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§5. The Tate Module of an Elliptic Curve 245

(4.15) Remark. In Gross and Zagier [1985], On Singular Moduli, other values of
j(τ) are given, for example,
√

1 + −163 6 6 2 2 2 2
j =−2 · 3 · 7 · 11 · 19 · 127 · 163.
2

§5. The Tate Module of an Elliptic Curve

Let denote a prime number throughtout this section. We have a sequence of reduc-
tions of Z by powers of , namely
r n
r 2
2
n
r 1
··· → Z/ n+1 → Z/ →· · · → Z/ → Z/ .
n
The projective limit of this sequence, denoted lim n Z/ consists of all a = (a n ) in
←−
n
the product Z/ such that r n (a n+1 ) = a n for all n ≥ 1. This projective limit
1≤n
is Z the ring of -adic integers. The field of fractions Z ⊗ Z Q is Q the field of
-adic numbers. This approach is carried out in detail in Serre, Course in Arithmetic,
Chapter II.
The sequence of reductions of Z can be described using subgroups of Q/Z rather
than quotients of Z. Let N (Q/Z) be the subgroup of all x in Q/Z with Nx = 0. Then
the equivalent sequence is the following where all the morphisms are multiplication
by :

n
··· → n+1(Q/Z) → (Q/Z) →· · · → 2(Q/Z) → (Q/Z).

A group having essentially the same structure as (Q/Z) is µ(k), the group of
roots of unity in an algebraically closed field k. The subgroup µ N (k) of x in µ(k)
with x N = 1 consists of the Nth roots of unity. It is cyclic and of order N when k is
algebraically closed and N is prime to the characteristic of k. Further, if k s denotes
the seperable algebraic closure of k, then the Galois group Gal(k s /k) operates on
n
µ(k s ) and each µ N (k s ). As above we have a sequence of cyclic groups of order ,
where is prime to the characteristic of k, namely

··· → µ n+1(k) → µ (k) →· · · → µ 2(k) → µ (k).
n

The inverse limit is denote Z (1)(k) or simple Z (1) and it is a Gal(k s /k)-module.
(5.1) Definition. Let k s be the separable algebraic closure of a field k of characteris-
∗
tic prime to . The Tate module of k is the inverse limit denoted Z (1)(k s ) or Z (1)
s
together with its action of Gal(k s /k).
The Tate module, or as it is sometimes called the Tate twist, is just Z as a limit
group, but it is more, namely a Galois module.
There is a corresponding construction for elliptic curves E over a field k of char-
acteristic different from using the division points N E(k s ) of E over the separable
algebraic closure k s of k. Again the Galois group Gal(k s /k) acts on E(k s ) and each
N E(k s ) from its action on the x-and y-coordinates of a point.