Fermat’s Last Theorem - Elliptic Curves: Number Theory and Cryptography - page 449

Combining the result of Eichler and Shimura with the Galois representations

discussed above, we obtain the following. If f = b n q n is a normalized
newform with rational integer coe cients, then there is a Galois representation

ρ f : G −→ GL 2 ( O )

such that

Trace( ρ f (Frob r )) = b r , det( ρ f (Frob r )) = r (15.8)

for all r N .

More generally, Eichler and Shimura showed that if f = b n q n is any
normalized newform (with no assumptions on its coe cients), then there is a
Galois representation

ρ f : G → GL 2 ( O )

satisfying (15.8).
Returning to the situation where the coe cients b n are in Z ,welet M be

the kernel of the ring homomorphism

T −→ F
T r −→ b r (mod ) .

Since the homomorphism is surjective (because 1 maps to 1) and F is a field,
M is a maximal ideal of T .Also, T / M = F .Since T r − b r ∈M ,themod

version of (15.8) says that

Trace( ρ f (Frob r )) ≡ T r mod M, det( ρ f (Frob r )) ≡ r mod M

for all r N . This has been greatly generalized by Deligne and Serre:

THEOREM 15.4
Let M be a m aximalidealof T and let be the characteristicof T /

M .There

existsasemisimp erepresentation

ρ M : G −→ GL 2 ( T /M )

su ch that

Trace( ρ M (Frob r )) ≡ T r mod M, det( ρ M (Frob r )) ≡ r mod M

for allprimes r

N .

The semisimplicity of ρ M means that either ρ M is irreducible or it is the
sum of two one-dimensional representations.
In general, let A be either O or a finite field. If

ρ : G −→ GL 2 ( A )