Fermat’s Last Theorem - Elliptic Curves: Number Theory and Cryptography - page 466

commutes.
The representations ρ such that g g 0 (mod p ) are examples of what are

known as deformations of the Galois representation for g 0 . The representa-
tion ρ universal is called a universal deformation.

≡

Example 15.3
We continue with Example 13.2. Let p = 5 and take the fixed set of primes
to be { 5 , 17 , 37 } . Thenitcanbeshownthat

R A O 5 [[ x ]] / ( x 2 − bx ) ,

where b/ 5 is a 5-adic unit and O 5 is the ring of 5-adic integers. This implies
that T A = Z 5 . The set A has two points, g 0 and g , corresponding to x =0
and x = b .

There exists an integer n , defined below, such that

n ≤ # T M ≤ # T A .

Moreover, a result of Flach shows that n · T A =0. Ifitcanbeshownthat

n =# T A ,then T A = T M .
In our example, n = 5. Since we know that T A = Z 5 ,wehave n =# T A .
Therefore, T A = T M .Itcanbeshownthat R A and R M are local complete
intersections. This yields R A = R M and A = M . Thisimpliesthat g is a
modular form.
In general, recall that we started with a semistable elliptic curve E . Associ-
ated to E is the 3-adic Galois representation ρ 3 ∞ . The theorem of Langlands-
Tunnell yields a modular form g 0 , and therefore a Galois representation

ρ 0 : G −→ GL 2 ( O 3 ) .

We have

ρ 0 (mod 3) ,
sothebasepoint ρ 0 is modular and semistable mod 3 (the notion of semistabil-
ity can be defined for general Galois r ep resen tati ons). Under the additional

ρ 3 ∞ ≡

assumption that ρ 3 restricted to Gal( Q / Q ( √ − 3)) is absolutely irreducible,

Wiles showed that if R M is a local complete intersection then n =# T A and
the map R A → R M is an isomorphism of local complete intersections. Finally,

in 1994, Wiles and Taylor used an ingenious argument to show that R M is a
local complete intersection, and therefore A = M .
What happens if ρ 3 does not satisfy the irreducibility assumption? Wiles
showed that there is a semistable elliptic curve E with the same mod 5
representation as E but whose mod 3 representation is irreducible. Therefore,
E is modular, so the mod 5 representation of E is modular. This means
that the mod 5 representation of E is modular. If the mod 5 representation,