Division polynomials

(2,5)-Curve
:f(x,y)≡-
2
y
+
5
x
+
λ
2
4
x
+
λ
4
3
x
+
λ
6
2
x
+
λ
8
x+
λ
10
=0

Preliminaries


Addition and Duplication Formulae


3-Torsion criterion

We call 3-torsion divisors such divisors
D
that
3D∼2∞=O
.There exist
2g
3
-1=80
3-torsion divisors on a curve of genus 2.All 3-torsion divisors are non-special, and so have two points in the support,that is
D
2Q
=(
x
1
,
y
1
)+(
x
2
,
y
2
)=(
αQ
2
,
αQ
4
,
βQ
3
,
βQ
5
)
.
3-Torsion divisors are defined by the equality
2
D
2Q
=-
D
2Q
, which in terms of the Mumford coordinates acquires the form
αQQ
2
=
αQ
2
,
αQQ
4
=
αQ
4
,​​
βQQ
3
=-
βQ
3
,
βQQ
5
=-
βQ
5

In Mumford coordinates

In[]:=
(*Equalitieswith
α
2
,
α
4
producetwoequations*)​​α2Tors3Eqγ=(
αQQ
2
/.αQQDuplSubs)-
αQ
2
​​α4Tors3Eqγ=(
αQQ
4
/.αQQDuplSubs)-
αQ
4
Out[]=
-3
αQ
2
-
2
γ
1
+2
γ
2
Out[]=
3
2
αQ
2
-3
αQ
4
+2
αQ
2
2
γ
1
-4
αQ
2
γ
2
+
2
γ
2
+2
γ
4
-
2
γ
1
λ
2
In[]:=
(*Equalitieswith
β
3
,
β
5
aretrivial*)​​(
βQQ
3
/.βQQDuplSubs/.{αQQ->αQ}/.γInαβQDuplSubs)+
βQ
3
//Factor​​(
βQQ
5
/.βQQDuplSubs/.{αQQ->αQ}/.γInαβQDuplSubs)+
βQ
5
//Factor
Out[]=
0
Out[]=
0

In
x
-,
y
-coordinates


Polynomials in
x
1
,
x
2


In
x
-,
y
-coordinates for computation


4-Torsion criterion

We call 4-torsion divisors such divisors
D
2Q
that (i)
4
D
2Q
∼2∞=O
, and (ii)
D
2Q
generates a cyclic group on order 4. Note, the 2-torsion divisors, which satisfy (i) but not (ii), are excluded.
There exist
2g
4
-
2g
2
=240
characteristics of order 4 which produce 4-torsion divisors on a curve of genus 2, we denote this set of characteristics by . All 4-torsion divisors are non-special, that is
D
2Q
=(
x
1
,
y
1
)+(
x
2
,
y
2
)
.At the same time, in each cyclic group
C
4
generated by
D
2Q
, one divisor is 2-torsion, and can be a special one. Each 2-torsion divisor arises in 8 cyclic groups
C
4
, which involve 16 characteristics.
We split  into two parts:​

spec
=[ε]∈
-1

([2ε])∼
D
1
=(
x
QQ
,
y
QQ
)
,
card

spec
=80
, ​

non-spec
=[ε]∈
-1

([2ε])∼
D
2
=(
αQQ
2
,
αQQ
4
,
βQQ
3
,
βQQ
3
)
,
card

non-spec
=160
.
4-Torsion divisors are defined by the equality
2
D
2Q
=-2
D
2Q
, which acquires the form
y
QQ
=0,on
D
2Q
∈

spec
,or
βQQ
3
=0,
βQQ
5
=0on
D
2Q
∈

non-spec
.


non-spec
: In Mumford coordinates

In[]:=
(*On

non-spec
:
βQQ
3
=0*)​​β3Tors4Eqγ=
γ
1
(
βQQ
3
/.βQQDuplSubs/.αQQDuplSubs)//Factor;​​β3Tors4Eqγ--2
αQ
4
-
2
αQ
2
+2
αQ
2
-
γ
2
+
2
γ
1

γ
2
-
2
γ
1
+
2
γ
1
(
γ
2
-
λ
2
)+
γ
4
//Factor
Out[]=
0
In[]:=
(*On

non-spec
:
βQQ
5
=0*)​​β5Tors4Eqγ=
γ
1
(
βQQ
5
/.βQQDuplSubs/.αQQDuplSubs)//Factor;​​β5Tors4Eqγ--2
αQ
4
+(3
αQ
2
-
γ
2
)(
αQ
2
-
γ
2
)+
2
γ
1
(2
αQ
2
-
λ
2
)+2
γ
4
2
αQ
2
-
γ
2
+
2
γ
1
-
γ
6
//Factor
Out[]=
0


non-spec
: In
x
-,
y
-coordinates



non-spec
: Polynomials in
x
1
,
x
2



non-spec
: In
x
-,
y
-coordinates for computation



spec
: In Mumford coordinates

In[]:=
yQQTors4Eqγ=(
y
QQ
//.xyQQDuplSpecSubs)
Out[]=
γ
5
+
γ
3
2
αQ
2
+
2
γ
1
-
λ
2
+
γ
1
2
2
αQ
2
+
2
γ
1
-
λ
2

In[]:=
yQQTors4Eq=
y
QQ
/.xyQQDuplSpecFullSubs//Simplify
Out[]=
1
32

5
DβQ
3
+8
2
DβQ
3
βQ
3
+4
3
DβQ
3
(5
αQ
2
-2
λ
2
)+32(2
αQ
2
βQ
3
+
βQ
5
-
βQ
3
λ
2
)+16
DβQ
3
6
2
αQ
2
+
αQ
4
-5
αQ
2
λ
2
+
2
λ
2



spec
: In
x
-,
y
-coordinates



spec
: Polynomials in
x
1
,
x
2


Example

Given curve


Auxiliary functions


Example: 3-torsion points

◼
  • Characteristics of order 3
    
    • ◼
  • 3-Torsion divisors from characteristics
    
    • All 3-torsion divisors are found, see xyTors3FromEqsSubs.

    Example: 4-torsion points

    All 4-torsion divisors are found,
    see xyTors4NonSpecFromEqsSubs, xyTors4SpecFromEqsSubs.