Fordham University

Finding equations of a 1-parameter of genus 5 Riemann surfaces with 48 automorphisms

Magaard, Shaska, Shpectorov, and Völklein give tables of smooth Riemann surfaces of genus \( 3 \leq g \leq 10\) with automorphism groups \(G\) satisfying \( \# G > 4(g-1)\). Their list is based on a computer search by Breuer.

They list a 1-parameter family of genus 5 Riemann surfaces with automorphism group (48,48) in the GAP library of small groups. The quotient of any surface in this family by its automorhism group has genus zero, and the quotient morphism is branched over four points with ramification indices (2,2,2,3).

We use Magma to compute equations of one member of this family, and give a conjectural description of this family.

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 autcv10c.txt used below. 
Magma V2.20-3     Sun Mar 20 2016 09:00:56 on Fordhamwinarski [Seed = 
1106872348]
Type ? for help.  Type -D to quit.
> load "autcv10c.txt";
Loading "autcv10c.txt"
> G:=SmallGroup(48,48);
> MatrixGens,MatrixSKG,Q,C:=RunExample(G,5,[2,2,2,3]);
Set seed to 0.


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


--------------------------------------
Class |   1  2  3  4  5  6  7  8  9 10
Size  |   1  1  3  3  6  6  8  6  6  8
Order |   1  2  2  2  2  2  3  4  4  6
--------------------------------------
p  =  2   1  1  1  1  1  1  7  3  3  7
p  =  3   1  2  3  4  5  6  1  8  9  2
--------------------------------------
X.1   +   1  1  1  1  1  1  1  1  1  1
X.2   +   1 -1  1 -1  1 -1  1 -1  1 -1
X.3   +   1  1  1  1 -1 -1  1 -1 -1  1
X.4   +   1 -1  1 -1 -1  1  1  1 -1 -1
X.5   +   2 -2  2 -2  0  0 -1  0  0  1
X.6   +   2  2  2  2  0  0 -1  0  0 -1
X.7   +   3  3 -1 -1  1  1  0 -1 -1  0
X.8   +   3 -3 -1  1 -1  1  0 -1  1  0
X.9   +   3 -3 -1  1  1 -1  0  1 -1  0
X.10  +   3  3 -1 -1 -1 -1  0  1  1  0



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

[2]     Order 2       Length 1      
        Rep G.2

[3]     Order 2       Length 3      
        Rep G.4

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

[5]     Order 2       Length 6      
        Rep G.1

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

[7]     Order 3       Length 8      
        Rep G.3

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

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

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


Surface kernel generators:  [ G.2 * G.4, G.1 * G.3 * G.5, G.1 * G.2 * G.3^2, 
G.3^2 * G.4 * G.5 ]
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 ]
S_1      =[ 0, 0, 0, 0, 1, 0, 0, 0, 0, 1 ]
H^0(C,K) =[ 0, 0, 0, 0, 1, 0, 0, 0, 0, 1 ]
I_2      =[ 1, 0, 0, 0, 0, 1, 0, 0, 0, 0 ]
S_2      =[ 2, 0, 0, 0, 0, 2, 1, 1, 1, 0 ]
H^0(C,2K)=[ 1, 0, 0, 0, 0, 1, 1, 1, 1, 0 ]
I_3      =[ 0, 1, 0, 1, 2, 0, 1, 0, 0, 2 ]
S_3      =[ 0, 2, 1, 2, 3, 0, 2, 1, 1, 4 ]
H^0(C,3K)=[ 0, 1, 1, 1, 1, 0, 1, 1, 1, 2 ]
I2timesS1=[ 0, 1, 0, 1, 2, 0, 1, 0, 0, 2 ]
Is clearly not generated by quadrics? false
Matrix generators for action on H^0(C,K):
Field K Cyclotomic Field of order 48 and degree 16
[
    [ 0  1  0  0  0]
    [ 1  0  0  0  0]
    [ 0  0 -1  0  1]
    [ 0  0  0 -1 -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],

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

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

    [-1 -1  0  0  0]
    [ 1  0  0  0  0]
    [ 0  0 -1  1  1]
    [ 0  0  0  0 -1]
    [ 0  0 -1  0  1]
]
Finding quadrics:
I2 contains a 1-dimensional subspace of CharacterRow 1
Dimension 2
Multiplicity 2
[
    x_0^2 - x_0*x_1 + x_1^2,
    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 6
Dimension 4
Multiplicity 2
[
    x_0^2 - x_1^2,
    x_0*x_1 - 1/2*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
]

The output above shows that the ideal contains quadrics from two isotypical subspaces of \(S_2\). Note that the matrices and polynomials shown above are over the rational numbers.

The first isotypical subspace, which corresponds to the character \( \chi_1\) in the character table shown above, yields a polynomial of the form 

c_1*(x_0^2 - x_0*x_1 + x_1^2)+c_2*(x_2^2 + x_2*x_4 + x_3^2 - x_3*x_4 + x_4^2)
Assume that \( c_1 \) and \( c_2\) are nonzero. Then after scaling \(x_0,x_1\) and dividing, we may assume that \( c_1=c_2=1\).

The second isotypical subspace corresponds to the character \( \chi_6\) in the character table shown above. Note that the matrix surface kernel generators have a block diagonal form with blocks of size \(2 \times 2\), and \(3 \times 3\). We therefore let the first two polynomials shown here generate one copy of \(V_{6}\) and use the FindParallelBases function to find an ordered bases of the span of the last two polynomials such that the action of \(G\) is given by the same matrices relative to both ordered bases. 

> GL5K:=Parent(MatrixSKG[1]);
> MatrixG:=sub<GL5K | MatrixSKG>;
> FindParallelBases(MatrixG,[Q[2][1],Q[2][2]],[Q[2][3],Q[2][4]]);
[x_2^2 - 2*x_2*x_3 + 2*x_2*x_4 + x_3^2 - 2*x_3*x_4]
[   -2*x_2*x_3 + x_2*x_4 - x_3*x_4 + 1/2*x_4^2]
This yields polynomials of the form 
c_3*(x_0^2 - x_1^2)+c_4*(x_2^2 - 2*x_2*x_3 + 2*x_2*x_4 + x_3^2 - 2*x_3*x_4),
c_3*(x_0*x_1 - 1/2*x_1^2) + c_4*(-2*x_2*x_3 + x_2*x_4 - x_3*x_4 + 1/2*x_4^2).
Assume that \( c_3 \) is nonzero. Then after dividing by \(c_3\), we may assume that \(c_3 = 1\).

Thus we have a conjectural description of the desired 1-parameter family: \[ \begin{array}{l} x_0^2 - x_0 x_1 + x_1^2 + x_2^2 + x_2x_4 + x_3^2 - x_3 x_4 + x_4^2 \\ x_0^2 - x_1^2+x_2^2 +c_4(- 2 x_2 x_3 + 2 x_2 x_4 + x_3^2 - 2 x_3 x_4) \\ x_0 x_1 - \frac{1}{2} x_1^2 + c_4 (-2 x_2 x_3 + x_2 x_4 - x_3 x_4 +\frac{1}{2} x_4^2) \end{array} \] In the next section we show that at least two different values of \(c_{4}\) yield equations of a smooth curve with the correct automorphisms.

Checking the equations in Magma

We check that for two different values of \(c_{4}\), we obtain a smooth curve with the correct automorphisms. This implies that a general member of the pencil is a smooth curve with the correct automorphisms. However, I have not shown that the two curves studied below are not isomorphic to each other; it is possible that we have described a point in the moduli space \( \mathcal{M}_{5}\) rather than a curve in \( \mathcal{M}_5\).

The value \(c_4=1\) yields a singular curve. Next we check the value \(c_{4}=17\): 

Magma V2.20-3     Sun Mar 20 2016 09:20:53 on Fordhamwinarski [Seed = 
3413327402]
Type ? for help.  Type -D to quit.
> K:=RationalField();
> P4<x_0,x_1,x_2,x_3,x_4>:=ProjectiveSpace(K,4);
> c_4:=17;
> X:=Scheme(P4,[
> x_0^2 - x_0*x_1 + x_1^2+x_2^2 + x_2*x_4 + x_3^2 - x_3*x_4 + x_4^2,
> x_0^2 - x_1^2+c_4*(x_2^2 - 2*x_2*x_3 + 2*x_2*x_4 + x_3^2 - 2*x_3*x_4),
> x_0*x_1 - 1/2*x_1^2 + c_4*(-2*x_2*x_3 + x_2*x_4 - x_3*x_4 + 1/2*x_4^2)
> ]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
8*$.1 - 4
2
> A:=Matrix([
> [-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]
> ]);
> B:=Matrix([
> [-1, -1, 0, 0, 0],
> [ 0, 1, 0, 0, 0],
> [ 0, 0, 0, 0, -1],
> [ 0, 0, 1, -1, -1],
> [ 0, 0, -1, 0, 0]
> ]);
> C:=Matrix([
> [-1, 0, 0, 0, 0],
> [ 1, 1, 0, 0, 0],
> [ 0, 0, -1, 0, 0],
> [ 0, 0, 0, 1, 0],
> [ 0, 0, 0, -1, -1]
> ]);
> Order(A);
2
> Order(B);
2
> Order(C);
2
> Order( (A*B*C)^(-1));
3
> GL5K:=GeneralLinearGroup(5,K);
> IdentifyGroup(sub<GL5K | A,B,C>);
<48, 48>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-x_1
-x_2 - x_4
-x_3 + x_4
x_4
and inverse
-x_0
-x_1
-x_2 - x_4
-x_3 + x_4
x_4
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-x_0 + x_1
x_3 - x_4
-x_3
-x_2 - x_3
and inverse
-x_0
-x_0 + x_1
x_3 - x_4
-x_3
-x_2 - x_3
> Automorphism(X,C);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0 + x_1
x_1
-x_2
x_3 - x_4
-x_4
and inverse
-x_0 + x_1
x_1
-x_2
x_3 - x_4
-x_4
Next, we check the value \(c_{4}=5+\zeta_{6}\): 

> K<z_6>:=CyclotomicField(6);
> P4<x_0,x_1,x_2,x_3,x_4>:=ProjectiveSpace(K,4);
> c_4:=5+z_6;
> X:=Scheme(P4,[
> x_0^2 - x_0*x_1 + x_1^2+x_2^2 + x_2*x_4 + x_3^2 - x_3*x_4 + x_4^2,
> x_0^2 - x_1^2+c_4*(x_2^2 - 2*x_2*x_3 + 2*x_2*x_4 + x_3^2 - 2*x_3*x_4),
> x_0*x_1 - 1/2*x_1^2 + c_4*(-2*x_2*x_3 + x_2*x_4 - x_3*x_4 + 1/2*x_4^2)
> ]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
8*$.1 - 4
2
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-x_1
-x_2 - x_4
-x_3 + x_4
x_4
and inverse
-x_0
-x_1
-x_2 - x_4
-x_3 + x_4
x_4
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-x_0 + x_1
x_3 - x_4
-x_3
-x_2 - x_3
and inverse
-x_0
-x_0 + x_1
x_3 - x_4
-x_3
-x_2 - x_3
> Automorphism(X,C);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0 + x_1
x_1
-x_2
x_3 - x_4
-x_4
and inverse
-x_0 + x_1
x_1
-x_2
x_3 - x_4
-x_4