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.
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