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considered either a hopeless pursuit or a backwater was revived by Ken Ribet’s work in linking it
to a central concern ofmodernnumber theory—the modularity conjecture. Ribet, one could say,
gave the theorem a little contemporary relevance; and Wiles produced not a ‘classical’ but a very
contemporary theorem.
What of his work-habits? They also can be seen as belonging to the late twentieth-century
setting. By the 1980s number theory had been a ‘professional’ study for 150 years; now, with
diminishing funding and constant demands for publication to justify the researcher’s existence,
it had become (like the rest of mathematics) intensely competitive. Wiles’s understanding that
news of his work on the conjecture might stimulate others to enter the field may appear paranoid,
but is perhaps not as unusual as Singh makes out with his images of loneliness, deviousness,
silence, and withdrawal. And there is a contrast with the more social ethic of 20 years earlier in
the chapters where he clarifies the crucial role of a number of others, most notably Taniyama,
Shimura, and Weil (for the fundamental conjecture on elliptic curves), and Frey and Ribet (for
the construction of a particular curve which links FLT to the Taniyama–Shimura–Weil conjec-
ture). The fact that the latter conjecture seemed no easier than FLT itself^14 is also important
if we are to have a balanced view of the history, rather than one centred on a 300-year-old
problem. It is hard: already in 1993 t-shirts were on sale in Cambridge which read: ‘Fermat’s
Last Theorem proved by Andrew Wiles at the Isaac Newton Institute’. Jaundiced mathematicians
might well object that no single one of the statements was true, but history books will print the
legend.^15
What kind of mathematics are we talking about? This is late twentieth-century number the-
ory of a very advanced kind. For all his ‘seclusion’, Wiles’s paper is packed with references to
work published during the previous five years. More generally, the key conjecture relates elliptic
curves and modular forms, and without a great deal of theory about both of these abstruse and
difficult mathematical objects developed in the previous century, there would certainly have been
no result.
Singh makes a brave attempt to explain the various objects in terms of analogies (the idea that
theL-series of a modular form is its ‘DNA’ is certainly striking, even if a number theorist would find
it strange—do they replicate?). Elliptic curves are, in the crudest sense, easy: they are defined by an
equation
y^2 =x^3 +ax+bwhere 4a^3 + 27 b^2 =0 (to ensure no double roots of the cubic, as Omar Khayyam knew). The
picture (Fig. 8) shows one such; but to do number theory one must (a) think ofaandbas rational
numbers, and (simultaneously) (b) consider all solutions of the equation with(x,y)inC, a four-
dimensional picture which we have difficulty drawing— although ‘intrinsically’, if we forget the
space it is in, it looks like a torus (Fig. 9)
To give more details, to try to explain modular forms, etc., would take us outside the aim of this
book, if not outside history; and other sources (e.g. van der Poorten 1996) can do it better than
I can—you are encouraged to consult them.
- In fact, Wiles’s paper provesenoughof the Taniyama conjecture to settle FLT, but not all of it; the full conjecture has since been
settled by Breuil, Conrad, Diamond, and Taylor. - John Ford:The ManWho Shot LibertyValance, 1962.