Fordham University

Equations of a genus 5 Riemann surface with automorphism group (192,181)

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 (192,181) 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 (2,3,8).

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 autcv10.txt used below. 
Magma V2.21-4     Wed Sep  2 2015 16:15:57 on ace-math01 [Seed = 1908636852]
Type ? for help.  Type -D to quit.
> load "autcv10.txt";
Loading "autcv10.txt"
> MatrixGens,MatrixSKG,Q,C:=RunExample(SmallGroup(192,181),5,[2,3,8]);
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  1  3  3 24 32     6     6 12 24 32  12  12  12  12
Order |   1  2  2  2  2  3     4     4  4  4  6   8   8   8   8
---------------------------------------------------------------
p  =  2   1  1  1  1  1  6     4     4  4  3  6   7   7   8   8
p  =  3   1  2  3  4  5  1     8     7  9 10  2  15  14  13  12
---------------------------------------------------------------
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   +   2  2  2  2  0 -1     2     2  2  0 -1   0   0   0   0
X.4   0   2 -2  2 -2  0 -1     0     0  0  0  1  Z1 -Z1 -Z1  Z1
X.5   0   2 -2  2 -2  0 -1     0     0  0  0  1 -Z1  Z1  Z1 -Z1
X.6   +   3  3  3  3  1  0    -1    -1 -1  1  0  -1  -1  -1  -1
X.7   +   3  3  3  3 -1  0    -1    -1 -1 -1  0   1   1   1   1
X.8   0   3  3 -1 -1  1  0-1+2*I-1-2*I  1 -1  0  -I  -I   I   I
X.9   0   3  3 -1 -1  1  0-1-2*I-1+2*I  1 -1  0   I   I  -I  -I
X.10  0   3  3 -1 -1 -1  0-1+2*I-1-2*I  1  1  0   I   I  -I  -I
X.11  0   3  3 -1 -1 -1  0-1-2*I-1+2*I  1  1  0  -I  -I   I   I
X.12  +   4 -4  4 -4  0  1     0     0  0  0 -1   0   0   0   0
X.13  +   6  6 -2 -2  0  0     2     2 -2  0  0   0   0   0   0
X.14  +   6 -6 -2  2  0  0     0     0  0  0  0  Z2 -Z2  Z2 -Z2
X.15  +   6 -6 -2  2  0  0     0     0  0  0  0 -Z2  Z2 -Z2  Z2


Explanation of Character Value Symbols
--------------------------------------

# denotes algebraic conjugation, that is,
#k indicates replacing the root of unity w by w^k

I = RootOfUnity(4)

Z1     = (CyclotomicField(8: Sparse := true)) ! [ RationalField() | 0, -1, 0, -1
]

Z2     = (CyclotomicField(8: Sparse := true)) ! [ RationalField() | 0, 1, 0, -1 
]


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

[2]     Order 2       Length 1      
        Rep G.7

[3]     Order 2       Length 3      
        Rep G.5 * G.7

[4]     Order 2       Length 3      
        Rep G.5

[5]     Order 2       Length 24     
        Rep G.1

[6]     Order 3       Length 32     
        Rep G.2

[7]     Order 4       Length 6      
        Rep G.3 * G.5 * G.6

[8]     Order 4       Length 6      
        Rep G.3

[9]     Order 4       Length 12     
        Rep G.3 * G.5

[10]    Order 4       Length 24     
        Rep G.1 * G.5 * G.6

[11]    Order 6       Length 32     
        Rep G.2 * G.7

[12]    Order 8       Length 12     
        Rep G.1 * G.3

[13]    Order 8       Length 12     
        Rep G.1 * G.3 * G.7

[14]    Order 8       Length 12     
        Rep G.1 * G.3 * G.5 * G.6

[15]    Order 8       Length 12     
        Rep G.1 * G.3 * G.5 * G.6 * G.7


Surface kernel generators:  [ G.1 * G.2^2 * G.3 * G.6, G.2^2 * G.4, G.1 * G.2 * 
G.3 * G.4 * 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, 0 ]

S_1      =[ 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ]

H^0(C,K) =[ 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ]

I_2      =[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]

S_2      =[ 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 1 ]

H^0(C,2K)=[ 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1 ]

I_3      =[ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0 ]

S_3      =[ 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 1 ]

H^0(C,3K)=[ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1 ]

I2timesS1=[ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0 ]

Is clearly not generated by quadrics? false
Matrix generators for action on H^0(C,K):
Field K Cyclotomic Field of order 192 and degree 64
[
    [1/3*(-2*z^56 + z^40 + z^24 + z^8) 1/3*(z^32 - 2) 0 0 0]
    [1/3*(-z^32 - 1) 1/3*(2*z^56 - z^40 - z^24 - z^8) 0 0 0]
    [0 0 -1 -z^48 0]
    [0 0 0 1 0]
    [0 0 0 -z^48 -1],

    [1/3*(z^32 - 2) 1/3*(-2*z^56 + z^40 + z^24 + z^8) 0 0 0]
    [1/3*(2*z^56 - z^40 - z^24 - z^8) 1/3*(-z^32 - 1) 0 0 0]
    [0 0 1 z^48 0]
    [0 0 -1 -z^48 - 1 z^48 - 1]
    [0 0 -1 0 z^48],

    [1/3*(2*z^32 - 1) 1/3*(-z^56 - z^40 - z^24 + 2*z^8) 0 0 0]
    [1/3*(z^56 - 2*z^40 - 2*z^24 + z^8) 1/3*(-2*z^32 + 1) 0 0 0]
    [0 0 -z^48 0 0]
    [0 0 1 0 -z^48]
    [0 0 -z^48 1 -z^48 - 1],

    [1/3*(2*z^32 - 1) 1/3*(2*z^56 - z^40 - z^24 - z^8) 0 0 0]
    [1/3*(-2*z^56 + z^40 + z^24 + z^8) 1/3*(-2*z^32 + 1) 0 0 0]
    [0 0 0 z^48 1]
    [0 0 -1 -z^48 - 1 z^48]
    [0 0 0 0 -z^48],

    [       -1         0         0         0         0]
    [        0        -1         0         0         0]
    [        0         0     -z^48 -z^48 + 1 -z^48 - 1]
    [        0         0  z^48 + 1      z^48 -z^48 + 1]
    [        0         0         0         0        -1],

    [        1         0         0         0         0]
    [        0         1         0         0         0]
    [        0         0      z^48  z^48 - 1  z^48 + 1]
    [        0         0         0        -1         0]
    [        0         0 -z^48 + 1  z^48 + 1     -z^48],

    [-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]
]
Matrix Surface Kernel Generators:
[
    [        0  z^32 - 1         0         0         0]
    [    -z^32         0         0         0         0]
    [        0         0        -1 -z^48 - 1  z^48 - 1]
    [        0         0         0         0     -z^48]
    [        0         0         0      z^48         0],

    [1/3*(-z^32 - 1) 1/3*(-z^56 + 2*z^40 + 2*z^24 - z^8) 0 0 0]
    [1/3*(z^56 + z^40 + z^24 - 2*z^8) 1/3*(z^32 - 2) 0 0 0]
    [0 0 -1 0 z^48]
    [0 0 -z^48 1 -z^48 - 1]
    [0 0 z^48 0 0],

    [1/3*(z^56 + z^40 + z^24 - 2*z^8) 1/3*(-2*z^32 + 1) 0 0 0]
    [1/3*(2*z^32 - 1) 1/3*(-z^56 + 2*z^40 + 2*z^24 - z^8) 0 0 0]
    [0 0 0 1 0]
    [0 0 z^48 - 1 -1 z^48]
    [0 0 z^48 -1 z^48 + 1]
]
Finding quadrics:
I2 contains a 3-dimensional subspace of CharacterRow 6
Dimension 6
Multiplicity 2
[
    x_0^2,
    x_0*x_1,
    x_1^2,
    x_2^2 - 2*z^48*x_2*x_4 - x_4^2,
    x_2*x_3 + x_2*x_4 - x_3*x_4,
    x_3^2 - 2*z^48*x_3*x_4 - x_4^2
]

This shows that the ideal contains quadrics from one isotypical subspace of \(S_2\). Note that the power of z\(=\zeta_{192}\) in the surface kernel generators and in the equations is always a multiple of 8. Therefore in the sequel we reduce these to z_24\(= \zeta_{24}\).

The first isotypical subspace corresponds to the character \( \chi_{6}\) 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 \( \operatorname{Span}\langle x_2^2 - 2 i x_2 x_4 - x_4^2, x_2 x_3 + x_2 x_4 - x_3 x_4, x_3^2 - 2 i x_3 x_4 - x_4^2\rangle \) generate one copy of \(V_{6}\) and use the FindParallelBases function to find an ordered basis of \( \operatorname{Span}\langle x_0^2,x_0 x_1,x_1^2 \rangle \) such that the action of \(G\) is given by the same matrices relative to these two ordered bases. 

> GL5K:=Parent(MatrixSKG[1]);
> MatG:=sub<GL5K | MatrixSKG>;
> FindParallelBases(MatG,[Q[1][4],Q[1][5],Q[1][6]],[Q[1][1],Q[1][2],Q[1][3]]); 
[ x_0^2 + (z^56 - z^40 - z^24)*x_0*x_1 - z^32*x_1^2]
[         1/2*(-3*z^56 - 3*z^40 + 3*z^24)*x_0*x_1]
[z^32*x_0^2 + (-z^56 + z^40 + z^24)*x_0*x_1 - x_1^2]
Therefore the candidate polynomials for this isotypical subspace are \[ \begin{array}{l} c_1(x_0^2+(\zeta_{24}^7 - \zeta_{24}^5 - \zeta_{24}^3) x_0 x_1 - \zeta_{24}^4 x_1^2)+ c_2(x_2^2 - 2 i x_2 x_4 - x_4^2) \\ c_1(1/2(-3 \zeta_{24}^7 - 3 \zeta_{24}^5 + 3 \zeta_{24}^3)x_0x_1) + c_2(x_2 x_3 + x_2 x_4 - x_3 x_4) \\ c_1(\zeta_{24}^4 x_0^2 + (-\zeta_{24}^7 + \zeta_{24}^5 + \zeta_{24}^3) x_0 x_1 - x_1^2) + c_2(x_3^2 - 2 i x_3 x_4 x_4^2) \end{array} \] By assuming that \( c_1 \neq 0\) and \( c_2 \neq 0\), after scaling \(x_0\) and \(x_1\), we may assume that \(c_1 = c_2 = 1\). Thus, we finally obtain the equations \[ \begin{array}{l} x_0^2+(\zeta_{24}^7 - \zeta_{24}^5 - \zeta_{24}^3) x_0 x_1 - \zeta_{24}^4 x_1^2+ x_2^2 - 2 i x_2 x_4 - x_4^2 \\ 1/2(-3 \zeta_{24}^7 - 3 \zeta_{24}^5 + 3 \zeta_{24}^3)x_0x_1 + x_2 x_3 + x_2 x_4 - x_3 x_4 \\ c_1(\zeta_{24}^4 x_0^2 + (-\zeta_{24}^7 + \zeta_{24}^5 + \zeta_{24}^3) x_0 x_1 - x_1^2) + c_2(x_3^2 - 2 i x_3 x_4 x_4^2) \end{array} \]

Checking the equations in Magma

We check that our equations give a smooth genus 5 curve with the desired automorphisms. 
Magma V2.20-3     Fri Mar 18 2016 09:16:35 on Fordham-David-Swinarski [Seed = 
4282537491]
Type ? for help.  Type -D to quit.
> K<z_24>:=CyclotomicField(24);
> P4<x_0,x_1,x_2,x_3,x_4>:=ProjectiveSpace(K,4);
> X:=Scheme(P4,[x_0^2+(z_24^7 - z_24^5 - z_24^3)*x_0*x_1 - z_24^4*x_1^2 + x_2^\
2 - 2*z_24^6*x_2*x_4 - x_4^2,
> 1/2*(-3*z_24^7 - 3*z_24^5 + 3*z_24^3)*x_0*x_1+ x_2*x_3 + x_2*x_4 - x_3*x_4,
> z_24^4*x_0^2 + (-z_24^7 + z_24^5 + z_24^3)*x_0*x_1 - x_1^2+x_3^2 - 2*z_24^6*\
x_3*x_4 -
> x_4^2]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
8*$.1 - 4
2
> A:=Matrix([
> [0,z_24^4-1,0,0,0],
> [-z_24^4,0,0,0,0],
> [0,0,-1,-z_24^6-1,z_24^6-1],
> [0,0,0,0,-z_24^6],
> [0,0,0,z_24^6,0]
> ]);
> B:=Matrix([
> [1/3*(-z_24^4 - 1), 1/3*(-z_24^7 + 2*z_24^5 + 2*z_24^3- z_24), 0, 0, 0],
> [1/3*(z_24^7 + z_24^5 + z_24^3 - 2*z_24), 1/3*(z_24^4 - 2), 0, 0,0],
> [0, 0, -1, 0 , z_24^6],
> [0, 0, -z_24^6, 1, -z_24^6 - 1],
> [0, 0, z_24^6, 0, 0]
> ]);
> Order(A);
2
> Order(B);
3
> Order(A*B);
8
> GL5K:=GeneralLinearGroup(5,K);
> IdentifyGroup(sub<GL5K | A,B>);
<192, 181>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
-z_24^4*x_1
(z_24^4 - 1)*x_0
-x_2
(-z_24^6 - 1)*x_2 + z_24^6*x_4
(z_24^6 - 1)*x_2 - z_24^6*x_3
and inverse
-z_24^4*x_1
(z_24^4 - 1)*x_0
-x_2
(-z_24^6 - 1)*x_2 + z_24^6*x_4
(z_24^6 - 1)*x_2 - z_24^6*x_3
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
1/3*(-z_24^4 - 1)*x_0 + 1/3*(z_24^7 + z_24^5 + z_24^3 - 2*z_24)*x_1
1/3*(-z_24^7 + 2*z_24^5 + 2*z_24^3 - z_24)*x_0 + 1/3*(z_24^4 - 2)*x_1
-x_2 - z_24^6*x_3 + z_24^6*x_4
x_3
z_24^6*x_2 + (-z_24^6 - 1)*x_3
and inverse
1/3*(z_24^4 - 2)*x_0 + 1/3*(-z_24^7 - z_24^5 - z_24^3 + 2*z_24)*x_1
1/3*(z_24^7 - 2*z_24^5 - 2*z_24^3 + z_24)*x_0 + 1/3*(-z_24^4 - 1)*x_1
(-z_24^6 + 1)*x_3 - z_24^6*x_4
x_3
-z_24^6*x_2 - z_24^6*x_3 - x_4

Favorite equations

We obtain nicer equations if we use the matrix surface kernel generators of Kuribayashi and Kimura. Wiman's equations are the nicest I know. See Genus-5-192-181.pdf for further discussion.