(i)

s

i=1

{mi 2( 2

i

+ pri )} ⊗ χi = 2χ∞ ∈ Km−1F for some χ∞ ∈ Km−1F .

(ii) For each ri ∈ F we deﬁne uri := (ri +

β)2/(℘(ri)

+ b) ∈ L. We let

αi :=

t2

+ auri

(ri + β)2( 2

i

+ pri )

, mi

i

+

t

ri + β

⊗ χi ∈ KmL(X).

and we set

τ =

s

i=1

αi ∈ KmL(X).

Then iL/F (γ) = dL(τ ) − 2(T∞, iL/F (χ∞)).

(iii) We deﬁne σ := NL(X)/F

(X)

(τ ). Then

σ = iF

(X)/F

(θ ) + 2σ

for some θ ∈ Tm ⊆ KmF and σ ∈ KmF (X).

Glancing at Diagram 1.1 one sees that θ is the end result of a chase starting

with γ. In our applications we are interested in the class of γ in H0

1(X,

ν(m)) and

the resulting class class of θ in νF (m)/NL/F (νL(m)). However it is necessary for

us to compute with elements in full K-theory rather than mod 2.

Theorem 2.1 follows from the next six lemmas. We begin by noting that The-

orem 2.1 (i) is given by Lemma 1.4 so we turn to part (ii). The next lemma gives

us information about the αi.

Lemma 2.2. For αi ∈ KmL(X) as deﬁned in Theorem 2.1 and where T∞

denotes the inﬁnite point of XL with respect to the parameter t = (y + β)/x, we

have:

(i) dL(αi) =

(

(Qri ; {mi(

i

+

√

pri )}) + (T∞; {(mi 2( 2

i

+ pri ))−1})

)

⊗ χi.

(ii) NLri

(X)/Fri

(X)(iLri (X)/L(X)αi)) ≡ 2{

√

pri (y+r)/x(℘(ri)+b), mi( i+

√

pri )}⊗χi

(mod iLri

(X)/F

(Tm) + 2iLri

(X)/F (X)

KmF (X)).

Proof. The calculation is given in Section 4.

Lemma 2.2 enables us to represent the cycle iL/F (γ) ∈

p∈XL

K1L(p) as

speciﬁed in part (ii) of Theorem 2.1.

Lemma 2.3. In the notation of Theorem 2.1, if we set

τ =

s

i=1

αi ∈ KmL(X)

then iL/F (γ) = dL(τ ) − 2(T∞, iL/F (χ∞)) where

∑s

i=1

(

{(mi 2( 2

i

+ pri ))} ⊗ χi

)

=

2χ∞ for χ∞ ∈ Km−1F as described in Lemma 1.4.

s

i=1

{mi 2( 2

i

+ pri )} ⊗ χi = 2χ∞ ∈ Km−1F for some χ∞ ∈ Km−1F .

(ii) For each ri ∈ F we deﬁne uri := (ri +

β)2/(℘(ri)

+ b) ∈ L. We let

αi :=

t2

+ auri

(ri + β)2( 2

i

+ pri )

, mi

i

+

t

ri + β

⊗ χi ∈ KmL(X).

and we set

τ =

s

i=1

αi ∈ KmL(X).

Then iL/F (γ) = dL(τ ) − 2(T∞, iL/F (χ∞)).

(iii) We deﬁne σ := NL(X)/F

(X)

(τ ). Then

σ = iF

(X)/F

(θ ) + 2σ

for some θ ∈ Tm ⊆ KmF and σ ∈ KmF (X).

Glancing at Diagram 1.1 one sees that θ is the end result of a chase starting

with γ. In our applications we are interested in the class of γ in H0

1(X,

ν(m)) and

the resulting class class of θ in νF (m)/NL/F (νL(m)). However it is necessary for

us to compute with elements in full K-theory rather than mod 2.

Theorem 2.1 follows from the next six lemmas. We begin by noting that The-

orem 2.1 (i) is given by Lemma 1.4 so we turn to part (ii). The next lemma gives

us information about the αi.

Lemma 2.2. For αi ∈ KmL(X) as deﬁned in Theorem 2.1 and where T∞

denotes the inﬁnite point of XL with respect to the parameter t = (y + β)/x, we

have:

(i) dL(αi) =

(

(Qri ; {mi(

i

+

√

pri )}) + (T∞; {(mi 2( 2

i

+ pri ))−1})

)

⊗ χi.

(ii) NLri

(X)/Fri

(X)(iLri (X)/L(X)αi)) ≡ 2{

√

pri (y+r)/x(℘(ri)+b), mi( i+

√

pri )}⊗χi

(mod iLri

(X)/F

(Tm) + 2iLri

(X)/F (X)

KmF (X)).

Proof. The calculation is given in Section 4.

Lemma 2.2 enables us to represent the cycle iL/F (γ) ∈

p∈XL

K1L(p) as

speciﬁed in part (ii) of Theorem 2.1.

Lemma 2.3. In the notation of Theorem 2.1, if we set

τ =

s

i=1

αi ∈ KmL(X)

then iL/F (γ) = dL(τ ) − 2(T∞, iL/F (χ∞)) where

∑s

i=1

(

{(mi 2( 2

i

+ pri ))} ⊗ χi

)

=

2χ∞ for χ∞ ∈ Km−1F as described in Lemma 1.4.