Full code

Curve

Branch Points and Differentials

In[]:=
genus=4;​​BPs={-18-2I,-16+5I,-11+3I,-10-I,-4+2I,-3+3I,3+3I,7-2I,13-I};​​Poly[x_]:=Collect[Product[x-BPs[[k]],{k,1,2genus+1}],x]​​fS=-
2
y
+Poly[x];​​f=
9
x
-
2
y
+
8
x
λ
2
+
7
x
λ
4
+
6
x
λ
6
+
5
x
λ
8
+
4
x
λ
10
+
3
x
λ
12
+
2
x
λ
14
+x
λ
16
+
λ
18
;​​λSubs=Solve[0==CoefficientList[f-fS,x]][[1]]
Out[]=
{
λ
2
39-10,
λ
4
217-288,
λ
6
-7585+826,
λ
8
-79138+82462,
λ
10
324058+455846,
λ
12
4126332-3930980,
λ
14
-14219032-29444932,
λ
16
-131012592+28208616,
λ
18
-101860560+245519280}
In[]:=
(*Monomialswhichgeneratefirstandsecondkinddifferentials*)​​Dif1K=Table[
i
x
,{i,0,genus-1}]//Reverse​​Dif2K=
4
x
,3
5
x
+2
4
x
λ
2
+
3
x
λ
4
,5
6
x
+4
5
x
λ
2
+3
4
x
λ
4
+2
3
x
λ
6
+
2
x
λ
8
,7
7
x
+6
6
x
λ
2
+5
5
x
λ
4
+4
4
x
λ
6
+3
3
x
λ
8
+2
2
x
λ
10
+x
λ
12
/.λSubs
Out[]=
{
3
x
,
2
x
,x,1}
Out[]=
{
4
x
,(217-288)
3
x
+(78-20)
4
x
+3
5
x
,(-79138+82462)
2
x
-(15170-1652)
3
x
+(651-864)
4
x
+(156-40)
5
x
+5
6
x
,(4126332-3930980)x+(648116+911692)
2
x
-(237414-247386)
3
x
-(30340-3304)
4
x
+(1085-1440)
5
x
+(234-60)
6
x
+7
7
x
}

Cuts

In[]:=
BPPlot=Graphics[Table[{PointSize[Large],Point[{Re[BPs[[k]]],Im[BPs[[k]]]}]},{k,1,Length[BPs]}],Axes->True,PlotRange{{-20,20},{-15,15}}];​​BPcutsM=Table[Graphics[{Thick,Line[{{Re[BPs[[2k-1]]],Im[BPs[[2k-1]]]},{Re[BPs[[2k]]],Im[BPs[[2k]]]}}]}],{k,1,genus}];​​BPcuts=Join[BPcutsM,{Graphics[{Thick,Line[{{Re[BPs[[2genus+1]]],Im[BPs[[2genus+1]]]},{20,-0.4}}]}]}];​​ZeroArg=ContourPlot[Arg[Poly[Rex+IImx]]==0,{Rex,-25,20},{Imx,-15,15},ContourStyle->{Thick,RGBColor[0.2,0,1]}];​​LCut=ContourPlot[Arg[Poly[Rex+IImx]]==0,{Rex,Re[BPs[[-1]]],20},{Imx,Im[BPs[[-1]]]-5,Im[BPs[[-1]]]+5},ContourStyle->{Thick,Black}];​​​​CutPlot=Show[BPPlot,ZeroArg,BPcutsM,LCut,PlotRange->{{-23,20},{-10,10}}]
Out[]=

First and Second Kind Periods

First Kind Integrals

In[]:=
PolyRoot[x_]:=
Abs[Poly[x]]
Exp[IIf[Arg[Poly[x]]<0,Arg[Poly[x]]/2+Pi,Arg[Poly[x]]/2]];​​dU=
Dif1K
-2PolyRoot[x]
;
In[]:=
AMapMInf=NIntegrate[dU,{x,-∞,BPs[[1]]},WorkingPrecision18];​​AMapSet=Table[NIntegrate[dU,{x,BPs[[i]],BPs[[i+1]]},WorkingPrecision18],{i,1,Length[BPs]-1}];​​AMapPInf=NIntegrate[dU,{x,BPs[[-1]],∞},WorkingPrecision18];​​AMapSign={-1,-1,-1,1,-1,-1,-1,1};
In[]:=
(ClpsC1=Sum[AMapSet[[2i-1]]AMapSign[[2i-1]],{i,1,genus}]+AMapPInf)//Chop​​(ClpsC2=AMapMInf+Sum[AMapSet[[2i]]AMapSign[[2i]],{i,1,genus}])//Chop
Out[]=
{0,0,0,0}
Out[]=
{0,0,0,0}
In[]:=
Chop[ClpsC1,
-17
10
]​​Chop[ClpsC1,
-18
10
]​​Chop[ClpsC2,
-16
10
]​​Chop[ClpsC2,
-17
10
]
Out[]=
{0,0,0,0}
Out[]=
{8.×
-18
10
+2.×
-18
10
,0,1.06×
-18
10
+1.45×
-18
10
,0}
Out[]=
{0,0,0,0}
Out[]=
{7.×
-17
10
-6.×
-17
10
,0,0,0}

Second Kind Integrals


First Kind Period Matrices


Second Kind Period Matrices


Legendre Relations


Verification

In[]:=
(*ThefunctionwhichcomputesapointoftheJacobianvarietyfromachracteristicCh*)​​Char2U[Ch_]:=Module[{tCh=Mod[Ch,2]},Sum[tCh[[2,i]]MωA[[All,i]],{i,1,genus}]+Sum[tCh[[1,i]]MωPA[[All,i]],{i,1,genus}]]​​​​InvMωA=Inverse[MωA];​​Acc=5;

Theta Function


Bolza Formula


Computation of ℘ - Functions

KCh=
1
1
1
1
0
1
0
1
;

℘ - Functions


Jacobi inversion problem on branch points


Jacobi inversion problem on arbitrary points


JIP: Example 1a

◼
  • Points
    
    • ◼
  • Entire rational functions from coordinates
    
    • ◼
  • Abel image of D and ℘-functions
    • All
    P
    i
    arechosenwithsigns=+1​​Signsofy=±
    (x)
    are{1,-1,-1,-1,1,-1,-1,-1,1,1}.Then
    sign:
    real
    onSheet1
    P
    1
    +
    +
    P
    2
    +
    +
    P
    3
    +
    -
    P
    4
    +
    +
    In[]:=
    uP1=AMapMInf+Sum[AMapSet[[i]]AMapSign[[i]],{i,1,2}]+NIntegrate[dU,{x,BPs[[3]],xPs[[1]]},WorkingPrecision18];​​uP2=AMapMInf+Sum[AMapSet[[i]]AMapSign[[i]],{i,1,3}]+NIntegrate[dU,{x,BPs[[4]],xPs[[2]]},WorkingPrecision18];​​uP3=-(AMapMInf+Sum[AMapSet[[i]]AMapSign[[i]],{i,1,5}]-NIntegrate[dU,{x,BPs[[6]],xPs[[3]]},WorkingPrecision18]);​​uP4=AMapMInf+Sum[AMapSet[[i]]AMapSign[[i]],{i,1,6}]+NIntegrate[dU,{x,BPs[[7]],xPs[[4]]},WorkingPrecision18];​​uP=uP1+uP2+uP3+uP4
    Out[]=
    {-1.18274997428733205+0.20563451583301787,0.073714132420250436-0.038375141967611938,-0.0047619330249629105+0.0026738674813980292,0.00059219099437290674-0.00006414985341453584}

    JIP: Example 1b

    CITE THIS NOTEBOOK

    Uniformization of a genus 4 hyperelliptic curve with arbitrary complex branch points​
    by Julia Bernatska​
    Wolfram Community, STAFF PICKS, August 8, 2024
    ​https://community.wolfram.com/groups/-/m/t/3243472