Fordham University

Equations of a genus 4 Riemann surface with automorphism group (72,40)

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 (72,40) 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 curve. 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 23:05:29 on Davids-MacBook-Pro-2 [Seed = 
3769253574]

+-------------------------------------------------------------------+
|       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(72,40);
> MatrixGens,MatrixSKG,Q,C:=RunExample(G,4,[2,4,6]);
Set seed to 0.


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


-----------------------------------
Class |   1  2  3  4  5  6  7  8  9
Size  |   1  6  6  9  4  4 18 12 12
Order |   1  2  2  2  3  3  4  6  6
-----------------------------------
p  =  2   1  1  1  1  5  6  4  6  5
p  =  3   1  2  3  4  1  1  7  3  2
-----------------------------------
X.1   +   1  1  1  1  1  1  1  1  1
X.2   +   1 -1  1  1  1  1 -1  1 -1
X.3   +   1  1 -1  1  1  1 -1 -1  1
X.4   +   1 -1 -1  1  1  1  1 -1 -1
X.5   +   2  0  0 -2  2  2  0  0  0
X.6   +   4  0 -2  0 -2  1  0  1  0
X.7   +   4 -2  0  0  1 -2  0  0  1
X.8   +   4  0  2  0 -2  1  0 -1  0
X.9   +   4  2  0  0  1 -2  0  0 -1



Conjugacy Classes of group G
----------------------------
[1]     Order 1       Length 1      
        Rep Id(G)

[2]     Order 2       Length 6      
        Rep G.2

[3]     Order 2       Length 6      
        Rep G.1

[4]     Order 2       Length 9      
        Rep G.3

[5]     Order 3       Length 4      
        Rep G.4 * G.5

[6]     Order 3       Length 4      
        Rep G.4

[7]     Order 4       Length 18     
        Rep G.1 * G.2 * G.3

[8]     Order 6       Length 12     
        Rep G.1 * G.5

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


Surface kernel generators:  [ G.1 * G.4^2, G.1 * G.2 * G.3 * G.4^2 * G.5^2, G.2 
* G.3 * G.4 * G.5^2 ]
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 ]
S_1      =[ 0, 0, 0, 0, 0, 1, 0, 0, 0 ]
H^0(C,K) =[ 0, 0, 0, 0, 0, 1, 0, 0, 0 ]
I_2      =[ 1, 0, 0, 0, 0, 0, 0, 0, 0 ]
S_2      =[ 1, 1, 0, 0, 0, 0, 0, 1, 1 ]
H^0(C,2K)=[ 0, 1, 0, 0, 0, 0, 0, 1, 1 ]
I_3      =[ 0, 0, 1, 0, 0, 1, 0, 0, 0 ]
S_3      =[ 0, 0, 1, 1, 1, 2, 1, 0, 1 ]
H^0(C,3K)=[ 0, 0, 0, 1, 1, 1, 1, 0, 1 ]
I2timesS1=[ 0, 0, 0, 0, 0, 1, 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  0 -1 -1]
    [ 0  0  0  1],

    [ 0  0  0  1]
    [ 0  0 -1 -1]
    [-1 -1  0  0]
    [ 1  0  0  0],

    [-1  0  0  0]
    [ 1  1  0  0]
    [ 0  0  1  1]
    [ 0  0  0 -1],

    [ 1  0  0  0]
    [ 0  1  0  0]
    [ 0  0 -1 -1]
    [ 0  0  1  0],

    [-1 -1  0  0]
    [ 1  0  0  0]
    [ 0  0  1  0]
    [ 0  0  0  1]
]
Matrix Surface Kernel Generators:
[
    [-1  0  0  0]
    [ 0 -1  0  0]
    [ 0  0  1  0]
    [ 0  0 -1 -1],

    [ 0  0 -1 -1]
    [ 0  0  0  1]
    [-1  0  0  0]
    [ 0 -1  0  0],

    [ 0  0 -1  0]
    [ 0  0  1  1]
    [ 1  1  0  0]
    [ 0 -1  0  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_2^2 - x_2*x_3 + x_3^2
]
Finding cubics:
I3 contains a 1-dimensional subspace of CharacterRow 3
Dimension 1
Multiplicity 1
[
    x_0^2*x_1 - x_0*x_1^2 + x_2^2*x_3 - x_2*x_3^2
]
I3 contains a 4-dimensional subspace of CharacterRow 6
Dimension 8
Multiplicity 2
[
    x_0^3 + x_1^3,
    x_0^2*x_1 - x_0*x_1^2 + x_1^3,
    x_0^2*x_2 - x_0*x_1*x_2 + x_1^2*x_2,
    x_0^2*x_3 - x_0*x_1*x_3 + x_1^2*x_3,
    x_0*x_2^2 - x_0*x_2*x_3 + x_0*x_3^2,
    x_1*x_2^2 - x_1*x_2*x_3 + x_1*x_3^2,
    x_2^3 + x_3^3,
    x_2^2*x_3 - x_2*x_3^2 + x_3^3
]

From the output, we see that this curve is cyclic trigonal. The canonical ideal is \[ x_0^2 - x_0 x_1 + x_1^2 + x_2^2 - x_2 x_3 + x_3^2, \quad x_0^2 x_1 - x_0 x_1^2 + x_2^2 x_3 - x_2 x_3^2 \]

Checking the equations in Magma

We check that these equations give a smooth genus 4 curve with the desired automorphisms. 
> 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_2^2 - x_2*x_3 + x_3^2,
> x_0^2*x_1 - x_0*x_1^2 + 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],
> [0,0,1,0],
> [0,0,-1,-1]
> ]);
> B:=Matrix([
> [0,0,-1,-1],
> [0,0,0,1],
> [-1,0,0,0],
> [0,-1,0,0]
> ]);
> Order(A);
2
> Order(B);
4
> Order( (A*B)^-1);
6
> GL4K:=GeneralLinearGroup(4,K);
> IdentifyGroup(sub<GL4K | A,B>);
<72, 40>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-x_1
x_2 - x_3
-x_3
and inverse
-x_0
-x_1
x_2 - x_3
-x_3
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
-x_2
-x_3
-x_0
-x_0 + x_1
and inverse
-x_2
-x_2 + x_3
-x_0
-x_1

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 5 and 6. We compute the number of fixed points of these group elements: 

> CCL:=Classes(G);
> SKG:=[ G.1 * G.4^2, G.1 * G.2 * G.3 * G.4^2 * G.5^2, G.2 
> * G.3 * G.4 * G.5^2 ];
> NumberOfFixedPoints(G,SKG,CCL[5][3]);
6
> NumberOfFixedPoints(G,SKG,CCL[6][3]);
0
Thus, class 5 yields a trigonal morphism. We compute the eigenvalues of the trigonal automorphism: 

> Eigenvalues(MatrixGens[4]*MatrixGens[5]);
{
    <z^12 - 1, 2>,
    <-z^12, 2>
}
The trigonal morphism corresponding to class five acts with eigenvalue \(\zeta_3\) on a two-dimensional subspace of the holomorphic differentials, and with eigenvalue \(\zeta_3^2\) on a two-dimensional subspace of the holomorphic differentials. Thus, in the notation of [AchterPries2007] we have \(r = 2, s=2\), so \(d_1 = 3, d_2 = 3\). Therefore, a basis of holomorphic differentials for this equation is given by \[ \{y \frac{dx}{y^2}, yx\frac{dx}{y^2}, g(x) \frac{dx}{y^2}, xg(x) \frac{dx}{y^2} \} \]

We change variables to map the quadric obtained above to the scroll quadric \(x_0x_3-x_1x_2\). This is accomplished by \[ \begin{array}{rcl} x_0 & \mapsto & x_0+\zeta_3^2 x_1\\ x_1 & \mapsto & x_2-\zeta_6^{-1} x_3\\ x_2 & \mapsto & x_2+\zeta_3^2 x_3\\ x_3 & \mapsto & -x_0 + \zeta_6^{-1} x_1 \end{array} \] This map carries the cubic generator onto \(x_1^3 - x_0^3 - x_3^3 -x_2^3\), which encodes \( y^3 x^3 - y^3 - g(x)^3 x^3-g(x)^3\). From this we obtain \( g(x) = x^3-1 \), and thus the trigonal equation is \( y^3 = (x^3-1)^2(x^3+1)\).