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14

Elliptic Curves over Local Fields

We return to the ideas of Chapter 5 where the torsion in E(Q) was studied using
¯
the reduction map E(Q) → E(F p ) at a prime p. Now we study reduction in terms
of E(Q) → E(Q p ). This leads to elliptic curves over any complete field K with a
discrete valuation where the congruences of 5(4.3) are interpreted using the formal
group introduced in Chapter 12 §7. We obtain a more precise version of 5(4.5).
For the reduction morphism we use an equation in normal form which is minimal
as in 5(2.2), that is, such that the coefficients a j are in the valuation ring R and
the valuation of the discriminant v( ) is minimal among such equation. When the
¯
reduced curve E is nonsingular, it can be studied within the theory of the cubic
equation, and the reduction map is a group homomorphism.
¯
When the reduced curve E is singular, we study the possible singular behavior,
and for this it is useful to introduce a second minimal model, the N´ eron minimal
#
model E . This concept depends on the notion of group scheme which we describe
briefly. Following Tate [1975, LN 476], we enumerate the possible singular reduc-
tions and the corresponding N´ eron models rather than embarking on a general theory
which would take us beyond the scope of this book. Included in the list of singular fi-
bres are various numerical invariants, like the conductor, which are used in the study
of the L-function of an elliptic curve over a global field.
Finally in this chapter we include an introduction to elliptic curves over R. This
is a treatment that I learned from Don Zagier which had been worked out for his
paper with B. Gross on the derivative of the L-function, see [1986].
The reader should refer to Silverman [1986], Chapter VII in this chapter.

§1. The Canonical p-Adic Filtration on the Points of an Elliptic
Curve over a Local Field

Using the notations 5(1.1) and a fixed uniformizing parameter π, we recall that the
is a filtration of the additive group of the local field K

2
K ⊃ R ⊃ Rπ ⊃ Rπ ⊃ ··· ⊃ Rπ m ⊃· · ·