At 01:51 22/08/2002, Andrew Coppin wrote:
IIRC, a pair of Japanise (?) students proposed that all elliptic curves are also modular forms. Someone else pointed out that, assuming Fermat was wrong, his equation x^n + y^n = z^n would be an elliptic equation, and the corresponding modular form is decidedly weird. Yet ANOTHER mathematition later proved that this modular form is in fact too weird to actually exist. In other words, assuming that EVERY elliptic curve REALLY IS a modular form, Fermat's Last Theorum MUST be true. Andrew Wiles proved that elliptic curves ARE modular, and as a side effect proved Fermat right.
An elliptic curve is basically a set of solutions to the equation y^2=Ax^3+Bx^2+Cx+D - where x and y are elements in some field (originally real numbers under addition and multiplication, but why restrict ourselves?) A modular function is one of the form f( (az + b)/(cz + d) ) = f(z) for complex a,b,c,d (it derives from linear fractional transforms (also known as bilinear) of the complex plane - those are the affine transforms without translation - and hence draws a lot on group theory.) The two theorems run: If FLT is false, then y^2=x(x-z^n)(x+y^n) is semistable, but not modular. (Conjectured in 1982 by Gerhard Frey; proven in 1986 by Ken Ribet via Jean-Pierre Serre). and All semistable elliptic curves are modular. (Conjectured by Yutaka Taniyama in 1955, later generalised by Goro Shimura and Andre Weil by dropping the "semistable" bit; proven in 1994 by Andrew Wiles). Most of that material can be found on this page on the subject, which leads up to the proof and covers things like elliptic curves and modular forms along the way: http://www.mbay.net/~cgd/flt/fltmain.htm As has already been noted, the given extended version of FLT is false, since counterexamples have been found. So the question has become: for _which_ pairs 1 < n < p do solutions exist? Morgan L. Owens "All a mathematician needs a computer for is emacs, TeX, and mail."