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15

Elliptic Curves over Global Fields and -Adic

Representations

In the previous two chapters we carried out the local study of elliptic curves, and
a substantial part of the theory was related to how the fundamental symmetry, the
Frobenius element, behaved on the curve modulo a prime. For an elliptic curve E
over a number field K (or more generally any global field), we have for each prime
a Frobenius element acting on the points of the curve. These Frobenius elements are
in Gal(K s /K), and this Galois group acts on the K s -valued points E(K s ),onthe
subgroup of N-division points N E = N E(K s ), and on the limit Tate modules T (E)
where N and are prime to the characteristic of K. In Chapters 12 and 13, the action
(E)
of Gal(K s /K) on the endomorphisms End K s (E) and the automorphisms Aut K s
over K s was considered in detail. As usual, K s denotes a separable algebraic closure
of K.
In this chapter we analyse in greater detail the action of Gal(K s /K) on N E,
T (E) and V (E) to give a general perspective and to relate to the general concept
of -adic representation. Taniyama [1957] first defined and investigated the notion of
an -adic representation. General properties of these representations have been used
by Faltings [1983] in the proof of the Mordell conjecture. Our aim is to give a brief
introduction to the theory.

§1. Minimal Discriminant Normal Cubic Forms over a Dedekind
Ring

Let R be a Dedekind ring with field of fraction K, and let v denote a finite place with
valuation ring R (v) ⊂ K and R ⊂ R (v) . Then R is the intersection of all R (v) as v
runs over the places of R. Of special interest is R = Z, K = Q,and R (v) = Z (p) for
any prime p.
Any elliptic curve E defined over K has a normal cubic equation with coefficients
in R, hence also in each R (v) . By the local theory considered in Chapter 10, we can
choose, as described in 10(1.2), a minimal equation

2
3
2
F v (x, y) = y + a 1,v xy + a 3,v y − x − a 2,v x − a 4,v x − a 6,v = 0