Klein's Fundamental 2-Form of Second Kind for the $C_{ab}$ Curves

0.35 MB·

by  Joe Suzuki

Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 017, 13 pages
Klein’s Fundamental 2-Form of Second Kind
for the C Curves
ab
Joe SUZUKI
Department of Mathematics, Osaka University, Machikaneyama Toyonaka,
Osaka 560-0043, Japan
E-mail: [email protected]
Received January 05, 2017, in final form March 11, 2017; Published online March 16, 2017
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https://doi.org/10.3842/SIGMA.2017.017
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0
2
Abstract. In this paper, we derive the exact formula of Klein’s fundamental 2-form of

r second kind for the so-called C curves. The problem was initially solved by Klein in the
a ab
M 19th century for the hyper-elliptic curves, but little progress had been seen for its extension
formorethan100years. Recently, ithasbeenaddressedbyseveralauthors, andwassolved

7 for subclasses of the C curves whereas they found a way to find its individual solution
ab
1 numerically. Theformulagivesastandardcohomologicalbasisforthecurves,andhasmany

applications in algebraic geometry, physics, and applied mathematics, not just analyzing
]
G sigma functions in a general way.
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Key words: C curves; Klein’s fundamental 2-form of second kind; cohomological basis;
ab
. symmetry
h
t
a 2010 Mathematics Subject Classification: 14H42; 14H50; 14H55
m
[

2 1 Introduction
v
1
3 Since Abel, Jacobi, Poincar´e, and Riemann established its framework, theories of Abelian and
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modular functions associated with algebraic curves have been of crucial importance in algebraic
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0 geometry, physics, and applied mathematics. Algebraic curves in this paper are intended as
. compact Riemann surfaces.
1
0 The study of the hyper-elliptic curves goes back to the beginning of the 20th century, and
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these appear in much detail in advanced text-books such as Baker [2] and Cassels and Flynn [9].
1
: However, little has been considered for more general curves than the hyper-elliptic curves. In
v
this paper, we consider a class of curves (C curves) in the form
i ab
X
ar (cid:88)ci,jxiyj = 0, (1.1)
i,j
where (i,j) range over ai+by ≤ ab, 0 ≤ i ≤ b, 0 ≤ j ≤ a for some mutually prime integers a, b,
and c are constants such that (1.1) implies (cid:80)ic xi−1yj (cid:54)= 0 or (cid:80)jc xiyj−1 (cid:54)= 0.
i,j i,j i,j
i,j i,j
This paper studies cohomologies of C curves in terms of which the Klein–Weierstrass con-
ab
struction of multivariate Abelian and sigma functions is made possible. It does not seek further
theories but focuses on the description of cohomologies of these curves.
According to the Riemann–Roch theorem, the entire holomorphic differentials make a vector
space over C of dimension g. For example, we can take them as
xi−1dx
du (x,y) = , (1.2)
i
2y
2 J. Suzuki
i = 1,2,...,g for the hyper-elliptic curve
2g+1
(cid:88)
y2 = c xi (1.3)
i
i=0
with c = c , c = 0, i = 0,...,2g+1.
i,0 i i,1
We consider a 2-form ((x,y),(z,w)) (cid:55)→ R((x,y),(z,w))dxdz over C ×C that has a (double)
pole only at (z,w) = (x,y) and is normalized:
lim R((x,y),(z,w))(x−z)2 = 1.
(z,w)→(x,y)
Such a 2-form can be written as
g
d (cid:88) dui(x,y)dri(z,w)
R((x,y),(z,w)) := Ω((x,y),(z,w))+ , (1.4)
dz dx dz
i=1
usingdifferentials{dr }g ofthesecondkindthathavepolesonlyatinfinityandameromorphic
i i=1
function Ω((x,y),(z,w)). If we further require symmetry:
R((x,y),(z,w)) = R((z,w),(x,y)),
then such an {dr } is unique modulo operation of Sp(2g,Z) and the space generated by {du }
i i
(see [5] for hyper-elliptic curves and [8, 11] for general curves). We call such a normalized
symmetric R((x,y),(z,w))dxdz Klein’s fundamental 2-form of second kind [14, 15].
For example, for the hyper-elliptic curve (1.3), such a 2-form can be obtained using the
canonical meromorphic differentials {dr }g
i i=1
2(cid:88)g−i zk
dr (z,w) = c (k+1−i) dz (1.5)
i k+1+i
2w
k=i
and 1-form
y+w
Ω((x,y),(z,w)) := dx.
2(x−z)y
It is known that the normalized symmetric bilinear R((x,y),(z,w)) can be expressed us-
ing (1.4) for C [17] and telescopic curves [1] although Ω((x,y),(z,w)) needs more extensions.
ab
However, for the canonical meromorphic differentials {dr }g , no exact formula has been given.
i i=1
For example, C and C curves and C curves were studied in [6] and [12], respectively. Re-
34 35 45
cently, the reference [7] addressed C curves with genus g. These results dealt with a specific
3,g+1
class of curves and failed to obtain the formula for general C curves with arbitrary mutually
ab
prime a, b. As a result, still many researchers are either numerically calculating or algorithmi-
cally computing {dr }g given {c } and {du }g . This paper extends (1.5) and obtains the
i i=1 i,j i i=1
general formula in a closed form for the C curves. It contains all the existing results and has
ab
many applications including algebraic expression of sigma functions [17], defining equations of
the Jacobian varieties, etc. [5, 10].
This paper is organized as follows: Section 2 sets up the holomorphic differentials {du }g
i i=1
and gives background of this paper. Section 3 states and proves the main result (formula) and
gives two typical examples both of which extend the previous cases. In the last section, we raise
an open problem.
Klein’s Fundamental 2-Form of Second Kind for the C Curves 3
ab
2 Background
Let a, b be mutually prime positive integers, and C the curve defined by
(cid:88)
F(x,y) := c xiyj = 0
i,j
i,j
with a unique point O at which the zero orders of x and y are a and b, respectively, where (i,j)
range over
D := {(i,j)|ai+bj ≤ ab, 0 ≤ i ≤ b, 0 ≤ j ≤ a},
and c are constants such that either (cid:80)ic xi−1yj (cid:54)= 0 or (cid:80)jc xiyj−1 (cid:54)= 0 for each (x,y) on
i,j i,j i,j
i,j i,j
the curve.
We consider the set of 1-forms [17]
xiyj
du = dx,
i,j ∂F(x,y)
∂y
where (i,j) range over
J(a,b) := {(i,j)|i,j ≥ 0, ai+bj ≤ ab−a−b}.
We know by the general theory that for g variable points (x ,y ),...,(x ,y ) on C, the sum
1 1 g g
of integrals from O to those g points
(cid:88) (cid:90) (xi,yi)
u = (u ) = du
i,j (i,j)∈J(a,b) i,j
O
(i,j)∈J(a,b)
fill the whole space Cg, where the weights of the variables u are ab−a(i+1)−b(j +1). If
i,j
we regard the weight of each coefficient c in (1.1) is ai+bi−ab, the weights assigned to the
i,j
differentials render F(x,y) homogeneous.
Proposition 2.1. The set J(a,b) has
(a−1)(b−1)
#J(a,b) = g =
2
elements, and the zero orders of {du } are nonnegative and different.
i,j
Definition 2.2. The 2-form R((x,y),(z,w))dxdz on C×C is called the fundamental 2-form of
the second kind if the following conditions are satisfied
1) it is symmetric: R((x,y),(z,w)) = R((z,w),(x,y)),
2) it has its only pole along the diagonal of C ×C, and
3) in the vicinity of each point, it is expanded in power series as
dt dt
xy zw
R((x,y),(z,w))dxdz = +O(1)
(t −t )2
x,y z,w
as (z,w) → (x,y), where t and t are local coordinates of points (x,y) and (z,w),
x,y z,w
respectively.
4 J. Suzuki
We shall look for a realization of R((x,y),(z,w)) in the form
G((x,y),(z,w))
R((x,y),(z,w)) = ,
(x−z)2∂F(x,y)∂F(z,w)
∂y ∂w
where G((x,y),(z,w)) is a symmetric polynomial in its variables.
Example 2.3 (hyper-elliptic curves [3, 4]). For the hyper-elliptic curve (1.3), in which a = 2
and b = 2g+1, where g is the genus of the curve C, then (1.1) expresses a hyper-elliptic curve,
the meromorphic function on C ×C
y+w
Ω((x,y),(z,w)) =
2y(x−z)
and differentials (1.2), there exists {dr }g such that (1.4) is symmetric. In fact,
i i=1
2g+1
(x−z) (cid:80) kc zk−1+2w(y+w)
d dw(x−z)+(y+w) k
Ω((x,y),(z,w)) = dz = k=1
dz 2y(x−z)2 4yw(x−z)2
2g+1 2g+1
x (cid:80) kc zk−1− (cid:80) (k−2)c zi+2c +2yw
k k 0
= k=1 k=1 . (2.1)
4yw(x−z)2
If we add
g 2g+1
(cid:88) (cid:88)
(x−z)2 xk−1 c (h−2k)zh−k−1
h
k=1 h=2k+1
to the numerator of (2.1), we obtain
G((x,y),(z,w)) = 2 (cid:88) ck2xk/2zk/2+ (cid:88) ck(cid:0)xk+21zk−21 +xk−21zk+21(cid:1)+2c0+2yw,
k: even k: odd
where
i/2
(cid:88)
−(k−2)zk +kzk−1x+(x−z)2 (k−2h)zk−h−1xh−1
h=1
(cid:98)k/2(cid:99)
(cid:88)
= −(k−2l)zk−l+1xl−1+(k−2l+2)zk−lxl +(x−z)2 (k−2h)zk−h−1xh−1
h=l
for l = 1,2,...,(cid:98)i/2(cid:99) has been applied ((cid:98)k/2(cid:99) = k/2 and (k −1)/2 when k is even and odd,
respectively), which means that by choosing
dri 2(cid:88)g+1 (h−2i)zh−i−1
(z,w) = ,
dz 2w
h=2i+1
we obtain
R((x,y),(z,w)) = R((z,w),(x,y))
and
G((x,y),(z,w))
lim R((x,y),(z,w))(x−z)2 = lim = 1.
(z,w)→(x,y) (z,w)→(x,y) 4yw
Klein’s Fundamental 2-Form of Second Kind for the C Curves 5
ab
3 The fundamental 2-form of the second kind for the C curves
ab
Hereafter,wedenoteF := ∂F(z,w),F := ∂F(z,w),H := F(y,z)−F(w,z),H := ∂H,andH := ∂H.
z ∂z w ∂w y−z z ∂z w ∂w
We find symmetric R((x,y),(z,w)) for the meromorphic function
H
Ω((x,y),(z,w)) := .
(x−z)F
y
To this end, if we note dw = −Fz, we have
dz Fw
dΩ (H +H dw)(x−z)+H (H F −H F )(x−z)+HF
= z wdz = z w w z w. (3.1)
dz (x−z)2F (x−z)2F F
y y w
j−1
Let g := (cid:80) c zi, g(cid:48) := ∂gj, and h := (cid:80) wiyj−1−i for j = 0,...,a. Then, one
j i,j j ∂z j
i: 0≤ai+bj≤ab i=0
checks
a a a
F = (cid:88)jwj−1g , H = (cid:88)h g , H = (cid:88)h g(cid:48), H = (cid:88) ∂hjg . (3.2)
w j j j z j j w ∂w j
j=0 j=0 j=0 j=0
Let
(cid:18) (cid:19)
I(i,j) := −wi∂hj +jwj−1h − ∂hiwj g(cid:48)g (x−z)+(cid:0)jwj−1h −jh wi(cid:1)g g
∂w i ∂y i j i j−1 i j
a
for i,j = 0,...,a. Then, from (3.2) and that the arithmetic is modulo F(z,w) = (cid:80) wjg = 0,
j
j=0
we have
 
a a a a
(cid:88)(cid:88)I(i,j) = (cid:88)−wig(cid:48)H +h g(cid:48)F − ∂hig(cid:48)(cid:88)wjgj(x−z)
i w i i w ∂y i
 
i=0 j=0 i=0 j=0
 
a a
(cid:88) (cid:88) 
+ h g F −wig jh g ,
i i w i j−1 j
 
i=0 j=0
which coincides with the numerator of (3.1).
We seek {dri,j} such that the 2-form
dz (i,j)∈J(a,b)
dΩ(z,w) (cid:88) dui,j dri,j
R((x,y),(z,w)) := +
dz dx dz
(i,j)∈J(a,b)
is symmetric, in other words, its numerator
a a
(cid:88)(cid:88) (cid:88)
I(u,v)+(x−z)2 u (x,y)r (z,w)
i,j i,j
u=0v=0 (i,j)∈J(a,b)
is symmetric, where
dr r (z,w)
i,j i,j
=
dz F
w
6 J. Suzuki
is the differential of the second kind given the differentials of the first kind
du u (x,y)
i,j i,j
=
dx F
y
with u (x,y) = xiyj.
i,j
(cid:40)
mod(α,β) mod(α,β) (cid:54)= 0,
Let mod(α,β) := α−(cid:98)α/β(cid:99)∗β, and mod(α,β) := for positive
α mod(α,β) = 0
integers α, β, and (cid:100)γ(cid:101) and (cid:98)γ(cid:99) are the smallest integer no less than γ and the largest integer
no more than γ, respectively for positive real γ. For example, mod(3,2) = mod(3,2) = 2,
2 = mod(4,2) (cid:54)= mod(4,2) = 0, etc.
Then, the following property plays an important role in the derivation of the main theorem:
Proposition 3.1. If m ≤ n, then
I(m,n)+I(n,m) = mwm−1yn−1g g(cid:48)(x−z)+mwm−1yn−1g g
m n m n
+mwn−1ym−1g(cid:48) g (x−z)+mwn−1ym−1g g
m n m n
n−1
− (cid:88) (cid:2){(n−k)g g(cid:48) +(k−m)g(cid:48) g }(x−z)+(n−m)g g (cid:3)
m n m n m n
k=m+1
×wk−1ym+n−k−1. (3.3)
Theorem 3.2. R((x,y),(z,w)) is symmetric when
(cid:88)(cid:88)(cid:88)
r (z,w) = c c u(s−i−1)zr+s−i−2wu−1 (3.4)
i,j r,u s,j+1
u≤j r s
(cid:88) (cid:88)(cid:88)
+ c c (j +1)(r−i−1)zr+s−i−2wv−1 (3.5)
r,j+1 s,v
j+1≤v r s
(cid:88) (cid:88) (cid:88)(cid:88)
+ c c {(i+1)(v−u)−(j +1−u)s
r,u s,v
u≤jj+2≤v r s
−(v−j −1)r}zr+s−i−2wu+v−j−2, (3.6)
where (r,s) range over s ≥ i+2 in D for the first term, over r ≥ i+2 in D for the second term,
and over (j +1−u)s+(v−j −1)r ≤ (i+1)(v−u) and r+s ≥ i+2 in D for the last term.
G((x,y),(z,w))
The symmetric value is given by R((x,y),(z,w)) = with
(x−z)2∂F(x,y)∂F(z,w)
∂x ∂z
a a
G((x,y),(z,w)) = (cid:88)G(12)+(cid:88)(cid:88)(cid:8)G(12)+G(34)+G(5)(cid:9),
u,u u,v u,v u,v
u=0 u=0u<v
(cid:88)(cid:88)
G(12) := u c c xrzsyu−1wv−1, (3.7)
u,v r,u s,v
r s
(cid:88)(cid:88)
G(34) := u c c xszryv−1wu−1, (3.8)
u,v r,u s,v
r s
and
v−1
G(u5,v) := − (cid:88) (cid:88)(cid:88)cr,ucs,v(cid:8)mod{(k−u)s+(v−k)r,v−u}z(cid:100)(k−u)vs−+u(v−k)r(cid:101)
k=u+1 r s
×x(cid:98)(k−u)vr−+u(v−k)s(cid:99)+mod{(k−u)r+(v−k)s,v−u}z(cid:98)(k−u)vs−+u(v−k)r(cid:99)
×x(cid:100)(k−u)vr−+u(v−k)s(cid:101)(cid:9)wk−1yu+v−k−1. (3.9)
Klein’s Fundamental 2-Form of Second Kind for the C Curves 7
ab
Proof. First of all, we prove
u(cid:88)(cid:88)c c (cid:8)t(12)(x,z)+(x−z)2∆(12)(x,z)(cid:9)wu−1yv−1 = G(12)
r,u s,v u,v
r s
for 0 ≤ u ≤ v ≤ a, where
t(12)(x,z) := szr+s−1(x−z)+zr+s
and
s−1
(cid:88)
∆(12)(x,z) := kzr+k−1xs−k−1.
k=1
In fact, we see
t(12)(x,z)+(x−z)2∆(12)(x,z)
s−l
(cid:88)
= −(s−l)zr+s−l+1xl−1+(s−l+1)zr+s−lxl +(x−z)2 kzr+k−1xs−k−1
k=1
for l = 1,...,s. In particular,
(cid:88)(cid:88)
uc c ∆(12)(x,z)yv−1wu−1
r,u s,v
r s
(cid:88) (cid:88)(cid:88)
= xiyj uc c (s−i−1)zr+s−i−2wu−1,
r,u s,j+1
(i,j)∈J(a,b) r s
where (r,s) range over s ≥ i + 1 in D. Thus, from the first and second terms in (3.3), we
obtain (3.4) and (3.7).
Similarly,
u(cid:88)(cid:88)c c (cid:8)t(34)(x,z)+(x−z)2∆(34)(x,z)(cid:9)wv−1yu−1 = G(34)
r,u s,v u,v
r s
for 0 ≤ u ≤ v ≤ a,
r−1
(cid:88)
t(34)(x,z) := rzr+s−1(x−z)+zr+s, ∆(34)(x,z) := kzs+k−1xr−k−1,
k=1
and
(cid:88)(cid:88)
uc c ∆(34)(x,z)yu−1wv−1
r,u s,v
r s
(cid:88) (cid:88)(cid:88)
= xiyj uc c (r−i−1)zr+s−i−2wv−1,
r,j+1 s,v
(i,j)∈J(a,b) r s
where (r,s) range over r ≥ i + 1 in D. Thus, from the third and fourth terms in (3.3), we
obtain (3.5) and (3.8).
On the other hand, we claim
t(5)(x,z)+(x−z)2∆(5)(x,z) = −{(p+1)(v−u)−(v−k)s−(k−u)r}zr+s−pxp
k k
−{(v−k)s+(k−u)r−p(v−u)}zr+s−p−1xp+1 (3.10)
8 J. Suzuki
for 0 ≤ u ≤ v ≤ a, where
t(5)(x,z) := {s(v−k)+r(k−u)}zr+s−1(x−z)+(v−u)zr+s,
k
p
∆(5)(x,z) := (x−z)2(cid:88){(v−k)s+(k−u)r−(v−u)h}zr+s−h−1xh−1,
k
h=1
where p is a unique integer such that
p(v−u) ≤ (v−k)s+(k−u)r < (p+1)(v−u). (3.11)
In fact, we see
t(5)(x,z)+(x−z)2∆(5)(x,z) = −{s(v−k)+r(k−u)−l(v−u)}zr+s−l+1xl−1
k k
+{s(v−k)+r(k−u)−(l−1)(v−u)}zr+s−lxl
p
(cid:88)
+(x−z)2 {(v−k)s+(k−u)r−(v−u)h}zr+s−h−1xh−1
h=l
for l = 1,2,...,p+1.
Similarly, we obtain
t(5) (x,z)+(x−z)2∆(5) (x,z) = −{(q+1)(v−u)−(v−k)r−(k−u)s}zr+s−qxq
u+v−k u+v−k
−{(v−k)r+(k−u)s−q(v−u)}zr+s−q−1xq+1 (3.12)
for 0 ≤ u ≤ v ≤ a, where
t(5) (x,z) := {r(v−k)+s(k−u)}zr+s−1(x−z)+(v−u)zr+s,
u+v−k
p
∆(5) (x,z) := (x−z)2(cid:88){(v−k)r+(k−u)s−(v−u)h}zr+s−h−1xh−1,
u+v−k
h=1
where q is a unique integer such that
q(v−u) ≤ (v−k)r+(k−u)s < (q+1)(v−u). (3.13)
From (3.11) and (3.13), we have two possibilities: p+q = r+s and p+q+1 = r+s. For
the former case, (3.10) and (3.12) are −(v−u)zqxp and −(v−u)zpxq, respectively; and for the
latter case, (3.10) and (3.12) are
−{(v−k)s+(k−u)r−q(v−u)}zq+1xp+{(v−k)s+(k−u)r−p(v−u)}zqxp+1
and
−{(v−k)r+(k−u)s−p(v−u)}zp+1xq +{(v−k)r+(k−u)s−q(v−u)}zpxq+1,
respectively.
Since
(cid:22) (cid:23) (cid:22) (cid:23)
(v−k)s+(k−u)r (v−k)r+(k−u)s
p = , q = ,
v−u v−u
(v−k)s+(k−u)r−p(v−u) = mod{(v−k)s+(k−u)r,v−u},
(v−k)r+(k−u)s−q(v−u) = mod{(v−k)r+(k−u)s,v−u},
Klein’s Fundamental 2-Form of Second Kind for the C Curves 9
ab
we have (3.9), which means we obtain (3.6) as well
v−1
(cid:88) (cid:88)(cid:88)c c ∆(5)(x,z)yu+v−k−1wk−1
r,u s,v k
k=u+1 r s
(cid:88) (cid:88)(cid:88)
= xiyj uc c {(i+1)(v−u)−(j +1−u)s
r,j+1 s,v
(i,j)∈J(a,b) r s
−(v−j −1)r}zr+s−i−2wv−1,
wherewehavechosenforeach(i,j) ∈ J(a,b),i = h−1andj = u+v−k−1,sothat1 ≤ i+1 ≤ p
and u+1 ≤ u+v−j −1 ≤ v−1. Hence, (u,v) and (s,r) need to satisfy
u ≤ j, j +2 ≤ v, (i+1)(v−u) ≥ (v−k)s+(k−u)r,
respectively. This completes the proof. (cid:4)
Corollary 3.3.
lim R((z,w),(x,y))(x−z)2 = 1.
(z,w)→(x,y)
Proof. Let f := (cid:80) c xr. Then, G(12), G(34), and G(5) converge to uf f , uf f ,
u r,u u,v u,v u,v u v u v
0≤ar+bu≤ab,r≥0
and −(v−u)(v−u−1)yu+v−2, respectively. Thus, G((x,y),(z,w)) converges to
a a a
(cid:88) (cid:88)(cid:88) (cid:88)(cid:88)
uf2y2u−2+ 2uf f yu+v−2− (v−u)(v−u−1)f f yu+v−2
u u v u v
u=0 u=0u<v u=0u<v
a a
= (cid:88)uf2y2u−2+(cid:88)(cid:88)(cid:8)(u+v)−(u−v)2(cid:9)f f yu+v−2
u u v
u=0 u=0u<v
a a
= 1 (cid:88)(cid:88)(cid:8)u+v−u2−v2+2uv(cid:9)f f yu+v−2
u v
2
u=0v=0
a a a a (cid:40) a (cid:41)2
(cid:88)(cid:88) (cid:88) (cid:88) (cid:88)
= uvf f yu+v−2 = uf yu−1 vf yv−1 = uf yu−1 = F2,
u v u v u y
u=0v=0 u=1 v=1 u=1
where
a a a a
(cid:88)(cid:88) (cid:88) (cid:88)
uf f yu+v−2 = yu−2 f yv = 0
u v v
u=0v=0 u=1 v=0
and
a a a a a a
(cid:88)(cid:88) (cid:88)(cid:88) (cid:88)(cid:88)
vf f yu+v−2 = u2f f yu+v−2 = v2f f yu+v−2 = 0
u v u v u v
u=0v=0 u=0v=0 u=0v=0
have been applied. On the other hand, the denominator of R((z,w),(x,y))(x−z)2 converges
to F2 as well. This completes the proof. (cid:4)
y
Example 3.4 (generalized hyper-elliptic curves). For the curve F(x,y) = y2+yf +f = 0,
1 0
G((x,y),(z,w)) = 2yw+f w+g y+f g
1 1 1 1
(cid:88) (cid:0) r+1 r−1 r−1 r+1(cid:1) (cid:88) r r
− cr,0 z 2 x 2 +z 2 x 2 − 2cr,0z2x2,
r: odd r: even
10 J. Suzuki
when
(cid:88) (cid:88)
r (z,w) = c c (r−i−1)zr+s−i−2
i,0 r,1 s,1
r≥i+2s≥0
2i+2
(cid:88) (cid:88)
+ c (r−i−1)zr−i−2+ c (2i+2−r)
r,1 r,0
r≥i+2 r=i+2
and converges to
2y2+2f y+f2+2(cid:0)f y+y2(cid:1) = (2y+f )2 = F2
1 1 1 1 y
as (z,w) → (x,y).
Example 3.5 (cyclic curves, super-elliptic curves [13]). For the curve F(x,y) = ya+f = 0,
0
a−1
G((x,y),(z,w)) = −(cid:88)cr,0(cid:88)(cid:8)mod((a−k)r,a)z(cid:100)(a−ak)r(cid:101)x(cid:98)kar(cid:99)
r k=1
+mod(kr,a)z(cid:98)(a−ak)r(cid:99)x(cid:100)kar(cid:101)}wk−1ya−k−1,
when
b
(cid:88)
r (z,w) = − c (ar−a−r−ai−rj)zr−2−iwa−2−j
i,j r,0
r=i+2
and converges to
−2f ya−2 = F2
0 y
as (z,w) → (x,y).
Example 3.6 (trigonal curves [6, 7]). For the curve F(x,y) = y3 −q(x)y −p(x) with p(x) =
g+1 (cid:98)2g+2(cid:99)
− (cid:80) c xr and q(x) = − (cid:80)3 c xr,
r,0 r,1
r=0 r=0
(cid:88) (cid:88) (cid:88) (cid:88)
G((x,y),(z,w)) = c xr c zs+ c xrw2+ c zsy2+3y2w2
r,1 s,1 r,1 s,1
r s r s
−(cid:88)c (cid:8)mod(2r,3)z(cid:100)2r/3(cid:101)x(cid:98)r/3(cid:99)+mod(r,3)z(cid:98)2r/3(cid:99)x(cid:100)r/3(cid:101)(cid:9)y
r,0
r
−(cid:88)c (cid:8)mod(r,3)z(cid:100)r/3(cid:101)x(cid:98)2r/3(cid:99)+mod(2r,3)z(cid:98)r/3(cid:99)x(cid:100)2r/3(cid:101)(cid:9)w
r,0
r
−(cid:88)c (cid:8)mod(r,2)z(cid:100)r/2(cid:101)x(cid:98)r/2(cid:99)+mod(r,2)z(cid:98)r/2(cid:99)x(cid:100)r/2(cid:101)(cid:9)wy
r,1
r
(cid:88) (cid:88) (cid:88) (cid:88)
= c xr c zs+ c xrw2+ c zsy2+3y2w2
r,1 s,1 r,1 s,1
r s r s
− (cid:88) c (cid:8)3z2r/3xr/3y+3z2r/3xr/3w(cid:9)
r,0
r=3m
− (cid:88) c (cid:8)2z(2r+1)/3x(r−1)/3y+z(2r−2)/3x(r+2)/3y
r,0
r=3m+1
+z(r+2)/3x(2r−2)/3w+2z(r−1)/3x(2r+1)/3w(cid:9)