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)
In lectures at the Newton Institute in June of 1993, Andrew Wiles announced a proof of a large part of the Taniyama-Shimura Conjecture and, as a consequence, Fermat's Last Theorem. This report for nonexperts discusses the mathematics involved in Wiles' lectures, including the necessary background and the mathematical history.
When Andrew John Wiles was 10 years old, he read Eric Temple Bell's The Last Problem and was so impressed by it that he decided that he would be the first person to prove Fermat's Last Theorem. This theorem states that there are no nonzero integers a, b, c, n with n > 2 such that a n + b n = c n. The object of this paper is to prove that all semistable… (More)
A genus one curve defined over Q which has points over Q p for all primes p may not have a rational point. It is natural to study the classes of Q-extensions over which all such curves obtain a global point. In this article, we show that every such genus one curve with semistable Jacobian has a point defined over a solvable extension of Q.
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)
Tate helped shape the great reformulation of arithmetic and geometry which has taken place since the 1950s. This is my article on Tate's work for the second volume in the book series on the Abel Prize winners. True to the epigraph, I have attempted to explain it in the context of the " great reformulation ". Contents 1 Hecke L-series and the cohomology of… (More)
The equation of Fermât has undoubtedly had a far greater influence on the development of mathematics than anyone could have imagined. AfterT847 most serious mathematical approaches to the problem followed the line introduced by Kummer. This approach involved a detailed analysis of the ideal class groups of cyclotomic fields. The class number formulas used… (More)
the fourteenth recipient of the 6 million NOK (about 750,000 USD) prize. A prize honoring the Norwegian mathematician Niels Henrik Abel was first proposed by the world-renowned mathematician Sophus Lie, also from Norway, and initially planned for the one-hundredth anniversary of Abel's birth in 1902, but the establishment of the Abel Prize had to wait… (More)