980s, implied Fermat's Last Theorem. Later on in(More)
A polynomial relation f (x, y) = 0 in two variables defines a curve C. If the coefficients of the polynomial are rational numbers then one can ask for solutions of the equation f (x, y) = 0 with x, y ∈ Q, in other words for rational points on the curve. If we consider a non-singular projective model C of the curve then over C it is classified by its genus.(More)
Hilbert delivered his famous lecture in which he described twenty-three problems that were to play an infl uential role in mathematical research. A century later, on May 24, 2000, at a meeting at the Collège de France, the Clay Mathematics Institute (CMI) announced the creation of a US$7 million prize fund for the solution of seven important classic(More)
Clozel, Harris and Taylor have recently proved a modularity lifting theorem of the following general form: if ρ is an ℓ-adic representation of the absolute Galois group of a number field for which the residual representation ρ comes from a modular form then so does ρ. This theorem has numerous hypotheses; a crucial one is that the image of ρ must be " big,(More)