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 = 2σ 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) = 2κ 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 − 2σ ) = 0 so there

exists θ ∈ KmL(X) with dL(θ) = σ − 2σ . 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)

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 = 2σ 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) = 2κ 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 − 2σ ) = 0 so there

exists θ ∈ KmL(X) with dL(θ) = σ − 2σ . 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)