by Izboldin’s Theorem.

Passing to Milnor K-theory mod 2, and using KnF/2KnF

∼

=

νF (n), we obtain

the following commutative diagram.

0 0

↓ ↓

νF (X)(m) →

p∈X

νF (p)(m − 1)

↓ ↓

νL(m) → νL(X)(m) →

q∈XL

νL(q)(m − 1) −→ νL(m − 1)

↓ ↓ ↓

νF (m) → νF

(X)

(m) →

p∈X

νF

(p)

(m − 1)

↓

H2

m+1(F

(X))

Diagram 1.2.

Aravire and Baeza have shown [AB] that the sequence

0 → νF (n)

iL/F

−→ νL(n)

NL/F

−→ νF (n)

µb

−→

Hn+1(F

) →

Hn+1(L)

is exact, where iL/F is scalar extension, NL/F is the norm, and µb : νF (n) →

Hn+1(F ) is deﬁned by µb(

df1

f1

∧ ··· ∧

dfn

fn

) = [b

df1

f1

∧ ··· ∧

dfn

fn

]. This means the

columns of Diagram 1.2 are exact. We will see in Lemma 1.7 below that the middle

row of Diagram 1.2 is exact as well, although the other rows of Diagram 1.2 need

not be. Analogous to

H1(X,

Km) we have the group

H1(X,

ν(m)) := cok

⎛

⎝νF

(X)(m) →

p∈X

νF (p)(m −

1)⎠

⎞

.

The main result of this paper is a description of

H0

1(X,

νF (m)) := ker

(

H1(X,

νF (m)) →

H1(XL,

νL(m))

)

.

We denote by π : X → PF

1

the degree 2 morphism given in aﬃne coordinates

by (x, y) → x. If N ∈ X and if π(N) corresponds to an irreducible polynomial

p(x) ∈ F [x] we shall say that N has π-degree d if d is the degree of p(x). We

label certain π-degree 2 and 1 points of the curve X as follows: Qr ↔ y = r,

Ps ↔ x = s, and Q∞ ↔ v

1

x

-adic valuation, for r, s ∈ F . Their function ﬁelds

are F (Qr) = F ( a/(℘(r) + b)), F (Ps) = F

(℘−1(b

+

as2)),

and F (Q∞) = F (

√

a).

Each are quadratic extensions of F , with those labeled by the Q’s being inseparable.

We denote pr := a/(℘(r) + b) and p∞ := a. Then for all r, including r = ∞ we

have F (Qr) = F (

√

pr). The Ps together with Q∞ comprise all π-degree 1 points

on X. The points Qr where r = ∞ have π-degree 2. (The seeming asymmetry here

is due to the arbitrary choice of the map π.)

In order to carry out calculations in Diagrams 1.1 and 1.2 it is necessary to

represent cycles in H1(X, Km) and H1(X, ν(m)) in a special way. This is given

next.

Passing to Milnor K-theory mod 2, and using KnF/2KnF

∼

=

νF (n), we obtain

the following commutative diagram.

0 0

↓ ↓

νF (X)(m) →

p∈X

νF (p)(m − 1)

↓ ↓

νL(m) → νL(X)(m) →

q∈XL

νL(q)(m − 1) −→ νL(m − 1)

↓ ↓ ↓

νF (m) → νF

(X)

(m) →

p∈X

νF

(p)

(m − 1)

↓

H2

m+1(F

(X))

Diagram 1.2.

Aravire and Baeza have shown [AB] that the sequence

0 → νF (n)

iL/F

−→ νL(n)

NL/F

−→ νF (n)

µb

−→

Hn+1(F

) →

Hn+1(L)

is exact, where iL/F is scalar extension, NL/F is the norm, and µb : νF (n) →

Hn+1(F ) is deﬁned by µb(

df1

f1

∧ ··· ∧

dfn

fn

) = [b

df1

f1

∧ ··· ∧

dfn

fn

]. This means the

columns of Diagram 1.2 are exact. We will see in Lemma 1.7 below that the middle

row of Diagram 1.2 is exact as well, although the other rows of Diagram 1.2 need

not be. Analogous to

H1(X,

Km) we have the group

H1(X,

ν(m)) := cok

⎛

⎝νF

(X)(m) →

p∈X

νF (p)(m −

1)⎠

⎞

.

The main result of this paper is a description of

H0

1(X,

νF (m)) := ker

(

H1(X,

νF (m)) →

H1(XL,

νL(m))

)

.

We denote by π : X → PF

1

the degree 2 morphism given in aﬃne coordinates

by (x, y) → x. If N ∈ X and if π(N) corresponds to an irreducible polynomial

p(x) ∈ F [x] we shall say that N has π-degree d if d is the degree of p(x). We

label certain π-degree 2 and 1 points of the curve X as follows: Qr ↔ y = r,

Ps ↔ x = s, and Q∞ ↔ v

1

x

-adic valuation, for r, s ∈ F . Their function ﬁelds

are F (Qr) = F ( a/(℘(r) + b)), F (Ps) = F

(℘−1(b

+

as2)),

and F (Q∞) = F (

√

a).

Each are quadratic extensions of F , with those labeled by the Q’s being inseparable.

We denote pr := a/(℘(r) + b) and p∞ := a. Then for all r, including r = ∞ we

have F (Qr) = F (

√

pr). The Ps together with Q∞ comprise all π-degree 1 points

on X. The points Qr where r = ∞ have π-degree 2. (The seeming asymmetry here

is due to the arbitrary choice of the map π.)

In order to carry out calculations in Diagrams 1.1 and 1.2 it is necessary to

represent cycles in H1(X, Km) and H1(X, ν(m)) in a special way. This is given

next.