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Journals and Conferences
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)
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 an + bn = cn. The object of this paper is to prove that all semistable… (More)
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 genus one curve defined over Q which has points over Qp 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.
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.
CLASS FIELD THEORY: CLASS FORMATIONS Tate’s theorem (see (4) above) shows that, in order to have a class field theory over a field k, all one needs is, for each system of fields ksep L K k; ŒLWk <1; L=K Galois, a G.L=K/-module CL and a “fundamental class” uL=K 2 H .G.L=K/;CL/ satisfying Tate’s hypotheses; the pairs .CL;uL=K/ should also satisfy certain… (More)
This article discusses many of the topics that I touched on during my Public Lecture at the National University of Singapore and my Lecture to Schools at Victoria Junior College. During the former lecture, I spoke in broad terms about the history of Fermat’s Last Theorem and about the connection between Fermat’s Last Theorem, and the conjecture—now a… (More)
Suppose that p, u, v, and w are integers, with p > 1. If u + v + w = 0, then uvw = 0. Professor Andrew Wiles of Princeton University deduced this form of Fermat’s Last Theorem at the conclusion of a series of three lectures during the June, 1993 workshop on Iwasawa theory, autmorphic forms, and p-adic representations at the Isaac Newton Institute for… (More)
Let Cδ,T be a reducible, multiply canonical homomorphism. B. Gödel’s classification of unconditionally pseudo-real triangles was a milestone in geometric combinatorics. We show that f̃ ∪∞ < log ( 1 א0 ) . It would be interesting to apply the techniques of  to analytically geometric monoids. It is well known that V 6= L′′.