Fermat’s Last Theorem - Elliptic Curves: Number Theory and Cryptography - page 465

There are two variables and three relations, so we do not have a complete
intersection. The tangent space is Z p ⊕ Z p . The inclusion S 4 ⊂ S 3 corresponds

to the ring homomorphism
O p [[ x, y ]] / ( x 2 − px, y 2 − py ) −→ O p [[ x, y ]] / ( x 2 − px, y 2

− py, xy )

and the map on tangent spaces is an isomorphism. However, S 3 = S 4 .The
problem is that the tangent space calculation does not notice the relation xy ,
which removed the point ( p, p )from S 3 to get S 4 . Therefore, the tangent
space thinks this point is still there and incorrectly predicts an isomorphism
between the three point space and the four point space.
The general fact we need is that if we have a surjective homomorphism of
rings that are local complete intersections, and if the induced map on tangent
spaces is an isomorphism, then the ring homomorphism is an isomorphism.

Deformations of Galois representations

Now let's return to our sets A and M . Corresponding to these two sets are
rings R A and R M .Wehave g 0 ∈ M ⊆ A .Let T A and T M be the tangent
spaces at g 0 . In the examples above, the base point g 0 would correspond to
x =0orto( x, y )=(0 , 0). Corresponding to the inclusion M ⊆ A ,thereare
surjective maps

R A −→ R M , T A −→ T M .

Therefore,

# T A .
The ring R M can be constructed using the Hecke algebra and the ring R A
is constructed using results about representability of functors. In fact, it was
shown that there is a representation

# T M ≤

ρ universal : G −→ GL 2 ( R A )

with the following property. Let

ρ : G −→ GL 2 ( O p )
be a representation and let g be the potential modular form attached to ρ .
Assume that ρ is unramified outside a fixed finite set of primes. If g ≡ g 0

(mod p ), then there exists a unique ring homomorphism

φ : R A −→ O p

such that the diagram

ρ universal

GL 2 ( R A )

G

φ

ρ

GL 2 ( O p )