Elliptic Curves over Q - Elliptic Curves: Number Theory and Cryptography - page 199

THEOREM 8.1
Let E be given by y 2 = x 3 + Ax + B with A, B

∈ Z .Let p be a primeand et

r be a positive integer. T hen

1. E r isasubgroupof E ( Q ) .

2. If ( x, y ) ∈ E ( Q ) ,then v p ( x ) < 0 ifand onlyif v p ( y ) < 0 .Inthiscase,
there existsaninteger r ≥ 1 su ch that v p ( x )= − 2 r and v p ( y )= − 3 r .

3. T he m ap

λ r : E r /E 5 r → Z p 4 r

p −r x/y (mod p 4 r ) ( x, y ) →

∞ → 0

isaninjective hom om orphism (w here Z p 4 r is a group under addition).

4. If ( x, y ) ∈ E r but ( x, y ) ∈ E r +1 ,then λ r ( x, y ) ≡ 0(mod p ) .

REMARK 8.2 The map λ r should be regarded as a logarithm for the
group E r /E 5 r since it changes the law of composition in the group to addition
in Z p 4 r , just as the classical logarithm changes the composition law in the
multiplicative group of positive real numbers to addition in R .

PROOF The denominator of x 3 + Ax + B equals the denominator of y 2 .
It is easy to see that the denominator of y is divisible by p if and only if
the denominator of x is divisible by p .If p j ,with j> 0, is the exact power
of p dividing the denominator of y ,then p 2 j is the exact power of p in the
denominator of y 2 . Similarly, if p k ,with k> 0, is the exact power of p dividing
the denominator of x , then denominator of x 3 + Ax + B is exactly divisible
by p 3 k . Therefore, 2 j =3 k . It follows that there exists r with j =3 r and
k =2 r . This proves (2). Also, we see that

{ ( x, y ) ∈ E r | v p ( x )= − 2 r, v p ( y )= − 3 r} = { ( x, y ) ∈ E r | v p ( x/y )= r}

is the set of points in E r not in E r +1 . Thisproves(4). Moreover,if λ r ( x, y ) ≡
0(mod p 4 r ), then v p ( x/y ) ≥ 5 r ,so( x, y ) ∈ E 5 r . Thisprovesthat λ r is
injective (as soon as we prove it is a homomorphism).
Let

t = x

1
y . y ,

s =

Dividing the equation y 2 = x 3 + Ax + B by y 3 yields

= x
y

3

+ A x
y

1
y

2

+ B 1
y

3

1
y

,