The Basic Theory - Elliptic Curves: Number Theory and Cryptography - page 66

denominators and remove common factors from the numerators and therefore
obtain a triple of integers with gcd=1. Therefore, P 2 ( Q )and P 2 ( Z ) will be
regarded as equal. Similarly, if R is a ring with

Z ⊆ R ⊆ Q ,

then P 2 ( R )= P 2 ( Z ).
In order to work with elliptic curves over R , we need to impose two condi-
tions on R .

R × 1. 2 ∈

2. If ( a ij )isan m n matrix such that ( a 11 ,a 12 ,...,a mn ) is primitive and

such that all 2 × 2 subdeterminants vanish (that is, a ij a k −a i a kj =0for
all i, j, k, ), then some R -linear combination of the rows is a primitive
n -tuple.

×

The first condition is needed since we'll be working with the Weierstrass equa-
tion. In fact, we should add the condition that 3 R × if we want to change

an arbitrary elliptic curve into Weierstrass form. Note that Z does not satisfy
the first condition. This can be remedied by working with

∈

x
2 k

Z (2) = { | x ∈ Z ,k≥ 0 }.

This is a ring. As pointed out above, P 2 ( Z (2) )equals P 2 ( Z ), so the introduc-
tion of Z (2) is a minor technicality.
The second condition is perhaps best understood when R is a field. In this
case, the primitivity of the matrix simply means that at least one entry is
nonzero. The vanishing of the 2 2 subdeterminants says that the rows are

proportional to each other. The conclusion is that some linear combination
of the rows (in this case, some row itself) is a nonzero vector.
When R = Z , the primitivity of the matrix means that the gcd of the
elements in the matrix is 1. Since the rows are assumed to be proportional,
there is a vector v and integers a 1 ,...,a m such that the i th row is a i v .The
m -tuple ( a 1 ,...,a m ) must be primitive since the gcd of its entries divides the
gcd of the entries of the matrix. Therefore, there is a linear combination of
the a i 's that equals 1. This means that some linear combination of the rows
of the matrix is v . The vector v is primitive since the gcd of its entries divides
the gcd of the entries of the matrix. Therefore, we have obtained a primitive
vector as a linear combination of the rows of the matrix. This shows that
Z satisfies the second condition. The same argument, slightly modified to
handle powers of 2, shows that Z (2) also satisfies the second condition.
In general, condition 2 says that projective modules over R of rank 1 are
free (see [74]). In particular, this holds for local rings, for finite rings, and for
Z (2) . These suce for our purposes.

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