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284 14. Elliptic Curves over Local Fields

I
If (2) is satisfied, then for infinitely many N the fixed subgroup N E(K s ) =
#
#
N E(K s ).For N strictly bigger than the index (E : E s,0 ) there is a subgroup of E s #
s
2
2
isomorphic to (Z/NZ) , and hence, E # has a subgroup isomorphic to (Z/NZ) .In
s,0
particular, E # is an elliptic curve and therefore proper. It suffices to show that the
s,0
#
scheme E is proper over Spec(R), and for this we use the next lemma which will
complete the proof of the theorem.
(3.3) Lemma. Let X be a smooth scheme over Spec(R) whose general fibre X η =
¯
X × R K is geometrically connected and whose special fibre Xis proper. Then Xis
¯
proper over Spec(R), and X is geometrically connected.
For the proof of the lemma see Serre–Tate [1968, p. 496]. It follows from basic
results on properness and correctedness for schemes.
Finally we consider the characteristic polynomial in the ramified case in a form
to be used in the study of the L-function. Let K be a finite extension of Q p with
a valuation v extending ord p on Q p and with residue class field k(v) and Nv =
f
¯
# k(v) = q = p . Let I be the inertia subgroup of G = Gal(K/K) = Gal(Q p /K)
¯
∼
and Fr the arithmetic Frobenius which generates G/I → Z.
ˆ
I
For each = p the element Fr has a well-defined action on T (E) independent
of the choice of representative in the coset modulo I, and its characteristic polyno-
mial

f E/K (T ) = det 1 − Fr | T (E) I T
has again integral coefficients independent of and is of degree ≤ 2. Its degree
depends on whether the reduction is good, multiplicative, or additive.
¯
(3.4) Remark. Let E denote the reduction of E over k(v). Then the cardinality
# E(k(v)) ns = qf E/K (1/q). For the three cases:
2
(1) Good reduction, we have f E/K (T ) = 1 − Tr(π)T + qT .
(2) Multiplicative reduction, we have

1 − T for the split case,
f E/K (T ) =
1 + T for the nonsplit case.
(3) Additive reduction, we have f E/K (T ) = 1.
Observe the cases (1), (2), and (3) are respectively equivalent to the degree of
f E/K (T ) beingofdegree2,1,and 0.

§4. Elliptic Curves over the Real Numbers

We recall the notation for 3(8.5) for a parametrization of the family of elliptic curves
over a field K. We will eventually specialize K to the real numbers R.