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276 14. Elliptic Curves over Local Fields

and one of the multiplicative group
2
∗
∗
K ⊃ R ⊃ 1 + Rπ ⊃ 1 + Rπ ⊃ ··· ⊃ 1 + Rπ m ⊃ ··· .
There are isomorphisms
n n+1
∗
∗
∗
∗
Rπ /Rπ → k + for n ≥ 0, v : K /R → Z, r : R /1 + Rπ → k ,
and
m
1 + Rπ /1 + Rπ m+1 → k + for m ≥ 1.
In 5(4.5) there are analogous results for the kernel E (1) (K) of the reduction homo-
morphism. Using the formal group associated to E, we will be able to give a more
precise picture of this p-adic filtration.
(1.1) Definition. Let E be an elliptic curve over K with reduction mapping
¯
¯
r : E(K) → E(k), and let E(k) ns the group of nonsingular points of E(k).The
canonical p-adic filtration of E is a sequence of subgroups of E(K)
(0) (1) (n)
E(K) ⊃ E (K) ⊃ E (K) ⊃· · · ⊃ E (K) ⊃· · · ,
where E (0) (K) = r −1 (E(k) ns ), E (1) (K) = r −1 (0), and for n ≥ 1 E (n) (K) is the set
of all (x, y) satisfying v(x) ≤−2n and v(y) ≤−3n.
From the equation in normal form with v(a j ) ≥ 0wehave v(x)< 0 if and only
if v(y)< 0 and in this case v(x) =−2n and v(y) =−3n for some n. In particular,
the two definitions of E (1) (K) are equivalent. It is also the case that these definitions
of E (m) (K) are equivalent to those given in 5(4.1).
For the filtration E(K) ⊃ E (0) (K) ⊃ E (1) (K) we have asserted that asserted
that E(K)/E (0) (K) is finite and that E (0) (K)/E (1) (K) is isomorphic to E(k) ns as
¯
in 5(3.4). An analysis of the structure of the quotients E (1) (K) was given in Chapter
5, §4 in terms of the structure of the quotients E (n) (K)/E (2n) (K). Using the formal
group law E of E where E (t 1 , t 2 ) is in R[[t 1 , t 2 ]], under the assumption that K is
complete, we can define a second group structure on Rπ by a + E b = E (a, b) in
Rπ, and this modified version of Rπ is isomorphic to E (1) (K). More precisely we
have the following theorem which uses the considerations of Chapter 12, §7.
(1.2) Theorem. Let E be an elliptic curve over a field K with a complete discrete
valuation. The function t(P) =−x(P)/y(P) is an isomorphism of E (1) (K) onto
Rπ where Rπ has the group structure given by a + E b = E (a, b) in terms of the
formal group law E of E.

Proof. The definition of E (t 1 , t 2 ) in 12(7.2) is in terms of the group law on the
elliptic curve E. Hence the function t(P) =−x(P)/y(P) is a group homomorphism
n
E (1) (K)/E (n) (K) → Rπ/Rπ for each n, and hence also in the limit E (1) (K) →
n
Rπ. By 5(4.5) it induces a monomorphism E (n) (K)/E (2n) (K) → Rπ /Rπ 2n for
each n, and this implies that E (1) (K) → Rπ is a monomorphism. Finally the map
is surjective, for given a value t in R, we can substitute into the power series for x
and y in t, see 12(7.1), to obtain a point on E (1) (K) having the given t value. This
proves the theorem.