Lời giải định lí Fermat

An elliptic curve over Q is said to be modular if it has a finite covering by a modular curve of the form X0(N ). Any such elliptic curve has the property that its Hasse-Weil zeta function has an analytic continuation and satisfies a functional equation of the standard type. If an elliptic curve over Q with a given j -invariant is modular then it is easy to see that all elliptic curves with the same j -invariant are modular (in which case we say that the j -invariant is modular). A well-known conjecture which grew out of the work of Shimura and Taniyama in the 1950’s and 1960’s asserts that every elliptic curve over Q is modular. However, it only became widely known through its publication in a paper of Weil in 1967 [We] (as an exercise for the interested reader!), in which, moreover, Weil gave conceptual evidence for the conjecture. Although it had been numerically verified in many cases, prior to the results described in this paper it had only been known that finitely many j -invariants were modular. In 1985 Frey made the remarkable observation that this conjecture should imply Fermat’s Last Theorem. The precise mechanism relating the two was formulated by Serre as the ε-conjecture and this was then proved by Ribet in the summer of 1986. Ribet’s result only requires one to prove the conjecture for semistable elliptic curves in order to deduce Fermat’s Last Theorem.

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