Fermat’s Last Theorem - Elliptic Curves: Number Theory and Cryptography - page 446

If m = is a prime, we call ρ the mod Galois representation attached to
E . We can also take m = n for n =1 , 2 , 3 ,... . By choosing an appropriate
sequence of bases, we obtain representations ρ n such that

(mod n ) ρ n ≡ ρ n +1

for all n . These may be combined to obtain

ρ ∞ : G −→ GL 2 ( O ) ,

where O denotes any ring containing the -adic integers (see Appendix A).
This is called the -adic Galois representation attached to E . An advantage of
working with ρ ∞ is that the -adic integers have characteristic 0, so instead
of congruences mod powers of , we can work with equalities.

Notation: Throughout this chapter, we will need rings that are finite ex-
tensions of the -adic integers. We'll denote such rings by O .Formany
purposes, we can take O to equal the -adic integers, but sometimes we need
slightly larger rings. Since we do not want to discuss the technical issues that
arise in this regard, we simply use O todenoteavaryingringthatislarge

enough for whatever is required. The reader will not lose much by pretending
that O is always the ring of -adic integers.

Suppose r is a prime of good reduction for E . There exists an element
Frob r ∈

G such th at the action of Frob r on E ( Q ) yields the action of the
Frobenius φ r on E ( F r )when E is reduced mod r (the element Frob r is not
unique, but this will not affect us). In particular, when = r , the matrices
describing the actions of Frob r and φ r on the -power torsion are the same
(use a basis and its reduction to compute the matrices). Let

a r = r +1 − # E ( F r ) .

From Proposition 4.11, we obtain that

(mod n ) , (mod n ) , Trace( ρ n (Frob r )) ≡ a r det( ρ n (Frob r )) ≡ r

and therefore

Trace( ρ ∞ (Frob r )) = a r , det( ρ ∞ (Frob r )) = r.

Recall that the numbers a r are used to produce the modular form f E attached
to E (see Section 14.2).
Suppose now that

ρ : G −→ GL 2 ( O )
is a representation of G . Under certain technical conditions (namely, ρ is
unramified at all but finitely many primes; see the end of this section), we
may choose elements Frob r (for the unramified primes) and define

a r = Trace( ρ (Frob r )) .