There has been a really amazing development today on Fermat's Last Theorem. Noam Elkies has announced a counterexample, so that FLT is not true after all! His spoke about this at the Institute today. The solution to Fermat that he constructs involves an incredibly large prime exponent (larger that 10^20), but it is constructive. The main idea seems to be a kind of Heegner point construction, combined with an really ingenious descent for passing from the modular curves to the Fermat curve. The really difficult part of the argument seems to be to show that the field of definition of the solution (which, a priori, is some ring class field of an imaginary quadratic field) actually descends to Q. I wasn't able to get all the details, which were quite intricate...
So it seems that the Shimura Taniyama conjecture is not true after
all. The experts think that it can still be salvaged, by
extending the concept of automorphic representation, and introducing a
notion of "anomalous curves" that would still give rise to a
This is an email I received indirectly from the late Gian-Carlo Rota dated April 2,
1994. The historical context: In June of 1993, Andrew Wiles announced
that he had proven Fermat's Last Theorem but later that year a subtle
bug was found which was not fixed until September of '94.
So in April of 1994
we didn't know whether Wiles' proof was valid and given the date was
April 2 and how well the email was written, many mathematicians
believed this message. But of course it was all an elaborate April Fools joke.