(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)
). Then
σ = iF
(X)/F
) +
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.
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