Fermat’s Last Theorem - Elliptic Curves: Number Theory and Cryptography - page 445

numbers a p under control. Frey predicted the following, which Ribet proved
in 1986:

THEOREM 15.1
E Frey cannot be m odular. T herefore, the conjecture of Taniyam a-Shimura-
We limp ies Ferm at's Last T heorem .

This result finally gave a theoretical reason for believing Fermat's Last
Theorem. Then in 1994, Wiles proved

THEOREM 15.2
Allsem istableelliptic curves over Q are m odular.

This result was subsequently extended to include all elliptic curves over Q .
See Theorem 14.4. Since the Frey curve is semistable, the theorems of Wiles
and Ribet combine to show that E Frey cannot exist, hence

THEOREM 15.3
Ferm at's Last T heorem istru e.

In the following three sections, we sketch some of the ideas that go into the
proofs of Ribet's and Wiles's theorems.

15.2 Galois Representations

Let E be an elliptic curve over Q and let m be an integer. From Theo-
rem 3.2, we know that

E [ m ] Z m ⊕ Z m .

Let { β 1 ,β 2 } be a basis of E [ m ]andlet σ ∈ G ,where

G =Gal( Q / Q ) .

Since σβ i ∈ E [ m ], we can write

σβ 1 = aβ 1 + cβ 2 , σ 2 = bβ 1 + dβ 2

with a, b, c, d ∈ Z m . We thus obtain a homomorphism

ρ m : G −→ GL 2 ( Z m )

ab
cd

.

σ −→