Fermat’s Last Theorem - Elliptic Curves: Number Theory and Cryptography - page 464

difference is a linear combination of the form A ( x, y )( x 2 px )+ B ( x, y )( y 2 − − py )

with A, B ∈O p [[ x, y ]]. The corresponding geometric object is

• •
(0 ,p )( p, p )

S 3 :

• •
(0 , 0) ( p, 0)

It can be shown that two power series give the same function on this set of
four points if they differ by a linear combination of x 2 − px and y 2 − py .The

tangent space is represented by

{ ax + by | a, b ∈O p } mod ( px, py ) ,

which means we are considering two linear polynomials to be congruent if
their difference is a linear combination of px and py . It is easy to see that

a 1 x + b 1 y ≡ a 2 x + b 2 y mod ( px, py ) ⇐⇒ a 1 ≡ a 2 , 1 ≡ b 2 (mod p ) .

Therefore, the tangent space is isomorphic to Z p ⊕ Z p .
The inclusion S 1 ⊂ S 3 corresponds to the ring homomorphism

O p [[ x, y ]] / ( x 2 px, y 2 −→ O p [[ x ]] / ( x 2 − − py ) − px ) .

The map on tangent spaces is the map Z p ⊕ Z p → Z p given by projection
onto the first factor.
In all three examples above, the rings are given by power series over O p .
The number of variables equals the number of relations and the resulting
ring is a finitely generated O p -module (this is easily verified in the three
examples). Such rings are called local complete intersections .Forsuch
rings, it is possible to recognize when a map is an isomorphism by looking at
the tangent spaces.
Before proceeding, let's look at an example that is not a local complete
intersection. Consider the ring

O p [[ x, y ]] / ( x 2 − px, y 2 − py, xy ) .

The corresponding geometric object is

•
(0 ,p )

S 4 :

• •
(0 , 0) ( p, 0)