Torsion divisors

Hyperelliptic genus 4

Preliminaries

Definition of the curve 


First Kind Integrals


Periods


θ-function definition


Characteristics of branch points


Origin of Jac()


Definition of ℘-functions


Points on Jacobian from characteristics


Functions for computing Abel pre-images


Reduced divisors


2-Torsion divisors

Characteristics of order 2


Classification of 2-torsion divisors


Odd and even characteristics of order 2


2-Torsion divisors
D
1

◼
  • Abel images of branch points
    
    • ◼
  • Computation from
    
    i,i+1
    
    • ◼
  • Computation from characteristics
    
    • ◼
  • List of 2-torsion divisors
    D
    1
    • There are exist nine 2-torsion divisors of the type
    D
    1
    =
    P
    1
    +3∞
    , where
    P
    1
    is one of branch points. As follows from comparison of uBPfromCharVals and uBPVals, we have
    In[]:=
    xyD1Vals=Table[{x->BPs[[i]],y->0},{i,1,Length[uD1Vals]}];
    The list of all divisors of the type
    D
    1
    with corresponding partitions
    {
    i
    1
    }
    , and characteristics
    [ϵM]
    and
    [ϵM]-[K](mod2)
    is printed below
    In[]:=
    DoPrint{i},", [ϵM]=",MatrixForm
    1
    2
    eChars[[i]],", ","[ϵM]-[K]=",MatrixForm
    1
    2
    Mod[eChars[[i]]-2KCh,2],",
    D
    1
    =",xyD1Vals[[i]],{i,1,Length[xyD1Vals]}
    {1}, [ϵM]=
    1
    2
    0
    0
    0
    0
    0
    0
    0
    , [ϵM]-[K]=
    0
    1
    2
    1
    2
    1
    2
    0
    1
    2
    0
    1
    2
    ,
    D
    1
    ={x{x-18-2},y0}

    2-Torsion divisors
    D
    2

    ◼
  • Abel images
    u(
    D
    2
    )
    
    • ◼
  • List of 2-torsion divisors
    D
    2
    • There are exist thirty-six 2-torsion divisors of the type
    D
    2
    =
    P
    1
    +
    P
    2
    +2∞
    , where
    P
    1
    and
    P
    2
    are branch points. Using uBPVals computed previously, we find
    In[]:=
    xyD2Vals=Table[Table[{x->BPs[[j]],y->0},{j,Comb2[[i]]}],{i,1,Length[Comb2]}];
    The list of all divisors of the type
    D
    2
    with corresponding partitions
    {
    i
    1
    ,
    i
    2
    }
    , and characteristics
    [ϵM]
    and
    [ϵM]-[K](mod2)
    is printed below
    In[]:=
    DoPrintComb2[[i]],", [ϵM]=",MatrixForm
    1
    2
    D2Chars[[i]],", ","[ϵM]-[K]=",MatrixForm
    1
    2
    Mod[D2Chars[[i]]-2KCh,2],",
    D
    2
    ={",TableForm[xyD2Vals[[i]]],"}",{i,1,Length[uD2Vals]}
    {1,2}, [ϵM]=
    0
    0
    0
    0
    1
    2
    0
    0
    0
    , [ϵM]-[K]=
    1
    2
    1
    2
    1
    2
    1
    2
    1
    2
    1
    2
    0
    1
    2
    ,
    D
    2
    ={
    x{x-18-2}
    y0
    x{x-16+5}
    y0
    }

    2-Torsion divisors
    D
    3

    ◼
  • Abel images
    u(
    D
    3
    )
    
    • ◼
  • List of 2-torsion divisors
    D
    3
    • There are exist eighty-four 2-torsion divisors of the type
    D
    3
    =
    P
    1
    +
    P
    2
    +
    P
    3
    +∞
    , where
    P
    1
    ,
    P
    2
    ,
    P
    3
    are branch points. Using uBPVals computed previously, we find
    In[]:=
    xyD3Vals=Table[Table[{x->BPs[[j]],y->0},{j,Comb3[[i]]}],{i,1,Length[Comb3]}];
    The list of all divisors of the type
    D
    3
    with corresponding partitions
    {
    i
    1
    ,
    i
    2
    ,
    i
    3
    }
    , and characteristics
    [ϵM]
    and
    [ϵM]-[K](mod2)
    is printed below
    In[]:=
    DoPrintComb3[[i]],", [ϵM]=",MatrixForm
    1
    2
    D3Chars[[i]],", ","[ϵM]-[K]=",MatrixForm
    1
    2
    Mod[D3Chars[[i]]-2KCh,2],",
    D
    3
    ={",TableForm[xyD3Vals[[i]]],"}",{i,1,Length[Comb3]}
    {1,2,3}, [ϵM]=
    0
    1
    2
    0
    0
    0
    0
    0
    0
    , [ϵM]-[K]=
    1
    2
    0
    1
    2
    1
    2
    0
    1
    2
    0
    1
    2
    ,
    D
    3
    ={
    x{x-18-2}
    y0
    x{x-16+5}
    y0
    x{x-11+3}
    y0
    }

    2-Torsion divisors
    D
    4

    ◼
  • Abel images
    u(
    D
    4
    )
    
    • ◼
  • List of 2-torsion divisors
    D
    4
    • There are exist 126 2-torsion divisors of the type
    D
    3
    =
    P
    1
    +
    P
    2
    +
    P
    3
    +
    P
    4
    , where
    P
    1
    ,
    P
    2
    ,
    P
    3
    ,
    P
    4
    are branch points. Using uBPVals computed previously, we find
    In[]:=
    xyD4Vals=Table[Table[{x->BPs[[j]],y->0},{j,Comb4[[i]]}],{i,1,Length[Comb4]}];
    The first 20 divisors of the type
    D
    4
    with corresponding partitions
    {
    i
    1
    ,
    i
    2
    ,
    i
    3
    ,
    i
    4
    }
    , and characteristics
    [ϵM]
    and
    [ϵM]-[K](mod2)
    is printed below
    In[]:=
    DoPrintComb4[[i]],", [ϵM]=",MatrixForm
    1
    2
    D4Chars[[i]],", ","[ϵM]-[K]=",MatrixForm
    1
    2
    Mod[D4Chars[[i]]-2KCh,2],",
    D
    3
    ={",TableForm[xyD4Vals[[i]]],"}",{i,1,20}
    {1,2,3,4}, [ϵM]=
    0
    0
    0
    0
    1
    2
    1
    2
    0
    0
    , [ϵM]-[K]=
    1
    2
    1
    2
    1
    2
    1
    2
    1
    2
    0
    0
    1
    2
    ,
    D
    3
    ={
    x{x-18-2}
    y0
    x{x-16+5}
    y0
    x{x-11+3}
    y0
    x{x-10-}
    y0
    }
    On the other hand, non-special 2-torsion divisors can be obtained from
    u[ϵM]
    by solving the Jacobi inversion problem.
    In[]:=
    InvAbelMap[uD4Vals[[1]]]//TableForm
    Out[]//TableForm=
    x-18.0000000000-2.0000000000
    y0.×
    -5
    10
    +0.×
    -5
    10
    
    x-16.0000000000+5.0000000000
    y0.×
    -5
    10
    +0.×
    -5
    10
    
    x-11.00000000000+3.00000000000
    y0.×
    -5
    10
    +0.×
    -5
    10
    
    x-10.0000000000-1.00000000000
    y0.×
    -6
    10
    +0.×
    -5
    10
    

    3-Torsion divisors

    Characteristics of order 3

    3-Torsion divisors are Abel pre-images of
    u[ϵM]
    with characteristics of order 3, that is
    3[ϵM]∼0
    .
    In[]:=
    (*Thelistofallcharacteristicsoforder3intheform3[ϵM],and[0](inthefirstposition)*)​​Chars3ϵM=Flatten[Table[{{i1,i2,i3,i4},{j1,j2,j3,j4}},{i1,0,2},{i2,0,2},{i3,0,2},{i4,0,2},{j1,0,2},{j2,0,2},{j3,0,2},{j4,0,2}],7];​​Chars3ϵM//Length​​
    2genus
    3
    Out[]=
    6561
    Out[]=
    6561
    In[]:=
    (*Characteristic[0],whichcorrespondstou[0]=0↦{4∞}isremoved*)​​AllOrd3Chars=Delete[Chars3ϵM,1];
    In[]:=
    (*Thefirst20characteristicsoforder3*)​​TableMatrixForm
    1
    3
    AllOrd3Chars[[i]],{i,1,20}
    Out[]=
    
    0
    0
    0
    0
    0
    0
    0
    1
    3
    ,
    0
    0
    0
    0
    0
    0
    0
    2
    3
    ,
    0
    0
    0
    0
    0
    0
    1
    3
    0
    ,
    0
    0
    0
    0
    0
    0
    1
    3
    1
    3
    ,
    0
    0
    0
    0
    0
    0
    1
    3
    2
    3
    ,
    0
    0
    0
    0
    0
    0
    2
    3
    0
    ,
    0
    0
    0
    0
    0
    0
    2
    3
    1
    3
    ,
    0
    0
    0
    0
    0
    0
    2
    3
    2
    3
    ,
    0
    0
    0
    0
    0
    1
    3
    0
    0
    ,
    0
    0
    0
    0
    0
    1
    3
    0
    1
    3
    ,
    0
    0
    0
    0
    0
    1
    3
    0
    2
    3
    ,
    0
    0
    0
    0
    0
    1
    3
    1
    3
    0
    ,
    0
    0
    0
    0
    0
    1
    3
    1
    3
    1
    3
    ,
    0
    0
    0
    0
    0
    1
    3
    1
    3
    2
    3
    ,
    0
    0
    0
    0
    0
    1
    3
    2
    3
    0
    ,
    0
    0
    0
    0
    0
    1
    3
    2
    3
    1
    3
    ,
    0
    0
    0
    0
    0
    1
    3
    2
    3
    2
    3
    ,
    0
    0
    0
    0
    0
    2
    3
    0
    0
    ,
    0
    0
    0
    0
    0
    2
    3
    0
    1
    3
    ,
    0
    0
    0
    0
    0
    2
    3
    0
    2
    3
    

    Computation of 3-torsion divisors

    3-Torsion divisors are non-special. They are computed as solutions of the Jacobi inversion problem from
    u[ϵM]
    with characteristics of order 3
    (*Thefirst20ofu[ϵM]withcharacteristicsoforder3*)​​uOrd3CharVals20=TableCharToU
    1
    3
    AllOrd3Chars[[i]],{i,1,20}
    (*Thefirst20of
    -1
    
    (u[ϵM])withcharacteristicsoforder3*)​​xyOrd3CharVals20=Table[InvAbelMap[uOrd3CharVals20[[i]]],{i,1,20}];
    The first 20 of 3-torsion divisors with corresponding characteristics
    [ϵM]
    are printed below
    In[]:=
    DoPrint"[ϵM]=",MatrixForm
    1
    3
    AllOrd3Chars[[i]],", D={",TableForm[xyOrd3CharVals20[[i]]],"}",{i,1,20}
    [ϵM]=
    0
    0
    0
    0
    0
    0
    0
    1
    3
    , D={
    x3.071090236+2.943662577
    y-3498.620+2983.038
    x4.533511092+1.534640861
    y10975.117-21422.481
    x5.562309200+0.226618576
    y-6685.007+29327.046
    x6.932285412-1.888871698
    y-192.786-10998.620
    }
    Cyclic groups
    C
    3
    ​
    The collection of 3-torsion divisors, of number 6560, split into 3280 pairs
    {D,2D=-D}
    . Together with the neutral element
    {4∞}
    each pair
    {D,2D=-D}
    forms a cyclic group
    C
    3
    . Evidently, the corresponding characteristics are
    [ϵM]
    ,
    2[ϵM]
    .

    4-Torsion divisors

    Characteristics of order 4

    4-Torsion divisors are Abel pre-images of
    u[ϵM]
    with characteristics of order 4, that is
    4[ϵM]∼0
    .
    In[]:=
    Chars4ϵM=Flatten[Table[{{i1,i2,i3,i4},{j1,j2,j3,j4}},{i1,0,3},{i2,0,3},{i3,0,3},{i4,0,3},{j1,0,3},{j2,0,3},{j3,0,3},{j4,0,3}],7];​​Chars4ϵM//Length​​
    2genus
    4
    Out[]=
    65536
    The list Chars4ϵM contains characteristics of order 4 and [0], which we remove from the list.

    Computation of 4-torsion divisors

    CITE THIS NOTEBOOK

    Computation of n-torsion divisors on a genus 4 hyperelliptic curve​
    by Julia Bernatska​
    Wolfram Community, STAFF PICKS, November 1, 2024
    ​https://community.wolfram.com/groups/-/m/t/3314103