Example 1
(3,4)-Curve
(3,4)-Curve
Vector of Riemann constants
Vector of Riemann constants
Thecurveisdefinedbytheequation-+y(4+5x+11)++3+7+16x+9=0
3
y
2
x
4
x
3
x
2
x
Periodsarepreviouslycomputed
Periods
Periods
Theta-function definition
Theta-function definition
Characteristics of order 2
Characteristics of order 2
Vanishing odd θ-functions w.r.t. v1
Vanishing odd θ-functions w.r.t.
v
1
InvMωA=Inverse[MωA];Acc=5;(*Computeθ[ε](0;τ)withoddcharacteristics*)DOddThetaList=TableDThetaChOddChars[[i]],{V1,V2,V3},τA,Acc,V1/.{V1->0,V2->0,V3->0},{i,1,Length[OddChars]};
∂
v
1
1
2
∂
v
1
Herev=uarenormalizedcoordinatesonJac()
-1
ω
[K] can not be singled out
DOddThetaList//TableForm
Out[]//TableForm=
-0.222968154117498+0.841731534511779 |
-0.436925131409659+0.752983537453354 |
0.223487042164506-0.841392488745694 |
0.437531782226707-0.752856369756969 |
0.100998947075545+0.735765867423013 |
0.591682074590233-0.448847993850597 |
-0.100929189141844-0.736002191966474 |
-0.591799854969513+0.449064426845801 |
0.058830051865581+0.058830051865580 |
0.355980169713770+0.355980169713768 |
-0.059304695492673-0.059304695492672 |
-0.355952336871781-0.355952336871779 |
-1.855765566223786-2.446096505872586 |
-3.041875842908558+0.417427010569896 |
-1.476774057918523-2.448465066750659 |
-2.775563202832327+0.687089301563201 |
1.579833231367252+1.579833231367253 |
1.844747749497915+1.844747749497916 |
-2.594274892909448-2.594274892909445 |
-2.328098906025485-2.328098906025485 |
2.607311589208755-0.266590334653883 |
0.642137005526967-1.642904368043174 |
-0.642137005526974-1.642904368043170 |
-2.607311589208756-0.266590334653879 |
0.233774661819535+1.758186654853675 |
2.527584807704605-0.541037054651932 |
-1.404701387334451-2.169843327769597 |
1.408529354880911-1.077922057596545 |
Vanishing odd θ-functions w.r.t. u1
Vanishing odd θ-functions w.r.t.
u
1
Criteria:θ[K]vanishestotheweigthedorder=(-1)(-1).
1
24
2
n
2
s
= 5 on (3, 4)-curve
In[]:=
1
24
2
n
2
s
Out[]=
5
InvMωA=Inverse[MωA];Acc=5;(*Computeθ[ε](0;τ)withoddcharacteristics*)DDUOddThetaList=TableDThetaChOddChars[[i]],InvMωA.{U1,U2,U5},τA,Acc,U1/.{U1->0,U2->0,U5->0},{i,1,Length[OddChars]};
∂
u
1
1
2
∂
u
1
∂
u
1
In[]:=
DDUOddThetaList//TableForm
Out[]//TableForm=
0.418965754002278-0.583770339901214 |
-0.236925300597791-0.925098139250597 |
0.821674654214512-0.486611680839797 |
0.116534440259777-0.709041491739669 |
-1.279823483502825+0.951473623512473 |
0.232178412600831+1.577765315212444 |
-0.906700499702771+0.773818479630065 |
0.093961777491176+1.188306366198945 |
0.725223280391731-0.725223280391731 |
-1.375624726523541+1.375624726523541 |
1.002669199078804-1.002669199078804 |
-1.008305308768233+1.008305308768232 |
0.817701755096032-1.033085932522258 |
0.152299612418370-1.308704524451464 |
0.159394540305415-0.550830270157562 |
0.276786858977162-0.502204779645302 |
-0.180208043351292+0.180208043351291 |
-1.× -15 10 -15 10 |
1.086273635588643-1.086273635588645 |
0.474982513395378-0.474982513395376 |
-0.373875966221430+0.903329443294439 |
0.362998710993960+1.549444189933544 |
0.362998710993960-1.549444189933545 |
-0.373875966221430-0.903329443294440 |
0.300457080033064+0.369108094868992 |
-0.633866501485503+1.370846135070406 |
0.521123296504262-1.417545899639046 |
-0.473454075619601+0.048543598125823 |
(*ChoosethenumberfromOddCharsofcharacteristic[K],suchthatθ[K](0;τ)=0*)DDUOddThetaVanishNumList={};Do[If[Chop[DDUOddThetaList[[i]]]==0,DDUOddThetaVanishNumList=Append[DDUOddThetaVanishNumList,i]],{i,1,Length[OddChars]}]DDUOddThetaVanishNumList//LengthKNum=DDUOddThetaVanishNumList[[1]];OddChars[[KNum]]//MatrixForm
∂
u
1
1
2
Out[]=
1
Out[]//MatrixForm=
1 2 | 0 | 1 2 |
0 | 1 2 | 1 2 |
(*Verification:theorderofvanishingofθ[K]is5*)DThetaChOddChars[[KNum]],InvMωA.{U1,U2,U5},τA,Acc,{U1,3}/.{U1->0,U2->0,U5->0}DThetaChOddChars[[KNum]],InvMωA.{U1,U2,U5},τA,Acc,{U1,5}/.{U1->0,U2->0,U5->0}
1
2
1
2
Out[]=
0.×-0.×
-15
10
-15
10
Out[]=
-26.0300270293482+26.0300270293482
Example 2
Hyperelliptic genus 4
Hyperelliptic genus 4
Vector of Riemann constants
Vector of Riemann constants
Thecurveisdefinedbytheequation-+(x+18+2)(x+16-5)(x+11-3)(x+10+)(x+4-2)(x+3-3)(x-3-3)(x-7+2)(x-13+)=0
2
y
Periodsarepreviouslycomputed
Periods
Periods
Theta-function definition
Theta-function definition
Characteristics of order 2
Characteristics of order 2
Vanishing even θ-functions w.r.t v1
Vanishing even θ-functions w.r.t
v
1
There are 10 branch points, with indices ,
S={1,2,3,4,5,6,7,8,9,0}
where 0 stands for ∞
Every characteristic is assigned by a partition ⋃ of , where is called multiplicity
ℐ
S
Half-integer characteristics consists of
| ℐ | # | ||||
0 | { i 1 i 2 i 3 i 4 |
| evennonsingular | θ[ε](0;τ)≠0 | ||
1 | { i 1 i 2 i 3 |
| oddnonsingular | θ[ε](0;τ)=0, ∂ v | ||
2 | { i 1 | 10 | evensingular | θ[ε](0;τ)=0, 2 ∂ v |
[K] is among even singular characteristics,
but can not be singled out by differentiating w.r.t. normalized coordinates v
◼
[K]
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
Vector of Riemann constants
by Julia Bernatska
Wolfram Community, STAFF PICKS, October 11, 2024
https://community.wolfram.com/groups/-/m/t/3296279
by Julia Bernatska
Wolfram Community, STAFF PICKS, October 11, 2024
https://community.wolfram.com/groups/-/m/t/3296279