In this section we record the technical details that comprise the proofs of Lem-
mas 1.3, 2.2, and 2.4. The first two lemmas have been used by other authors but
are given here for completeness.
Lemma 4.1. Suppose that p(x) F [x] is irreducible of degree d and E =
F [x]/(p(x)). Then K1E is generated by elements of the form {f(x)} where the
degrees deg(f(x))
d
2
.
Proof. Let V E be the F -subspace of all elements represented by residue classes
of f(x) where the degrees deg(f(x))
d
2
. Then dimF (V ) = d/2 + 1 if d is even
and is (d + 1)/2 if d is odd. For any nonzero element h E, let Lh : E E be
the injective F -linear map given by multiplication by h, that is Lh(g) = hg for all
g E. As dimF (E) = d and dimF (V ) d/2 we know Lh
−1(V
) V = {0}. This
means there are nonzero g1, g2 V with Lh(g1) = g2. So, in E we have h = g2/g1,
which proves the lemma.
Lemma 4.1 generalizes to quadratic extensions of F [x]/(p(x)) as follows.
Lemma 4.2. Suppose p(x) F [x] is irreducible of degree d, E1 := F [x]/(p(x)),
g(x) ℘(E1), and y =
℘−1(g(x))
E2 =
E1[℘−1(g(x))].
If d is even, then K1E2 is
generated by elements of the form {f0(x)+yf1(x)} where the degrees deg(f0(x))
d
2
and deg(f1(x))
d
2
1. If d is odd, then we have the same result except we require
deg(f0(x))
d−1
2
, deg(f1(x))
d−1
2
.
Proof. We let V E2 be the F -subspace of all elements represented by classes
of the form f0(x) + yf1(x) where f0(x) and f1(x) have degrees with bounds as
specified. Then, when d is even, dimF (V ) = (d/2 + 1) + (d/2) = d + 1 and when d
is odd, dimF (V ) = ((d 1)/2 + 1) + ((d 1)/2 + 1) = d + 1 also. As in the proof of
Lemma 4.1, for h = 0 E2, let Lh : E2 E2 be the injective F -linear map given
by multiplication by h. Since dimF (E2) = 2d we find Lh
−1
(V ) V = {0}. This
means there are nonzero g1, g2 V with Lh(g1) = g2. So, in E2 we have h = g2/g1
which proves the lemma.
We now can give the proof of Lemma 1.3.
Lemma 1.3. The group
H1(X,
Km) is generated by classes of cycles with sup-
port among the points with π-degree 1. Alternately, any class can be represented by
a cycle with support among the points Qr, where r F or Q∞. By a change of vari-
ables, we can move away from Q∞. The same applies to classes in H1(X, ν(m)).
Proof. We first show for any element δ ⊕p∈XKm−1F (p) that there exists µ
KmF (X) such that the support of δ lies above π-degree 1 points, that is,
supp(δ dF µ) {p X | π(p) PF
1
has degree 1}.
For this we consider two types of points in X, the split points and the non-split
points. If N X and if π(N) = N we say N is split if F (N) = F (N) and N is
non-split otherwise. If N is non-split then {N} =
π−1(N)
and F (N) is a quadratic
extension of F (N). In this case if vN (x) 0, we have y =
℘−1(b
+
ax2)
F (N)
and F (N)/F (N) is a separable extension. The infinite point Q∞ is a π-degree
1 non-split point in this notation because at infinity, v∞(x) = v∞(y) = −1 with
y/x =

a F (Q∞), so we have a proper inseparable quadratic extension of residue
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