|
Magaard, Shaska, Shpectorov, and Völklein give tables of smooth Riemann surfaces of genus \( g \leq 10\) with automorphism groups \(G\) satisfying \( \# G > 4(g-1)\). Their list is based on a computer search by Breuer.
They list a genus 7 Riemann surface with automorphism group \( (120,34) \cong S_5\) in the GAP library of small groups. The quotient of this surface by its automorphism group has genus zero, and the quotient morphism is branched over three points with ramification indices (2,4,5). This Riemann surface is known as Bring's curve.
We use Magma to compute equations of this Riemann surfcae. The main tools are the Eichler trace formula and black-box commands in Magma for obtaining matrix generators of a representation of a finite group having a specified character.
Magma V2.21-7 Thu May 26 2016 12:21:49 on Davids-MacBook-Pro-2 [Seed =
1986397060]
+-------------------------------------------------------------------+
| This copy of Magma has been made available through a |
| generous initiative of the |
| |
| Simons Foundation |
| |
| covering U.S. Colleges, Universities, Nonprofit Research entities,|
| and their students, faculty, and staff |
+-------------------------------------------------------------------+
Type ? for help. Type -D to quit.
> load "autcv10e.txt";
Loading "autcv10e.txt"
> G:=SmallGroup(120,34);
> MatrixGens,MatrixSKG,Q,C:=RunExample(SmallGroup(120,34),4,[2,4,5]);
Set seed to 0.
Character Table of Group G
--------------------------
-----------------------------
Class | 1 2 3 4 5 6 7
Size | 1 10 15 20 30 24 20
Order | 1 2 2 3 4 5 6
-----------------------------
p = 2 1 1 1 4 3 6 4
p = 3 1 2 3 1 5 6 2
p = 5 1 2 3 4 5 1 7
-----------------------------
X.1 + 1 1 1 1 1 1 1
X.2 + 1 -1 1 1 -1 1 -1
X.3 + 4 -2 0 1 0 -1 1
X.4 + 4 2 0 1 0 -1 -1
X.5 + 5 1 1 -1 -1 0 1
X.6 + 5 -1 1 -1 1 0 -1
X.7 + 6 0 -2 0 0 1 0
Conjugacy Classes of group G
----------------------------
[1] Order 1 Length 1
Rep Id(G)
[2] Order 2 Length 10
Rep (1, 2)
[3] Order 2 Length 15
Rep (1, 2)(3, 4)
[4] Order 3 Length 20
Rep (1, 2, 3)
[5] Order 4 Length 30
Rep (1, 2, 3, 4)
[6] Order 5 Length 24
Rep (1, 2, 3, 4, 5)
[7] Order 6 Length 20
Rep (1, 2, 3)(4, 5)
Surface kernel generators: [
(1, 2),
(1, 3, 5, 4),
(1, 4, 5, 3, 2)
]
Is hyperelliptic? false
Is cyclic trigonal? false
Multiplicities of irreducibles in relevant G-modules:
I_1 =[ 0, 0, 0, 0, 0, 0, 0 ]
S_1 =[ 0, 0, 1, 0, 0, 0, 0 ]
H^0(C,K) =[ 0, 0, 1, 0, 0, 0, 0 ]
I_2 =[ 1, 0, 0, 0, 0, 0, 0 ]
S_2 =[ 1, 0, 0, 1, 1, 0, 0 ]
H^0(C,2K)=[ 0, 0, 0, 1, 1, 0, 0 ]
I_3 =[ 0, 1, 1, 0, 0, 0, 0 ]
S_3 =[ 0, 1, 2, 0, 0, 1, 1 ]
H^0(C,3K)=[ 0, 0, 1, 0, 0, 1, 1 ]
I2timesS1=[ 0, 0, 1, 0, 0, 0, 0 ]
Is clearly not generated by quadrics? true
Matrix generators for action on H^0(C,K):
Field K Cyclotomic Field of order 120 and degree 32
[
[ 0 -1 -1 0]
[ 1 -1 -1 0]
[-1 1 1 1]
[ 0 0 -1 -1],
[-1 0 0 0]
[ 0 -1 0 0]
[ 0 0 -1 -1]
[ 0 0 0 1]
]
Matrix Surface Kernel Generators:
[
[-1 0 0 0]
[ 0 -1 0 0]
[ 0 0 -1 -1]
[ 0 0 0 1],
[ 1 -1 -1 0]
[ 1 0 0 0]
[ 0 -1 0 0]
[ 0 0 -1 -1],
[ 0 -1 0 0]
[ 0 0 1 1]
[ 1 -1 -1 -1]
[-1 1 1 0]
]
Finding quadrics:
I2 contains a 1-dimensional subspace of CharacterRow 1
Dimension 1
Multiplicity 1
[
x_0^2 + x_0*x_1 + x_1^2 - x_1*x_2 + x_2^2 - x_2*x_3 + x_3^2
]
Finding cubics:
I3 contains a 1-dimensional subspace of CharacterRow 2
Dimension 1
Multiplicity 1
[
x_0^2*x_1 + x_0*x_1^2 + x_1^2*x_2 - x_1*x_2^2 + x_2^2*x_3 - x_2*x_3^2
]
I3 contains a 4-dimensional subspace of CharacterRow 3
Dimension 8
Multiplicity 2
[
x_0^3 - x_3^3,
x_0^2*x_1 + x_0*x_1^2 + x_1*x_2*x_3 - x_1*x_3^2 - x_2^2*x_3 + x_2*x_3^2 +
1/3*x_3^3,
x_0^2*x_2 + x_0*x_2^2 - x_0*x_2*x_3 + x_0*x_3^2 + x_2^2*x_3 - x_2*x_3^2 +
4/3*x_3^3,
x_0^2*x_3 + x_0*x_1*x_3 + x_1^2*x_3 - x_1*x_2*x_3 + x_2^2*x_3 - x_2*x_3^2 +
x_3^3,
x_0*x_1*x_2 - x_0*x_2^2 + x_0*x_2*x_3 - x_0*x_3^2 + x_1*x_2*x_3 - x_1*x_3^2
- x_2^2*x_3 + x_2*x_3^2 - 2/3*x_3^3,
x_1^3 - 3*x_1*x_2*x_3 + 3*x_1*x_3^2 + 3*x_2^2*x_3 - 3*x_2*x_3^2 + x_3^3,
x_1^2*x_2 - x_1*x_2^2 - x_1*x_2*x_3 + x_1*x_3^2 + 2*x_2^2*x_3 - 2*x_2*x_3^2
+ 4/3*x_3^3,
x_2^3 - 3*x_2^2*x_3 + 3*x_2*x_3^2 - 2*x_3^3
]
The output above indicates that the ideal is \[ \begin{array}{l} x_0^2 + x_0 x_1 + x_1^2 - x_1 x_2 + x_2^2 - x_2 x_3 + x_3^2 \\ x_0^2 x_1 + x_0 x_1^2 + x_1^2 x_2 - x_1 x_2^2 + x_2^2 x_3 - x_2 x_3^2 \end{array} \]
> K:=RationalField();
> P3<x_0,x_1,x_2,x_3>:=ProjectiveSpace(K,3);
> X:=Scheme(P3,[
> x_0^2 + x_0*x_1 + x_1^2 - x_1*x_2 + x_2^2 - x_2*x_3 + x_3^2,
> x_0^2*x_1 + x_0*x_1^2 + x_1^2*x_2 - x_1*x_2^2 + x_2^2*x_3 - x_2*x_3^2
> ]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
6*$.1 - 3
2
> A:=Matrix([
> [-1,0,0,0],
> [0,-1,0,0],
> [0,0,-1,-1],
> [0,0,0,1]
> ]);
> B:=Matrix([
> [1,-1,-1,0],
> [1,0,0,0],
> [0,-1,0,0],
> [0,0,-1,-1]
> ]);
> Order(A);
2
> Order(B);
4
> Order( (A*B)^-1);
5
> GL4K:=GeneralLinearGroup(4,K);
> IdentifyGroup(sub<GL4K | A,B>);
<120, 34>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations :
-x_0
-x_1
-x_2
-x_2 + x_3
and inverse
-x_0
-x_1
-x_2
-x_2 + x_3
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations :
x_0 + x_1
-x_0 - x_2
-x_0 - x_3
-x_3
and inverse
-x_2 + x_3
x_0 + x_2 - x_3
-x_1 + x_2 - x_3
-x_3