in this case.
We conclude this section with a lemma that is the generalization of the Milnor
sequence for rational function fields modulo 2.
Lemma 1.7. The sequence
0 νL(m) νL(X)(m)
dL
⊕q∈XL νL(q)(m 1)
ΣN
νL(m 1) 0
is exact.
Proof. As L is algebraically closed in L(X) we have νL(m) νL(X)(m) is injective.
Suppose that θ νL(X)(m) and dL(θ) = 0 ⊕q∈XL νL(q)(m 1). If
˜
θ KmL is
a lift of θ under KmL νL(m) then
dL(˜)
θ = for some σ ⊕q∈XL Km−1L(q).
Therefore ΣN(2σ) = 2ΣN(σ) = 0 KmL and as KmL has no 2-torsion we find
ΣN(σ) = 0. As XL is rational
0 KmL KmL(X)
dL
⊕q∈XL Km−1L(q)
ΣN
−→ Km−1L 0
is exact. So there exists θ1 KmL(X) with dL(θ1) = σ. From this it follows
that
dL(˜
θ 2θ1) = 0 ⊕q∈XL Km−1L(q). Again using the exactness of the K-
theory sequence, there exists λ KmL with iL(X)/L(λ) =
˜
θ 2θ1. So we find
θ = iL(X)/L(λ) where λ is the class of λ in νL(m). This gives the exactness at
νL(X)(m).
For exactness at ⊕q∈XL νL(q)(m 1) we suppose σ ⊕q∈XL Km−1L(q) and
ΣN(σ) = for κ KmL (so, ΣN(σ) = 0 where by σ we denote the class of σ
in ⊕q∈XL νL(q)(m 1).) As ΣN : ⊕q∈XL Km−1L(q) Km−1L is surjective there
exists σ ⊕q∈XL Km−1L(q) with ΣN(σ ) = κ. Then ΣN(σ ) = 0 so there
exists θ KmL(X) with dL(θ) = σ . So dL(θ) = σ ⊕q∈XL νL(q)(m 1) and
the exactness at ⊕q∈XL νL(q)(m 1) follows. Finally, the exactness at νL(m 1) is
clear from the exactness at Km−1L which gives the lemma.
2. K-Theory Computations.
The objective of this section is to prove Theorem 2.1 below. In order to help the
reader follow the flow of the proof some of the technical aspects of the calculation
are deferred to Section 4. We set t = (y + β)/x and note that it is a parameter
for the rational field L(X). We let T∞ be the point on XL corresponding to v
1
t
on
L(X). Since (y + β)(y + β + 1) =
ax2
and x =
t/(t2
+ a) one checks that v
1
t
(x) = 1,
v
1
t
(y + β) = 0, v
1
t
(y + β + 1) = 2. This means that in the affine x-y coordinates,
T∞ = (0, β + 1) and that π(T∞) is the point given by x = 0 on PF 1 .
We denote by Tm be the subgroup of KmF generated by all symbols of the
form {u} χ where u NL/F L aNL/F L and χ Km−1F . Of course, as L/F
is quadratic, 2KmF Tm. For each r F we recall pri := a/(℘(ri) + b) and we
denote Fri = F (

pri ) and Lri = L(
√i
pri ). So the points Qri have function fields
F (Qri ) = Fri .
Theorem 2.1. After a change of variables, any class [γ] H0
1(X,
ν(m)) can
be represented by
γ =
s
i=1
(Qri ; {mi(
i
+

pri )} χi) ⊕p∈XKm−1F (p)
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