of PF 1 corresponding to the irreducible polynomial x2 − (℘(r) + b)/a, and therefore

are split points.

We study summands of δ one at a time. There are two steps.

Step 1. For the ﬁrst step we assume N is a non-split point with π(N) = N cor-

responding to p(x) where the degree of p(x) is d ≥ 2. By Lemma 4.2, we know

that when d is even, K1F (N) = K1F (N)[y] is generated by elements of the form

{f0(x) + yf1(x)} where deg(f0(x)) ≤ d

2

and deg(f1(x)) ≤ d

2

− 1, and when d

is odd we have such generators with deg(f0(x)) ≤

d−1

2

and deg(f1(x)) ≤

d−1

2

.

We call such generators low-degree generators. Because F (N)/F (N) is quadratic,

Km−1F (N) = K1F (N) ⊗ Km−2F (N). Let η = {f0(x) + yf1(x)} ⊗ η0 where

{f0(x) + yf1(x)} is a low-degree generator and η0 ∈ Km−2F (N) is a sum symbols

{g1,j(x), g2,j (x), . . . gm−2,j (x)} with each gi,j (x) ∈ F [x] having degree less than d.

Such η generate Km−1F (N) and for such an (N; η) we can express

dF ({p(x)} ⊗ {f0(x) + yf1(x)} ⊗ η0) = (N; η) +

i

(Si; γSi ) +

j

(Ti, γTi )

where the Si ∈ X are points with π(Si) corresponding to irreducible factors of

f0(x)2

+ f0(x)f1(x) +

(abx2

+

b)f1(x)2

and the Ti ∈ X are points with π(Ti) corre-

sponding to irreducible factors of the gi,j (x). This means the π-degrees of each Ti

are less than d.

Now,

deg(f0(x)2 +f0(x)f1(x)+(abx2 +b)f1(x)2)

≤ d when d is even and ≤ d+1

when d is odd. Moreover, if qi(x) is one of its irreducible factors of degree d or d +1

then we have that h = f0(x) · f1(x)

−1

∈ F [x]/(qi(x)) satisﬁes ℘(h) = b + abx2.

So Si corresponding to such a qi(x) is a split point. This observation shows that

when N is a non-split point of degree d ≥ 2, we can rewrite any (N, η) modulo the

image im(dF : KmF (x) →

p∈X

Km−1F (p)) as a sum of elements with support

over split points of degree at most d when d is even or at most degree d + 1 when

d is odd, together with other points of π-degree strictly less than d.

Step 2. Now suppose that S is a split point on X of π-degree d ≥ 2. This means

that π(S) = S corresponds to an irreducible polynomial p(x) and

℘−1(b

+

abx2)

∈

F [x]/(p(x)) = F (S). Since a, b]] = 0 ∈ WqF (S), by Springer’s Theorem it must

happen that the degree of p is even, say d = 2d0. We suppose r(x) ∈ F [x] and

℘(r(x)) = b +

abx2

∈ F (S). Let Ve ⊂ F [x] be the (e + 1)-dimensional subspace

of polynomials of degree at most e and view Ve ⊂ F (S). Consider the linear map

T : Vd0 → F (S) deﬁned by T (f(x)) = r(x) · f(x). By dimension count there exists

nonzero f(x) ∈ Vd0 with T (f(x)) = g(x) ∈ Vd0−1. Then as p(x) |

f(x)2

+f(x)g(x)+

g(x)2(b

+

ax2)

and

deg(f(x)2

+ f(x)g(x) +

g(x)2(b

+

ax2))

≤ 2d0 = d we see that

p(x) =

λ(f(x)2

+ f(x)g(x) +

g(x)2(b

+

ax2))

for some scalar λ ∈ F .

As F (S) = F (S), for η ∈ Km−1F (S) we can express η as a sum of a product of

symbols {hi(x)} where by Lemma 4.1 we can assume hi(x) ∈ F [x] and deg(hi(x))

d/2. If we let η ∈ Km−1F (x) denote the same sum of symbols but where the hi(x)

are lifted to the polynomials hi(x), we then ﬁnd that

d({f(x) + g(x)y} ⊗ η) = (S; η) +

j

(Uj ; θj ))

are split points.

We study summands of δ one at a time. There are two steps.

Step 1. For the ﬁrst step we assume N is a non-split point with π(N) = N cor-

responding to p(x) where the degree of p(x) is d ≥ 2. By Lemma 4.2, we know

that when d is even, K1F (N) = K1F (N)[y] is generated by elements of the form

{f0(x) + yf1(x)} where deg(f0(x)) ≤ d

2

and deg(f1(x)) ≤ d

2

− 1, and when d

is odd we have such generators with deg(f0(x)) ≤

d−1

2

and deg(f1(x)) ≤

d−1

2

.

We call such generators low-degree generators. Because F (N)/F (N) is quadratic,

Km−1F (N) = K1F (N) ⊗ Km−2F (N). Let η = {f0(x) + yf1(x)} ⊗ η0 where

{f0(x) + yf1(x)} is a low-degree generator and η0 ∈ Km−2F (N) is a sum symbols

{g1,j(x), g2,j (x), . . . gm−2,j (x)} with each gi,j (x) ∈ F [x] having degree less than d.

Such η generate Km−1F (N) and for such an (N; η) we can express

dF ({p(x)} ⊗ {f0(x) + yf1(x)} ⊗ η0) = (N; η) +

i

(Si; γSi ) +

j

(Ti, γTi )

where the Si ∈ X are points with π(Si) corresponding to irreducible factors of

f0(x)2

+ f0(x)f1(x) +

(abx2

+

b)f1(x)2

and the Ti ∈ X are points with π(Ti) corre-

sponding to irreducible factors of the gi,j (x). This means the π-degrees of each Ti

are less than d.

Now,

deg(f0(x)2 +f0(x)f1(x)+(abx2 +b)f1(x)2)

≤ d when d is even and ≤ d+1

when d is odd. Moreover, if qi(x) is one of its irreducible factors of degree d or d +1

then we have that h = f0(x) · f1(x)

−1

∈ F [x]/(qi(x)) satisﬁes ℘(h) = b + abx2.

So Si corresponding to such a qi(x) is a split point. This observation shows that

when N is a non-split point of degree d ≥ 2, we can rewrite any (N, η) modulo the

image im(dF : KmF (x) →

p∈X

Km−1F (p)) as a sum of elements with support

over split points of degree at most d when d is even or at most degree d + 1 when

d is odd, together with other points of π-degree strictly less than d.

Step 2. Now suppose that S is a split point on X of π-degree d ≥ 2. This means

that π(S) = S corresponds to an irreducible polynomial p(x) and

℘−1(b

+

abx2)

∈

F [x]/(p(x)) = F (S). Since a, b]] = 0 ∈ WqF (S), by Springer’s Theorem it must

happen that the degree of p is even, say d = 2d0. We suppose r(x) ∈ F [x] and

℘(r(x)) = b +

abx2

∈ F (S). Let Ve ⊂ F [x] be the (e + 1)-dimensional subspace

of polynomials of degree at most e and view Ve ⊂ F (S). Consider the linear map

T : Vd0 → F (S) deﬁned by T (f(x)) = r(x) · f(x). By dimension count there exists

nonzero f(x) ∈ Vd0 with T (f(x)) = g(x) ∈ Vd0−1. Then as p(x) |

f(x)2

+f(x)g(x)+

g(x)2(b

+

ax2)

and

deg(f(x)2

+ f(x)g(x) +

g(x)2(b

+

ax2))

≤ 2d0 = d we see that

p(x) =

λ(f(x)2

+ f(x)g(x) +

g(x)2(b

+

ax2))

for some scalar λ ∈ F .

As F (S) = F (S), for η ∈ Km−1F (S) we can express η as a sum of a product of

symbols {hi(x)} where by Lemma 4.1 we can assume hi(x) ∈ F [x] and deg(hi(x))

d/2. If we let η ∈ Km−1F (x) denote the same sum of symbols but where the hi(x)

are lifted to the polynomials hi(x), we then ﬁnd that

d({f(x) + g(x)y} ⊗ η) = (S; η) +

j

(Uj ; θj ))