Fermat’s Last Theorem - Elliptic Curves: Number Theory and Cryptography - page 467

restricted to Gal( Q / Q ( √ 5)), is absolutely irreducible, then the above result
of Wiles, with 5 in place of 3, shows that E is modular.
There are only finitely many ell ip tic cur ves over Q for which both the mod 3

representation (res tr icted t o Gal( Q / Q ( √ − 3))) and the mod 5 representation

(restricted to Gal( Q / Q ( √ 5))) are not absolutely irreducible. These finitely
many exceptions can be proved to be modular individually.
Therefore, semistable elliptic curves over Q are modular. Eventually, the
argument was extended by Breuil, Conrad, Diamond, and Taylor to include
all elliptic curves over Q (Theorem 14.4).

The integer n is defined as follows. Let g 0 = b m q m and let

L ( g 0 ,s )= ∞

1 − b −s + 1 − 2 s − 1 ,

b m m −s =

m =1 primes ∈ S

where S is a finite set of bad primes (in our example, S = { 5 , 17 , 37 } ). Write

1 − b X + X 2 =(1 − α X )(1 − β X ) .

The symmetric square L -function is defined to be
L (Sym 2 g 0 ,s )=

(1 − α 2 −s )(1 − β 2 −s )(1 − α β −s ) − 1 .

∈ S

There exists a naturally defined transcendental number Ω (similar to the pe-
riods considered in Section 9.4), defined by a double integral, such that

L (Sym 2 g 0 , 2)
Ω

= r = a rational number .

The number n is defined to be the p -part of r (that is, n is a power of p such
that r equals n times a rational number with numerator and denominator
prime to p ).
The formula that Wiles proved is therefore that L (Sym 2 g 0 , 2) / Ωequals# T A
times a rational number prime to p . This means that the order of an algebraic
object, namely T A , is expressed in terms of the value of an analytic function,
in this case the symmetric square L -function. This formula is therefore of
a nature similar to the analytic class number of algebraic number theory,
which expresses the class number in terms of an L -series, and the conjecture
of Birch and Swinnerton-Dyer (see Section 14.2), which expresses the order
of the Shafarevich-Tate group of an elliptic curve in terms of the value of its
L -series.