Full code

Created by Julia Bernatska, University of Connecticut

Curve

Branch Points and Differentials


Contour Γ

In[]:=
ΔContColor=RGBColor[0.2,0,1];​​BPPlot=Graphics[Table[{PointSize[Large],Black,Point[{Re[BPs[[k]]],Im[BPs[[k]]]}]},{k,1,Length[BPs]}],Axes->True,PlotRange{{-10,10},{-10,10}},Axes->False];​​ZeroArgΔ=ContourPlot[Arg[DiscrSF[Rex+IImx]]==0,{Rex,-10,10},{Imx,-10,10},ContourStyle{ΔContColor,Thick},PlotPoints100];​​Show[ZeroArgΔ,BPPlot,PlotRange->{{-6,3},{-4,4}}]
Out[]=

Solutions for
y


y Values


Contour Υ

In[]:=
ΥPContColor=RGBColor[{0.2,0.8,0.2}];​​ZeroArgΥp=ContourPlot[Arg[Υp[Rex+IImx]]==0,{Rex,-6,3},{Imx,-4,4},ContourStyle->ΥPContColor,PlotPoints150]
Out[]=

Density plots of
arg
y
a

In[]:=
DensArgyP1=DensityPlot[Arg[ySolP1[Rex+IImx]],{Rex,-6,2.5},{Imx,-4,4},PlotLegendsAutomatic,PlotPoints->70];​​DensArgyP2=DensityPlot[Arg[ySolP2[Rex+IImx]],{Rex,-6,2.5},{Imx,-4,4},PlotLegendsAutomatic,PlotPoints->70];​​DensArgyP3=DensityPlot[Arg[ySolP3[Rex+IImx]],{Rex,-6,2.5},{Imx,-4,4},PlotLegendsAutomatic,PlotPoints->70];
In[]:=
Show[DensArgyP1,ZeroArgΔ,ZeroArgΥp,BPPlot,PlotRange->{{-6,2.5},{-4,4}},AspectRatio->Automatic]
Out[]=
In[]:=
Show[DensArgyP2,ZeroArgΔ,ZeroArgΥp,BPPlot,PlotRange->{{-6,2.5},{-4,4}},AspectRatio->Automatic,PlotLegendsAutomatic]
Out[]=
In[]:=
Show[DensArgyP3,ZeroArgΔ,ZeroArgΥp,BPPlot,PlotRange->{{-6,2.5},{-4,4}},AspectRatio->Automatic,PlotLegendsAutomatic]
Out[]=
Thelistofconnectionsa-bfromthedensityplotsis​​
e
1
e
2
e
3
e
4
e
5
e
6
e
7
e
8
1-3
2-3
1-2
1-2
1-2
1-2
2-3
1-3

Cuts

ContoursΓandΥwithconnectionof
y
a
In[]:=
LCut=ContourPlot[Arg[DiscrSF[Rex+IImx]]==0,{Rex,-7,-4},{Imx,-0.5,0.5},ContourStyle->{Thick,Black}];​​RCut=ContourPlot[Arg[DiscrSF[Rex+IImx]]==0,{Rex,0.8,3},{Imx,-0.5,0.5},ContourStyle->{Thick,Black}];​​BPcuts=Graphics[{Thick,Black,Line[{{Re[BPs[[2]]],Im[BPs[[2]]]},{Re[BPs[[3]]],Im[BPs[[3]]]}}]}],Graphics[{Thick,Black,Line[{{Re[BPs[[4]]],Im[BPs[[4]]]},{Re[BPs[[5]]],Im[BPs[[5]]]}}]}],GraphicsBlack,Thick,Circle
1
2
Re[BPs[[6]]+BPs[[7]]],
1
2
Im[BPs[[6]]+BPs[[7]]],0.1,
1
2
Abs[BPs[[6]]-BPs[[7]]],
1
2
Pi,
3
2
Pi,Graphics[{Thick,Black,Line[{{-10,0},{Re[BPs[[1]]],Im[BPs[[1]]]}}]}],Graphics[{Thick,Black,Line[{{Re[BPs[[8]]],Im[BPs[[8]]]},{10,0}}]}];
In[]:=
Show[ZeroArgΔ,BPPlot,BPcuts,LCut,RCut,PlotRange->{{-6,3},{-4,4}}]
Out[]=

First and Second Kind Periods

Riemann surface

Riemannsurfacewithcuts(black),andcontoursΓ(blue)andΥ(green)
Continuouspath

γ
throughallbranchpoints(orange,dashed),​​and
a
simplerpathγ,obtainedbycontinuousdefirmationof

γ
(orange,solid)

First Kind Period matrices

-Periods

Vector of Riemann constants

Computation of ℘ - Functions

JIP: Example 3a

JIP: Example 3b

CITE THIS NOTEBOOK

Uniformization of a genus 3 trigonal curve​
by Julia Bernatska​
Wolfram Community, STAFF PICKS, August 20, 2024
​https://community.wolfram.com/groups/-/m/t/3252458