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§7. Expansions Near the Origin and the Formal Group 249

where zero on E has coordonates (t, s) = (0, 0),and t is a local parameter at 0 on
E. This (t, s)-plane version of the cubic equation generates a formal series relations
for s as a function of t by iterated substitution of the six-term expression for s into
each term of s on the right-hand side of the above equation. The formal expansion
has the following form:

3
3
2
5
4
s = t + a 1 t + (a + a 2 )t + (a + 2a 1 a 2 + a 3 )t 6
1 1
4 2 2 7
+ (a + 3a a 2 + 3a 1 a 3 + a + a 4 )t + ···
1
1
2
3 2
= t (1 + A 1 t + A 2 t +· · · ),
where A n is a polynomial of weight n in the a i with positive integer coefficients.
(7.1) Proposition. In terms of the local uniformizing parameter t =−x/yat 0 on
E the following formal expansions hold in Z[a 1 , a 2 , a 3 , a 4 , a 6 ][[t]]:
2
x = t −2 − a 1 t −1 − a 2 − a 3 t − (a 4 + a 1 a 3 )t − ··· ,
x −3 −2 −1
y =− =−t + a 1 t + a 2 t + a 3 +· · · ,
t

2
3
3
2
ω = 1 + a 1 t + (a + a 2 )t + (a + 2a 1 a 2 + a 3 )t +· · · dt.
1 1
i
The coefficients of t are isobaric of weight i + 2,i + 3, and i, respectively, for x, y,
ω. Further, if the coefficients a j are in a ring R, then the expansions lie in R[[t]].
3
Proof. The formal expansions for x and y come from the expansion s = t (1+A 1 t+
··· ), y =−1/s,and x−ty, and it is clear that x and y are in Z[a 1 , a 2 , a 3 , a 4 , a 6 ][[t]].
The formal expansion for ω is derived from this in two ways:
dx −2t −3 + ···
ω = = −3 dt,
2y + a 1 x + a 3 −2t + ···
which has coefficients in Z[1/2, a 1 , a 2 , a 3 , a 4 , a 6 ][[t]], and
dy 3t −4 + ···
ω = = dt,
2
3x + 2a 2 x + a 4 − a 1 y 3t −4 + ···
which has coefficients in Z[1/3, a 1 , a 2 , a 3 , a 4 , a 6 ][[t]]. This proves the assertion
about the coefficients of the above expression.
The expansion x(t) and y(t) in the previous proposition are an algebraic analogue

of the complex analytic expansions of ℘(u) and ℘ (u). In Chapter 13 we will see that
these formal expressions have p-adic convergence properties.
We can go further to analyze the group law near 0 formally also. The line joining
two points (t 1 , s 1 ) and (t 2 , s 2 ) in the (t, s)-plane has slope given by

3
4
t − t 3 t − t 4
s 2 − s 1 2 1 2 1
λ = = + A 1 +· · ·
t 2 − t 1 t 2 − t 1 t 2 − t 1