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 − dµ 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

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 − dµ 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