Dear Colleague,
It is always difficult to tell exactly what was the reason or
motivation behind some mathematical statement (theorem or conjecture).
I can only tell that when the conjecture was stated I was deeply
interested in the following problem : how can one for a specific
elliptic curve over Q prove that it satisfies Taniyama-Weil conjecture
(this question is now useless, since it has been proved by various
authors (Wiles, Taylor, Conrad, Breuil, Diamond)
that Taniyama-Weil conjecture holds for all elliptic curve.
Th interest in the question was motivated by a theorem of Goldfeld on
which I had reported at the Bourbaki seminar : it says that the
imaginary quadratic fields having a given class number can be
effectively determined if one has some elliptic curves satisfying
Taniyama-Weil conjecture and such that its L-function has a zero of
sufficiently large order at 1.
To rpoduce such curves, I worked out a method with Mestre (called the
graph method), but at the end, to check the Taniyama-Weil conjecture
for these
curves, we needed bounds on the coefficients of the minimal Weierstrass
equation of the Weil curves having the same first coefficients in its
L-function.
This in turn led me to teh study of problems related with abc and with
Szpiro's conjecture.
I unfortunately cannot be much more precise. I can only say that, if
of course Fermat Last Theorem was immediately recognised as one of the
consequences of the abc, it was not the original motivation to state
it. (However notice that a proof of Taniyama-Weil conjecture for a
large category of elliptic curves was the main step in the proof by
Wiles of Fermat Last Theorem).
Best regards,
Joseph Oesterlé