Fermat’s Last Theorem - Elliptic Curves: Number Theory and Cryptography - page 450

is a semisimple representation, then we say that ρ is modular of level N if
there exists a homomorphism

π : T −→ A

such that

Trace( ρ (Frob r )) = π ( T r ) , det( ρ (Frob r )) = π ( r )

for all r N . This says that ρ is equivalent to a representation coming from
one of the above constructions.
When A = T /M , the homomorphism π is the map T → T /M .

When f = b n q n is a normalized eigenform and A = O , recall that
T r ( f )= b r f for all r . This gives a homomorphism π : T →O (it is possible
to regard the coe cients b r as elements of a su ciently large O ).

The way to obtain maximal ideals M of T is to use a normalized eigenform

to get a map T O to a finite field. The kernel of the map

from T to the finite field is a maximal ideal

→O ,thenmap

.
When A is a finite field, the level N of the representation ρ is not unique.
In fact, a key result of Ribet (see Section 15.3) analyzes how the level can be
changed. Also, in the definition of modularity in this case, we should allow
modular forms of weight k ≥ 2 (this means that the factor ( cz + d ) 2 in (15.5)
is replaced by ( cz + d ) k ). However, this more general situation can be ignored
for the present purposes.
If ρ is a mod ul ar representation of some level, and c ∈ G is complex conju-
gation (regard Q as a subfield of C )thenitcanbeshownthatdet( ρ ( c )) = − 1.
This says that ρ is an odd representation. A conjecture of Serre [105], which
was a motivating force for much of the work described in this chapter, pre-
dicts that (under certain mild hypotheses) odd representations in the finite
field case are modular (where we need to allow modular forms of weight k

M

2
in the definition of modularity). Serre also predicts the level and the weight
of a modular form that yields the representation.
Finally, there is a type of representation, called finite, that plays an impor-
tant role in Ribet's proof. Let p be a prime. We can regard the Galois group
for the p -adics as a subgroup of the Galois group for Q :

≥

G p =Gal( Q p / Q p ) ⊂ G =Gal( Q / Q ) .

There is a natural map from G p to Gal( F p / F p ). The kernel is denoted I p and
is called the inertia subgroup of G p :

G p /I p Gal( F p / F p ) . (15.9)

A representation

ρ : G → GL 2 ( F )
is said to be unramified at p if ρ ( I p )=1,namely, I p is contained in the
kernel of ρ .If p = and ρ is unramified at p ,then ρ is said to be finite at p .