580 XIII Connections with Number Theory
Moreover, if we putC=−(A+B),thenC=0and(A,C)=(B,C)=1. The linear
change of variables
X→ 4 X, Y→ 8 Y+ 4 XreplacesEA,Bby the elliptic curveWA,Bdefined by
Y^2 +XY−{X^3 +(B−A− 1 )X^2 / 4 −ABX/ 16 },which has discriminant
∆=(ABC)^2 / 28.Our hypotheses ensure that the coefficients ofWA,Bare integers and that∆is a nonzero
integer. It may be shown thatWA,Bis actually a minimal model forEA,B. Moreover,
when we reduce modulo any primewhich divides∆, the singular point which arises
is a node. ThusWA,Bis semi-stable and its conductorNis the product of the distinct
primes dividingABC.
Fermat’s last theorem will be proved, for any primep≥5, if we show that such an
elliptic curve cannot exist ifA,B,Care allp-th powers. Ifpis large, one reason for
suspecting that such an elliptic curve cannot exist is that the discriminant is then very
large compared with the conductor. Another reason, which does not depend on the size
ofp, was suggested by Frey (1986). Frey gave a heuristic argument thatWA,Bcould
not then be modular, which would contradict theTW-conjecture.
Frey’s intuition was made more precise by Serre (1987). LetGbe the group of
all automorphisms of the field of all algebraic numbers. With any modular form for
Γ 0 (N)one can associate a 2-dimensional representation ofGover a finite field. Serre
showed that Fermat’s last theorem would follow from theTW-conjecture, together with
a conjecture about lowering the level of such ‘Galois representations’ associated with
modular forms. The latter conjecture was called Serre’sε-conjecture, because it was a
special case of a much more general conjecture which Serre made.
Serre’sε-conjecture was proved by Ribet (1990), although the proof might be de-
scribed as being of orderε−^1. Now, for the first time, the falsity of Fermat’s last the-
orem would have a significant consequence: the falsity of theTW-conjecture. Since
WA,Bis semi-stable with the normalizations made above, to prove Fermat’s last the-
orem it was actually enough to show that any semi-stable elliptic curve was modular.
As stated in§5, this was accomplished by Wiles (1995) and Taylor and Wiles (1995).
We will not attempt to describe the proof since, besides Fermat’s classic excuse, it is
beyond the scope of this work.
Fermat’s last theorem contributed greatly to the development of mathematics, but
Fermat was perhaps lucky that his assertion turned out to be correct. After proving
Fermat’s assertion forn=3, that the cube of a positive integer could not be the sum
of two cubes of positive integers, Euler asserted that, also for anyn≥4, ann-th power
of a positive integer could not be expressed as a sum ofn− 1 n-th powers of positive
integers. A counterexample to Euler’s conjecture was first found, forn=5, by Lander
and Parkin (1966):
275 + 845 + 1105 + 1335 = 1445.