Divisors - Elliptic Curves: Number Theory and Cryptography - page 340

to E . A natural choice is u P = x − x 0 =0when y 0 =0and u P = y when

2 , 8) on the curve y 2 = x 3 + 72. The line y 0 = 0. For example, let P =( −

x +2=0

passes through P ,sowetake u P ( x, y )= x + 2. The function

f ( x, y )= x + y − 6

vanishes at P . Let's find its order of vanishing at P . The equation for the
curve can be rewritten as

( y +8)( y − 8) = ( x +2) 3 − 6( x +2) 2 + 12( x +2) .

Therefore,
f ( x, y )=( x +2)+( y − 8) = ( x +2) 1+ ( x +2) 2

.

− 6( x +2)+12

y +8

The function in parentheses is finite and does not vanish at P ,soord P ( f )=1.
The function

t ( x, y )= 3

4 ( x +2) − y +8
comes from the tangent line to E at P .Wehave
t ( x, y )=( x +2) 3

( x +2) 2 − 6( x +2)+12

y +8

4 −

4( y +8) − 4( x +2) 2 + 24( x +2)+3( y − 8)

( x +2)

=

.

( x +2) 2
4( y +8)

4( x +2)+24+3 ( x +2) 2

− 6( x +2)+12
y +8

= −

The expression in parentheses is finite and does not vanish at P ,soord P ( t )=
2. In general, the equation of a tangent line will yield a function that vanishes
to order at least 2 (equal to 2 unless 3 P = ∞ in the group law of E ,inwhich
case the order is 3).

Example 11.3
The point at infinity is a little harder to deal with. If the elliptic curve E is
given by

y 2 = x 3 + Ax + B,

a uniformizer at ∞ is u ∞ = x/y . Thischoiceismotivatedbythecomplex
situation: The Weierstrass function ℘ gives the x -coordinate and

2 ℘ gives
the y -coordinate. Recall that the point 0 ∈ C /L corresponds to ∞ on E .
Since ℘ has a double pole at 0 and ℘ has a triple pole at 0, the quotient ℘/℘
has a simple zero at 0, hence can be used as a uniformizer at 0 in C /L .

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