Fordham University

Equations of a genus 4 Riemann surface with automorphism group (72,42)

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 4 Riemann surface with automorphism group (72,42) 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 (2,3,12).

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.

Obtaining candidate polynomials in Magma

We use some Magma code developed by David Swinarski during a visit to the University of Sydney in June/July 2011. Here is the file autcv10e.txt used below. 
Magma V2.21-7     Sun Apr 24 2016 18:32:51 on Davids-MacBook-Pro-2 [Seed = 
1325748620]

+-------------------------------------------------------------------+
|       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(72,42);        
> MatrixGens,MatrixSKG,Q,C:=RunExample(G,4,[2,3,12]);
Set seed to 0.


Character Table of Group G
--------------------------


-------------------------------------------------------------------
Class |   1  2  3    4    5   6   7  8  9   10   11  12  13  14  15
Size  |   1  3  6    1    1   8   8  8  6    3    3   6   6   6   6
Order |   1  2  2    3    3   3   3  3  4    6    6   6   6  12  12
-------------------------------------------------------------------
p  =  2   1  1  1    5    4   7   6  8  2    5    4   4   5  10  11
p  =  3   1  2  3    1    1   1   1  1  9    2    2   3   3   9   9
-------------------------------------------------------------------
X.1   +   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  -1
X.3   0   1  1  1    J -1-J-1-J   J  1  1    J -1-J-1-J   J-1-J   J
X.4   0   1  1 -1 -1-J    J   J-1-J  1 -1 -1-J    J  -J 1+J  -J 1+J
X.5   0   1  1 -1    J -1-J-1-J   J  1 -1    J -1-J 1+J  -J 1+J  -J
X.6   0   1  1  1 -1-J    J   J-1-J  1  1 -1-J    J   J-1-J   J-1-J
X.7   +   2  2  0    2    2  -1  -1 -1  0    2    2   0   0   0   0
X.8   0   2  2  0-2-2*J  2*J  -J 1+J -1  0-2-2*J  2*J   0   0   0   0
X.9   0   2  2  0  2*J-2-2*J 1+J  -J -1  0  2*J-2-2*J   0   0   0   0
X.10  +   3 -1 -1    3    3   0   0  0  1   -1   -1  -1  -1   1   1
X.11  +   3 -1  1    3    3   0   0  0 -1   -1   -1   1   1  -1  -1
X.12  0   3 -1  1-3-3*J  3*J   0   0  0 -1  1+J   -J   J-1-J  -J 1+J
X.13  0   3 -1  1  3*J-3-3*J   0   0  0 -1   -J  1+J-1-J   J 1+J  -J
X.14  0   3 -1 -1-3-3*J  3*J   0   0  0  1  1+J   -J  -J 1+J   J-1-J
X.15  0   3 -1 -1  3*J-3-3*J   0   0  0  1   -J  1+J 1+J  -J-1-J   J


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 3      
        Rep G.4

[3]     Order 2       Length 6      
        Rep G.1

[4]     Order 3       Length 1      
        Rep G.2^2

[5]     Order 3       Length 1      
        Rep G.2

[6]     Order 3       Length 8      
        Rep G.2 * G.3

[7]     Order 3       Length 8      
        Rep G.2^2 * G.3

[8]     Order 3       Length 8      
        Rep G.3

[9]     Order 4       Length 6      
        Rep G.1 * G.4

[10]    Order 6       Length 3      
        Rep G.2^2 * G.4

[11]    Order 6       Length 3      
        Rep G.2 * G.4

[12]    Order 6       Length 6      
        Rep G.1 * G.2

[13]    Order 6       Length 6      
        Rep G.1 * G.2^2

[14]    Order 12      Length 6      
        Rep G.1 * G.2 * G.4

[15]    Order 12      Length 6      
        Rep G.1 * G.2^2 * G.4


Surface kernel generators:  [ G.1, G.2 * G.3^2 * G.4 * G.5, G.1 * G.2^2 * G.3^2 
* G.5 ]
Is hyperelliptic?  false
Is cyclic trigonal?  true
Multiplicities of irreducibles in relevant G-modules:
I_1      =[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
S_1      =[ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ]
H^0(C,K) =[ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ]
I_2      =[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
S_2      =[ 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0 ]
H^0(C,2K)=[ 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0 ]
I_3      =[ 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ]
S_3      =[ 0, 2, 0, 0, 1, 0, 0, 0, 1, 2, 1, 0, 0, 1, 1 ]
H^0(C,3K)=[ 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1 ]
I2timesS1=[ 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 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 72 and degree 24
[
    [      -1        0        0        0]
    [       0       -1        0        0]
    [       0    -z^12        0 z^12 - 1]
    [       0 z^12 - 1    -z^12        0],

    [z^12 - 1        0        0        0]
    [       0    -z^12        0        0]
    [       0        0    -z^12        0]
    [       0        0        0    -z^12],

    [        1         0         0         0]
    [        0         1  z^12 - 1         0]
    [        0      z^12        -1 -z^12 + 1]
    [        0 -z^12 + 1         0         0],

    [        1         0         0         0]
    [        0        -1 -z^12 + 1      z^12]
    [        0         0         0  z^12 - 1]
    [        0         0     -z^12         0],

    [       1        0        0        0]
    [       0        1 z^12 - 1    -z^12]
    [       0        0       -1        0]
    [       0        0        0       -1]
]
Matrix Surface Kernel Generators:
[
    [      -1        0        0        0]
    [       0       -1        0        0]
    [       0    -z^12        0 z^12 - 1]
    [       0 z^12 - 1    -z^12        0],

    [ z^12 - 1         0         0         0]
    [        0         0         1         0]
    [        0 -z^12 + 1      z^12         0]
    [        0         1         0     -z^12],

    [    z^12        0        0        0]
    [       0        0        0       -1]
    [       0       -1        0        0]
    [       0    -z^12        1 z^12 - 1]
]
Finding quadrics:
I2 contains a 1-dimensional subspace of CharacterRow 6
Dimension 1
Multiplicity 1
[
    x_1^2 + z^12*x_1*x_2 + (-z^12 + 1)*x_1*x_3 + (z^12 - 1)*x_2^2 - z^12*x_3^2
]
Finding cubics:
I3 contains a 1-dimensional subspace of CharacterRow 2
Dimension 2
Multiplicity 2
[
    x_0^3,
    x_1^2*x_2 + z^12*x_1^2*x_3 + z^12*x_1*x_2^2 + x_1*x_3^2
]
I3 contains a 1-dimensional subspace of CharacterRow 5
Dimension 1
Multiplicity 1
[
    x_0*x_1^2 + z^12*x_0*x_1*x_2 + (-z^12 + 1)*x_0*x_1*x_3 + (z^12 - 
        1)*x_0*x_2^2 - z^12*x_0*x_3^2
]
I3 contains a 3-dimensional subspace of CharacterRow 10
Dimension 6
Multiplicity 2
[
    x_1^3,
    x_1^2*x_2 + z^12*x_1*x_2^2 + 1/3*(-2*z^12 + 2)*x_3^3,
    x_1^2*x_3 + (-z^12 + 1)*x_1*x_3^2 - 2/3*z^12*x_3^3,
    x_1*x_2*x_3 + (-z^12 + 1)*x_2*x_3^2 + 1/3*x_3^3,
    x_2^3 + x_3^3,
    x_2^2*x_3 + z^12*x_2*x_3^2 + 1/3*(2*z^12 - 2)*x_3^3
]

From the output, we see that this curve is cyclic trigonal. The ideal contains the quadric \[ x_1^2 + \zeta_{6} x_1 x_2 + (-\zeta_{6} + 1) x_1 x_3 + (\zeta_6 - 1) x_2^2 - \zeta_6 x_3^2 \] The extra cubic generator is of the form \[ c_1(x_0^3) +c_2(x_1^2 x_2 + \zeta_6 x_1^2 x_3 + \zeta_6 x_1 x_2^2 + x_1 x_3^2) \] Assume that \(c_1\) and \(c_2\) are nonzero. Then after scaling \(x_0\) and dividing, we may assume that \(c_1=c_2=1\).

Checking the equations in Magma

We check that these equations give a smooth genus 4 curve with the desired automorphisms. 

> K<z_6>:=CyclotomicField(6);
> z_3:=z_6^2;
> P3<x_0,x_1,x_2,x_3>:=ProjectiveSpace(K,3);
> X:=Scheme(P3,[
> x_1^2 + z_6*x_1*x_2 + (-z_6 + 1)*x_1*x_3 + (z_6 - 1)*x_2^2 - z_6*x_3^2,
> x_0^3+x_1^2*x_2 + z_6*x_1^2*x_3 + z_6*x_1*x_2^2 + x_1*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,-z_6,0,z_3],
> [0,z_3,-z_6,0]
> ]);
> B:=Matrix([
> [z_3,0,0,0],
> [0,0,1,0],
> [0,-z_3,z_6,0],
> [0,1,0,-z_6]
> ]);
> Order(A);
2
> Order(B);
3
> Order( (A*B)^-1);
12
> GL4K:=GeneralLinearGroup(4,K);
> IdentifyGroup(sub<GL4K | A,B>);
<72, 42>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-x_1 - z_6*x_2 + (z_6 - 1)*x_3
-z_6*x_3
(z_6 - 1)*x_2
and inverse
-x_0
-x_1 - z_6*x_2 + (z_6 - 1)*x_3
-z_6*x_3
(z_6 - 1)*x_2
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
(z_6 - 1)*x_0
(-z_6 + 1)*x_2 + x_3
x_1 + z_6*x_2
-z_6*x_3
and inverse
-z_6*x_0
(-z_6 + 1)*x_1 + x_2 - z_6*x_3
z_6*x_1 + x_3
(z_6 - 1)*x_3

Cyclic trigonal approach

To study the symmetry of the branch locus, we look for a trigonal morphism. From the list of conjugacy classes above, we know that the order 3 elements belong to classes 4-8. We compute the number of fixed points of these group elements: 

> CCL:=Classes(G);
> SKG:=[ G.1, G.2 * G.3^2 * G.4 * G.5, G.1 * G.2^2 * G.3^2* G.5 ];
> NumberOfFixedPoints(G,SKG,CCL[4][3]);                           
6
> NumberOfFixedPoints(G,SKG,CCL[5][3]);
6
> NumberOfFixedPoints(G,SKG,CCL[6][3]); 
3
> NumberOfFixedPoints(G,SKG,CCL[7][3]);
3
> NumberOfFixedPoints(G,SKG,CCL[8][3]);
0
Thus, classes 4 and 5 both yield trigonal morphisms. We compute the quotient groups in each case: 

> IdentifyGroup(quo<G | CCL[4][3]>);
<24, 12>
> IdentifyGroup(quo<G | CCL[5][3]>);
<24, 12>
> Eigenvalues(MatrixGens[2]^2);
{
    <z^12 - 1, 3>,
    <-z^12, 1>
}
> Eigenvalues(MatrixGens[2]);  
{
    <-z^12, 3>,
    <z^12 - 1, 1>
}
In both cases, the quotient group is \(S_4\). The trigonal morphism corresponding to class four acts with eigenvalue \(\zeta_3\) on a three-dimensional subspace of the holomorphic differentials, and with eigenvalue \(\zeta_3^2\) on a one-dimensional subspace of the holomorphic differentials. Thus, in the notation of [AchterPries2007] we have \(r = 3, s=1\), so \(d_1 = 6, d_2 = 0\).

One locus of six points on the sphere with symmetry \(S_4\) is given by the octahedron. Thus, we might guess that a trigonal equation for this Riemann surface is \( y^3 = x(x^4-1)\). Then a basis of holomorphic differentials for this equation is given by \[ \{y \frac{dx}{y^2}, \frac{dx}{y^2}, x \frac{dx}{y^2}, x^2 \frac{dx}{y^2} \} \] Mapping these sections to the variables \(x_0,x_1,x_2,x_3\) leads to the following equations for the canonical ideal: \[ x_1 x_3-x_2^2, x_0^3-x_2 x_3^2+x_1^2 x_2. \]

Indeed, the coordinate change \[ \begin{array}{rcl} x_1 & \mapsto & \frac{1}{2}(-\zeta_{24}^5 + \zeta_{24})*(x_1+x_3),\\ x_2 & \mapsto & \frac{1}{4}(-\zeta_{24}^7 + \zeta_{24}^5 + \zeta_{24}^3)x_1+\frac{1}{2}(-\zeta_{24}^7 + \zeta_{24}^3)x_2+\frac{1}{4}(\zeta_{24}^7 + \zeta_{24}^5 - \zeta_{24}^3)x_3\\ x_3 & \mapsto & \frac{1}{4}(\zeta_{24}^7 - \zeta_{24})x_1-\frac{1}{2}\zeta_{24}^7x_2+\frac{1}{4}(-\zeta_{24}^7 - \zeta_{24})x_3 \end{array} \] together with rescaling \(x_0\), transforms the equations obtained above to these equations.