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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 5 Riemann surface with automorphism group (96,195) 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 (2,4,6).
We use Magma to compute equations of this Riemann surface. 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 V2.21-4 Fri Sep 4 2015 05:16:50 on ace-math01 [Seed = 1274511259]
Type ? for help. Type -D to quit.
> load "autcv10.txt";
Loading "autcv10.txt"
> MatrixGens,MatrixSKG,Q,C:=RunExample(SmallGroup(96,195),5,[2,4,6]);
Set seed to 0.
Character Table of Group G
--------------------------
--------------------------------------------------------
Class | 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Size | 1 1 2 3 3 6 12 8 12 12 12 8 8 8
Order | 1 2 2 2 2 2 2 3 4 4 4 6 6 6
--------------------------------------------------------
p = 2 1 1 1 1 1 1 1 8 4 5 2 8 8 8
p = 3 1 2 3 4 5 6 7 1 9 10 11 3 2 3
--------------------------------------------------------
X.1 + 1 1 1 1 1 1 1 1 1 1 1 1 1 1
X.2 + 1 1 -1 1 1 -1 1 1 -1 1 -1 -1 1 -1
X.3 + 1 1 1 1 1 1 -1 1 -1 -1 -1 1 1 1
X.4 + 1 1 -1 1 1 -1 -1 1 1 -1 1 -1 1 -1
X.5 + 2 2 2 2 2 2 0 -1 0 0 0 -1 -1 -1
X.6 + 2 -2 0 -2 2 0 0 2 0 0 0 0 -2 0
X.7 + 2 2 -2 2 2 -2 0 -1 0 0 0 1 -1 1
X.8 0 2 -2 0 -2 2 0 0 -1 0 0 0 1+2*J 1-1-2*J
X.9 0 2 -2 0 -2 2 0 0 -1 0 0 0-1-2*J 1 1+2*J
X.10 + 3 3 3 -1 -1 -1 -1 0 1 1 -1 0 0 0
X.11 + 3 3 -3 -1 -1 1 -1 0 -1 1 1 0 0 0
X.12 + 3 3 3 -1 -1 -1 1 0 -1 -1 1 0 0 0
X.13 + 3 3 -3 -1 -1 1 1 0 1 -1 -1 0 0 0
X.14 + 6 -6 0 2 -2 0 0 0 0 0 0 0 0 0
Explanation of Character Value Symbols
--------------------------------------
J = RootOfUnity(3)
Conjugacy Classes of group G
----------------------------
[1] Order 1 Length 1
Rep Id(G)
[2] Order 2 Length 1
Rep G.3
[3] Order 2 Length 2
Rep G.2
[4] Order 2 Length 3
Rep G.3 * G.5
[5] Order 2 Length 3
Rep G.5
[6] Order 2 Length 6
Rep G.2 * G.5
[7] Order 2 Length 12
Rep G.1
[8] Order 3 Length 8
Rep G.4
[9] Order 4 Length 12
Rep G.1 * G.2 * G.3 * G.5
[10] Order 4 Length 12
Rep G.1 * G.5
[11] Order 4 Length 12
Rep G.1 * G.2 * G.3
[12] Order 6 Length 8
Rep G.2 * G.4^2
[13] Order 6 Length 8
Rep G.3 * G.4
[14] Order 6 Length 8
Rep G.2 * G.4
Surface kernel generators: [ G.1 * G.4^2, G.1 * G.2 * G.3 * G.5, G.2 * G.3 *
G.4^2 * G.5 * G.6 ]
Is hyperelliptic? false
Is cyclic trigonal? false
Multiplicities of irreducibles in relevant G-modules:
I_1 =[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
S_1 =[ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0 ]
H^0(C,K) =[ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0 ]
I_2 =[ 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
S_2 =[ 1, 1, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 1 ]
H^0(C,2K)=[ 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1 ]
I_3 =[ 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0 ]
S_3 =[ 0, 0, 0, 1, 0, 2, 0, 2, 1, 1, 3, 0, 2, 1 ]
H^0(C,3K)=[ 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1 ]
I2timesS1=[ 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0 ]
Is clearly not generated by quadrics? false
Matrix generators for action on H^0(C,K):
Field K Cyclotomic Field of order 96 and degree 32
[
[ 0 -z^16 + 1 0 0 0]
[ z^16 0 0 0 0]
[ 0 0 0 1 1]
[ 0 0 1 0 1]
[ 0 0 0 0 -1],
[ 1 0 0 0 0]
[ 0 -1 0 0 0]
[ 0 0 -1 0 0]
[ 0 0 0 -1 0]
[ 0 0 0 0 -1],
[-1 0 0 0 0]
[ 0 -1 0 0 0]
[ 0 0 1 0 0]
[ 0 0 0 1 0]
[ 0 0 0 0 1],
[z^16 - 1 0 0 0 0]
[ 0 -z^16 0 0 0]
[ 0 0 0 1 1]
[ 0 0 -1 0 -1]
[ 0 0 0 -1 0],
[ 1 0 0 0 0]
[ 0 1 0 0 0]
[ 0 0 -1 0 0]
[ 0 0 0 -1 0]
[ 0 0 1 1 1],
[ 1 0 0 0 0]
[ 0 1 0 0 0]
[ 0 0 0 1 0]
[ 0 0 1 0 0]
[ 0 0 -1 -1 -1]
]
Matrix Surface Kernel Generators:
[
[ 0 z^16 0 0 0]
[-z^16 + 1 0 0 0 0]
[ 0 0 1 0 0]
[ 0 0 0 -1 0]
[ 0 0 -1 0 -1],
[ 0 -z^16 + 1 0 0 0]
[ -z^16 0 0 0 0]
[ 0 0 -1 0 -1]
[ 0 0 0 -1 -1]
[ 0 0 1 1 1],
[ z^16 0 0 0 0]
[ 0 z^16 - 1 0 0 0]
[ 0 0 -1 -1 -1]
[ 0 0 0 0 -1]
[ 0 0 0 1 1]
]
Finding quadrics:
I2 contains a 1-dimensional subspace of CharacterRow 1
Dimension 1
Multiplicity 1
[
x_2^2 - x_2*x_4 + x_3^2 - x_3*x_4 + x_4^2
]
I2 contains a 2-dimensional subspace of CharacterRow 5
Dimension 4
Multiplicity 2
[
x_0^2,
x_1^2,
x_2^2 - x_2*x_4 + x_3^2 - x_3*x_4 - 1/2*x_4^2,
x_2*x_3 - 1/2*x_2*x_4 - 1/2*x_3*x_4 + 1/4*x_4^2
]
This shows that the ideal contains quadrics from two isotypical subspaces of \(S_2\). Note that the power of z\(=\zeta_{96}\) in the matrix surface kernel generators and in the equations is always a multiple of 16. Therefore in the sequel we reduce these to z_6\(= \zeta_{6}\).
The first isotypical subspace corresponds to the character \( \chi_{1}\) in the character table shown above. It yields the candidate polynomial \[ x_2^2 - x_2 x_4 + x_3^2 - x_3*x_4 + x_4^2 \]
The second isotypical subspace corresponds to the character \( \chi_{5}\) in the character table shown above. Note that the matrices generating the action have a block form with one \( 2 \times 2 \) block in the upper left and one \( 3 \times 3\) block in the lower right. We therefore let \( \operatorname{Span}\langle x_2^2 - x_2 x_4 + x_3^2 - x_3 x_4 - 1/2 x_4^2, x_2 x_3 - 1/2 x_2 x_4 - 1/2 x_3 x_4 + 1/4 x_4^2 \rangle \) generate one copy of \(V_{5}\) and use the FindParallelBases function to find an ordered basis of \( \operatorname{Span}\langle x_0^2,x_1^2 \rangle \) such that the action of \(G\) is given by the same matrices relative to both ordered bases.> GL5K:=Parent(MatrixSKG[1]);
> MatG:=sub;
> FindParallelBases(MatG,[Q[2][3],Q[2][4]],[Q[2][1],Q[2][2]]);
[ x_0^2 - z^16*x_1^2]
[1/6*(-2*z^16 + 1)*x_0^2 + 1/6*(-z^16 + 2)*x_1^2] Magma V2.20-3 Fri Mar 18 2016 10:04:49 on Fordham-David-Swinarski [Seed =
1775449141]
Type ? for help. Type -D to quit.
> K<z_6>:=CyclotomicField(6);
> P4<x_0,x_1,x_2,x_3,x_4>:=ProjectiveSpace(K,4);
> X:=Scheme(P4,[x_2^2 - x_2*x_4 + x_3^2 - x_3*x_4 + x_4^2,
> x_0^2 - z_6*x_1^2+x_2^2 - x_2*x_4 + x_3^2 - x_3*x_4 - 1/2*x_4^2,
> 1/6*(-2*z_6 + 1)*x_0^2 + 1/6*(-z_6 + 2)*x_1^2+ x_2*x_3 - 1/2*x_2*x_4 -
> 1/2*x_3*x_4 + 1/4*x_4^2]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
8*$.1 - 4
2
> A:=Matrix([
> [0,z_6,0,0,0],
> [-z_6+1,0,0,0,0],
> [0,0,1,0,0],
> [0,0,0,-1,0],
> [0,0,-1,0,-1]
> ]);
> B:=Matrix([
> [0,-z_6+1, 0, 0, 0],
> [-z_6, 0, 0, 0,0],
> [0,0, -1, 0, -1],
> [0, 0, 0, -1,-1],
> [0,0, 1, 1, 1]
> ]);
> Order(A);
2
> Order(B);
4
> Order(A*B);
6
> GL5K:=GeneralLinearGroup(5,K);
> IdentifyGroup(sub<GL5K | A,B>);
<96, 195>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations :
(-z_6 + 1)*x_1
z_6*x_0
x_2 - x_4
-x_3
-x_4
and inverse
(-z_6 + 1)*x_1
z_6*x_0
x_2 - x_4
-x_3
-x_4
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations :
-z_6*x_1
(-z_6 + 1)*x_0
-x_2 + x_4
-x_3 + x_4
-x_2 - x_3 + x_4
and inverse
z_6*x_1
(z_6 - 1)*x_0
x_3 - x_4
x_2 - x_4
x_2 + x_3 - x_4