Fermat’s Last Theorem - Elliptic Curves: Number Theory and Cryptography - page 444

of E Frey . A conjecture of Brumer and Kramer predicts that a semistable
elliptic curve over Q whose minimal discriminant is an th power will have
a point of order . Mazur's Theorem (8.11) says that an elliptic curve over
Q cannothaveapointoforder when 11. Moreover, if the 2-torsion is

rational, as is the case with E Frey , then there are no points of order when
≥ 5. Since Δ is almost an th power, we expect E Frey to act similarly to
a curve that has a point of order . Such curves cannot exist when ≥ 5,
so E Frey should act like a curve that cannot exist. Therefore, we expect that
E Frey does not exist. The problem is to make these ideas precise.
Recall (see Chapter 14) that the L -series of an elliptic curve E over Q is
defined as follows. For each prime p of good reduction, let

≥

a p = p +1 − # E ( F p ) .

Then

L E ( s )=( ∗ )
p

(1 − a p p −s + p 1 − 2 s ) − 1 = ∞

a n

n s ,

n =1

where (*) represents the factors for the bad primes (see Section 14.2) and the
product is over the good primes. Suppose E ( Q ) contains a point of order .
By Theorem 8.9, E ( F p ) contains a point of order for all primes p = such
that E has good reduction at p . Therefore, | # E ( F p ), so

a p ≡ p +1 (mod ) (15.4)

for all such p . This is an example of how the arithmetic of E is related to
properties of the coe cients a p . We hope to obtain information by studying
these coe cients.
In particular, we expect a congruence similar to (15.4) to hold for E Frey .
In fact, a close analysis (requiring more detail than we give in Section 13.3) of
Ribet'sproofshowsthat E Frey is trying to satisfy this congruence. However,
the irreducibility of a certain Galois representation is preventing it, and this
leads to the contradiction that proves the theorem.
The problem with this approach is that the numbers a p at first seem to
be fairly independent of each other as p varies. However, the Conjecture of
Taniyama-Shimura-Weil (now Theorem 14.4) claims that, for an elliptic curve
E over Q ,

f E ( τ )= ∞ a n q n

n =1

(where q = e 2 πiτ ) is a modular form for Γ 0 ( N )forsome N (see Section 14.2).
In this case, we say that E is modular . This is a fairly rigid condition and
can be interpreted as saying that the numbers a p have some coherence as p
varies. For example, it is likely that if we change one coecient a p ,then
the modularity will be lost. Therefore, modularity is a tool for keeping the