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§9. Division Polynomial 273

(9.2) Division polynomial over fields of characteristic unequal to 2 or 3. Let N
2
3
be an odd number, and consider the curve E = E a,b defined by y = x + ax + b
over a field k of characteristic p = 2. We introduce the polynomial

ψ N (x) = (x − x(P))
P∈(E[N]−{0})/{±1}
2
of degree (N − 1)/2 which can be viewed as an element in Z[x, a, b]. When N is
even, y appears only with odd power in ψ N (x, y). When N is odd, y appears only
with even power in ψ N (x, y).If N is odd, then on the curve we can eliminate y so
that ψ N (x, y) becomes ψ N (x).
(9.3) Recurrence formulas. We have the following examples

4 2 2
ψ 0 = 0, ψ 1 = 1, ψ 2 = 2y, ψ 3 = 3x + 6ax + 12bx − a ,
6 4 3 2 2 2 2
ψ 4 = 4y(x + 5ax + 20bx − 5a x − 4abx − 8b − a ),
and the following recurrence formulas
3
ψ 2N+1 = ψ N+2 ψ − ψ N−1 ψ N+1 , N ≥ 2
N
N
2 2
ψ 2N = (ψ N+2 ψ − ψ N−2 ψ ) , N > 2.
N−1 N+1
2y
(9.4) Formula for multiplication by N. In terms of the division polynomials we
have
2 2
ψ N+2 ψ − ψ N−2 ψ
ψ N−1 ψ N+1 N−1 N+1
[N](x, y) = x − 2 , 3 .
ψ 4yψ
N N