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250 12. Endomorphisms of Elliptic Curves

2 2 3 2 2 3
= t + t 1 t 2 + t + A 1 t + t t 1 + t 2 t + t 1 +· · · ,
2
1
2
2
1
where the coefficients A j are determined in the above expansion of s in terms of t.
Now let v = s 1 −λt 1 = s 2 −λt 2 ,and substitute s = λt +v into the Weierstrass cubic
to obtain a cubic equation in t with roots t 1 and t 2 . The negative of the coefficient of
2
t in the cubic is the sum of the three roots t 1 , t 2 ,and t 3 can be expressed as a power
series E in t 1 and t 2 with coefficients in Z[a 1 , a 2 , a 3 , a 4 , a 6 ]. A calculation,which
is left to the reader,yields the following result.
(7.2) Proposition. If (t 1 , s 1 )+(t 2 , s 2 ) = (t 3 , s 3 ) on the elliptic curve E in the (t, s)-
plane, then formally t 3 = E (t 1 , s 2 ) has the form
2
2
a 1 λ + a 3 λ − a 2 ν − 2a 4 λν − 3a 6 λ ν
t 3 =−t 1 − t 2 +
2
1 + a 2 λ + a 4 λ + a 6 λ 3

2 2 3 3
= t 1 + t 2 − a 1 t 1 t 2 − a 2 t t 2 + t 1 t − 2a 3 t t 2 + t 1 t +
1 2 1 2
2 2
+ (a 1 a 2 − 3a 3 )t t +· · · ,
1 2
where E (t 1 , s 2 ) is in Z[a 1 , a 2 , a 3 , a 4 , a 6 ][[t 1 , t 2 ]].
Observe that if the coefficients a j of the model for E lie in a ring R,then t 3 =
E (t 1 , t 2 ) is in R[[t 1 , t 2 ]].
(7.3) Definition. A formal group law F(X, Y) is a formal series F(X, Y) in
R[[X, Y]] satisfying:

(1) F(X, 0) = X, F(0, Y) = Y,
(2) (associativity) F(X, F(Y, Z)) = F(F(X, Y), Z) in R[X, Y, Z].
Further, F(X, Y) is a commutative formal group law provided F(X, Y) = F(Y, X).

There exists θ(X) ∈ XR[[x]] with F(X,θ(X)) = 0.
The formal series E (t 1 , t 2 ) arising from the group law on an elliptic curve is a
formal group law. The formal additive group law is F(X, Y) = X +Y and the formal
multiplicative group law is F(X, Y) = X + Y + XY.
Associated with the formal group F(X, Y) over R is a sequence of formal series
[m](X) over R defined inductively by

[1](X) = X and [m](X) = F(X, [m − 1](X)) for m > 1.
In general [m](X) = mX+··· (higher-order terms). This is the formal multiplication
by m in the formal group F(X, Y). For example,if F = E for an elliptic curve E,
then we have

2 3 4
[2](X) = 2X − a 1 X − 2a 2 X + (a 1 a 2 − 7a 3 )X +· · ·
and

2
2
4
3
[3](X) = 3X − 3a 1 X + (a − 8a 2 )X + 3(4a 1 a 2 − 13a 3 )X + ··· .
1