Full code

The σ-function associated with the canonical (2,5)-curve,
wgtσ=-3
, has the form
σ(u;λ)=
∑
-
k
1
-3
k
3
+4
i
4
+6
i
6
+8
i
8
+10
i
10
=-3
A
{
k
3
,
k
1
,
i
4
,
i
6
i
8
,
i
10
}
,where​​
A
{
k
1
,
k
3
,
i
4
,
i
6
i
8
,
i
10
}
=
k
3
u
3
k
1
u
1
i
4
λ
4
i
6
λ
6
i
8
λ
8
i
10
λ
10
.
In section Recurrence relations the function producing
A
{
k
3
,
k
1
,
i
4
,
i
6
i
8
,
i
10
}
from a given set of indices
{
k
1
,
k
3
,
i
4
,
i
6
i
8
,
i
10
}
is defined as a solution of the recurrence relations obtained from the heat equations which annihilate the σ-function.
The function σ25Canon[n], defined in section Definition of σ-function series, returns the partial sum of the series representation of the σ-function associated with the canonical (2,5)-curve up to the weight
-n
.
Next, in chapter Canonical
(2,5)
-curve the partial sums with
n=13,23
of the series for the σ-function associated with the canonical curve

(2,5)
are computed as an example, and verified to satisfy the fundamental cubic relations, and the main identities for 4-index ℘-functions associated with

(2,5)
.
In chapter
(2,5)
-Curve with term
μ
2
4
x
the partial sums with
n=5,9,13,23
of the series for the σ-function associated with a more general (2,5)-curve


(2,5)
are obtained. The σ-function associated with


(2,5)
is obtained from the σ-function associated with

(2,5)
by applying the transformation from

(2,5)
to


(2,5)
. The partial sums satisfy the fundamental cubic relations, and the main identities for 4-index ℘-functions associated with


(2,5)
.
In chapter
(2,6)
-Curve from (2,5)-curve the σ-function associated with a (2,6)-curve

(2,6)
is obtained from the σ-function associated with


(2,5)
by applying the transformation from


(2,5)
to

(2,6)
. Partial sums with
n=5,9,13
of the series for the σ-function associated with

(2,6)
are computed. The partial sums satisfy the identities for 4-index ℘-functions associated with

(2,6)
.

Recurrence relations


Definition of σ-function series (canonical curve)


Canonical
(2,5)
-curve


(2,5)
:f(x,y;λ)≡-
2
y
+
5
x
+
λ
4
3
x
+
λ
6
2
x
+
λ
8
x+
λ
10
=0​​Theseriesexpansionforσ(u;λ),u=(
u
1
,
u
3
),associatedwith

(2,5)
isgivenbelow.
For verification we use the fundamental cubic relations and main identities for 4-index ℘-functions. The latter are defined in the standard way.
℘
{i,j}
(u)=-
2
∂
∂
u
i
∂
u
j
logσ(u),
℘
{i,j,k}
(u)=-
3
∂
∂
u
i
∂
u
j
∂
u
k
logσ(u),etc.

σ : First terms

In[]:=
(*Rationallimit*)​​σ25RatLim=σ25Canon[3]
Out[]=
-
3
u
1
3
+
u
3
(*wgt
u
i
=-i*)​​σ25CanonSer13=Seriesσ25Canon[13]/.
u
i_
:>
i
t
u
i
,{t,0,13}
Out[]=
-
3
u
1
3
+
u
3
3
t
+-
7
u
1
λ
4
1260
-
1
12
4
u
1
u
3
λ
4
7
t
+
9
u
1
λ
6
5670
-
1
90
6
u
1
u
3
λ
6
-
1
6
3
u
1
2
u
3
λ
6
+
1
6
3
u
3
λ
6
9
t
+-
17
11
u
1
2
λ
4
1663200
-
8
u
1
u
3
2
λ
4
10080
+
11
u
1
λ
8
24948
-
8
u
1
u
3
λ
8
1260
-
1
30
5
u
1
2
u
3
λ
8
-
1
6
2
u
1
3
u
3
λ
8
11
t
+-
67
13
u
1
λ
4
λ
6
48648600
-
10
u
1
u
3
λ
4
λ
6
37800
-
7
u
1
2
u
3
λ
4
λ
6
2520
-
1
72
4
u
1
3
u
3
λ
4
λ
6
+
13
u
1
λ
10
347490
+
10
u
1
u
3
λ
10
5670
-
1
126
7
u
1
2
u
3
λ
10
-
1
18
4
u
1
3
u
3
λ
10
-
1
3
u
1
4
u
3
λ
10
13
t
+
14
O[t]
In[]:=
σ25CanonSer23=Seriesσ25Canon[23]/.
u
i_
:>
i
t
u
i
,{t,0,23}

Verification

◼
  • Fundamental cubic relations
    • 2
    ℘
    {1,1,1}
    =4
    3
    ℘
    {1,1}
    +4
    ℘
    {1,1}
    ℘
    {1,3}
    +4
    ℘
    {3,3}
    +4
    λ
    4
    ℘
    {1,1}
    +4
    λ
    6
    ,​​
    ℘
    {1,1,1}
    ℘
    {1,1,3}
    =4
    2
    ℘
    {1,1}
    ℘
    {1,3}
    +2
    2
    ℘
    {1,3}
    -2
    ℘
    {1,1}
    ℘
    {3,3}
    +2
    λ
    4
    ℘
    {1,3}
    +2
    λ
    8
    ,​​
    2
    ℘
    {1,1,3}
    =4
    ℘
    {1,1}
    2
    ℘
    {1,3}
    -4
    ℘
    {1,3}
    ℘
    {3,3}
    +4
    λ
    10
    In[]:=
    C25CanonFundCubRels=4
    λ
    6
    +4
    λ
    4
    ℘
    {1,1}
    +4
    3
    ℘
    {1,1}
    +4
    ℘
    {1,1}
    ℘
    {1,3}
    +4
    ℘
    {3,3}
    -
    2
    ℘
    {1,1,1}
    ,2
    λ
    8
    +2
    λ
    4
    ℘
    {1,3}
    +4
    2
    ℘
    {1,1}
    ℘
    {1,3}
    +2
    2
    ℘
    {1,3}
    -2
    ℘
    {1,1}
    ℘
    {3,3}
    -
    ℘
    {1,1,1}
    ℘
    {1,1,3}
    ,4
    λ
    10
    +4
    ℘
    {1,1}
    2
    ℘
    {1,3}
    -4
    ℘
    {1,3}
    ℘
    {3,3}
    -
    2
    ℘
    {1,1,3}
    ;
    In[]:=
    C25CanonFundCubRels/.
    ℘
    {i_,j_}
    -
    -i-j
    t
    D[Log[σ25CanonSer13],
    u
    i
    ,
    u
    j
    ],
    ℘
    {i_,j_,k_}
    -
    -i-j-k
    t
    D[Log[σ25CanonSer13],
    u
    i
    ,
    u
    j
    ,
    u
    k
    ]//Factor
    Out[]=
    {
    5
    O[t]
    ,
    3
    O[t]
    ,
    1
    O[t]
    }
    In[]:=
    C25CanonFundCubRels/.
    ℘
    {i_,j_}
    -
    -i-j
    t
    D[Log[σ25CanonSer23],
    u
    i
    ,
    u
    j
    ],
    ℘
    {i_,j_,k_}
    -
    -i-j-k
    t
    D[Log[σ25CanonSer23],
    u
    i
    ,
    u
    j
    ,
    u
    k
    ]//Factor
    Out[]=
    {
    15
    O[t]
    ,
    13
    O[t]
    ,
    11
    O[t]
    }
    ◼
  • Identities for 4-index ℘-functions
    

    • (2,5)
    -Curve with term
    μ
    2
    4
    x

    
    
    (2,5)
    :
    
    f
    (x,y;μ)≡-
    2
    y
    +
    5
    x
    +
    μ
    2
    4
    x
    +
    μ
    4
    3
    x
    +
    μ
    6
    2
    x
    +
    μ
    8
    x+
    μ
    10
    =0
    Theseriesexpansionfor
    
    σ
    (
    
    u
    ;μ),
    
    u
    =(
    
    u
    1
    ,
    
    u
    3
    ),associatedwith
    
    
    (2,5)
    isgivenby​​
    
    σ
    (
    
    u
    ;μ)=exp
    1
    2
    t
    
    u
    (
    -1
    D
    C)
    
    u
    σ(A
    
    u
    ;λ(μ))
    ,where​​A=
    1
    1
    5
    μ
    2
    0
    1
    ,C=
    2
    5
    μ
    2
    1
    25
    2
    μ
    2
    -
    1
    25
    2
    μ
    2
    1
    5
    μ
    2
    μ
    4
    -
    7
    25
    2
    μ
    2
    ,D=
    1
    0
    -
    1
    5
    μ
    2
    1
    .
    Notethat
    -1
    D
    =
    t
    A
    ,dueto
    A
    B
    C
    D
    ∈Sp(,4).
    
    ℘
    {i,j}
    (
    
    u
    )=-
    2
    ∂
    ∂
    
    u
    i
    ∂
    
    u
    j
    log
    
    σ
    (
    
    u
    ),
    
    ℘
    {i,j,k}
    (
    
    u
    )=-
    3
    ∂
    ∂
    
    u
    i
    ∂
    
    u
    j
    ∂
    
    u
    k
    log
    
    σ
    (
    
    u
    ),etc.

    Transformation
    

    σ : First terms

    In[]:=
    σ25T5=Exp[QuadFormC25](σ25Canon[5]/.uC25Sub/.λToμSubs);​​σ25Ser5=Seriesσ25T5/.
    u
    i_
    :>
    i
    t
    u
    i
    ,{t,0,5}//Simplify

    Verification

    ◼
  • Fundamental cubic relations

    • Transformation to (2,5)-curve

    σ : First terms

    Verification

    ◼
  • Identities for 4-index ℘-functions

    • σ : First terms