512
PARTISAN REVIEW
other modular forms. In each case he was able to identifY an elliptic equa–
tion that provided a perfect match.
Not long after this discovery Taniyama committed suicide (for never–
explained reasons), and Shimura continued on alone to accumulate more
and more evidence for the one-to-one correspondence between the terms
of the two types of series. This led him to promulgate what came to be
known as the Taniyama-Shimura conjecture, namely: "To every modular
form corresponds some elliptic equation." During the late sixties, many
mathematicians became interested in this strange correspondence and
found more and more cases that confirmed the Taniyama-Shimura con–
jecture. But since no one had worked out a general proof, it remained
merely as unproved a conjecture as had Fermat's Last Theorem.
While Wiles was preoccupied with elliptic equations, not much was
happening on the Last Theorem front. But in 1984, the German mathe–
matician Gerhard Frey breathed new life into the problem by claiming
that whoever could prove the Taniyama-Shimura conjecture would there–
by also prove Fermat's Last Theorem. For the purpose of his argument,
Frey assumed that the Last Theorem was false (i.e., that the Fermat equa–
tion
an
+
b
11
=
en does
have a whole-number-solution for some
n
>
2).
In that case it would be possible to rearrange the Fermat equation as an
elliptic equation. But this elliptic equation would have such strange prop–
erties that it could not possibly have a modular correspondent. According
to the Taniyama-Shimura conjecture, however, every elliptic equation
must
have a modular correspondent; therefore if the Taniyama-Shimura
conjecture were true, the Last Theorem would have to be true as well.
Unfortunately, Frey had not proven that the properties of the elliptic
equation resulting from the rearranged Fermat equation
would
really be too
strange for the existence of a modular correspondent. However, by the
summer of 1986, Kenneth Ribet of the Universi ty of California at
Berkeley had managed to plug the hole in Frey's argument and had shown
that the rearranged Fermat equation could not, in fact, have a modular cor–
respondent. So the way was now open for the final assault on the Last
Theorem. All that was needed now to prove its truth was to prove the
truth of the Taniyama-Shimura conjecture.
Mter completing his Ph.D. in Cambridge, Andrew Wiles moved to
America to join the mathematics faculty of Princeton University. His pro–
ductive thesis studies of elliptic equations had made his name known in
the profession as an up-and-coming young mathematician, and, provided
he kept up the good work at Princeton, he faced a bright professional
future. But as soon as he learned of Ribet's proof of Frey's claim that proof
of the Taniyama-Shimura conjecture would be tantamount to proving
Fermat's Last Theorem, Wiles set out to prove that conjecture, dropping
