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Magaard, Shaska, Shpectorov, and Völklein list smooth curves of genus \( g \leq 10\) with automorphism groups \(G\) satisfying \( \# G > 4(g-1)\). Their list is based on a computer search by Thomas Breuer.
They list a genus 7 Riemann surface with automorphism group (54,3) 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 (2,6,9).
We use Magma to compute equations of this curve.
Here is the file autcv10e.txt used below.
Magma V2.21-7 Sat May 7 2016 07:14:35 on Davids-MacBook-Pro-2 [Seed =
3952170679]
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| Simons Foundation |
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Type ? for help. Type -D to quit.
> load "autcv10e.txt";
Loading "autcv10e.txt"
> G:=SmallGroup(54,3);
> MatrixGens,MatrixSKG,Q,C:=RunExample(G,7,[2,6,9]);
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 9 1 1 2 2 2 9 9 2 2 2 2 2 2
Order | 1 2 3 3 3 3 3 6 6 9 9 9 9 9 9
---------------------------------------------------------------------------
p = 2 1 1 4 3 6 5 7 3 4 13 12 16 18 10 17
p = 3 1 2 1 1 1 1 1 2 2 7 7 7 7 7 7
---------------------------------------------------------------------------
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 J -1-J J -1-J 1 1+J -J J 1 1 -1-J -1-J -1-J
X.4 0 1 1 J -1-J J -1-J 1-1-J J J 1 1 -1-J -1-J -1-J
X.5 0 1 1 -1-J J -1-J J 1 J-1-J -1-J 1 1 J J J
X.6 0 1 -1 -1-J J -1-J J 1 -J 1+J -1-J 1 1 J J J
X.7 + 2 0 2 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1
X.8 0 2 0 2*J-2-2*J 2*J-2-2*J 2 0 0 -J -1 -1 1+J 1+J 1+J
X.9 0 2 0-2-2*J 2*J-2-2*J 2*J 2 0 0 1+J -1 -1 -J -J -J
X.10 0 2 0 2*J-2-2*J -J 1+J -1 0 0 Z1 Z2#4 Z2 Z1#2 Z1#5 Z1#8
X.11 + 2 0 2 2 -1 -1 -1 0 0 Z2 Z2 Z2#2 Z2#2 Z2#4 Z2
X.12 0 2 0 2*J-2-2*J -J 1+J -1 0 0 Z1#7 Z2 Z2#2 Z1#5 Z1#8 Z1#2
X.13 0 2 0-2-2*J 2*J 1+J -J -1 0 0 Z1#8 Z2#4 Z2 Z1#7 Z1#4 Z1
X.14 + 2 0 2 2 -1 -1 -1 0 0 Z2#2 Z2#2 Z2#4 Z2#4 Z2 Z2#2
X.15 + 2 0 2 2 -1 -1 -1 0 0 Z2#4 Z2#4 Z2 Z2 Z2#2 Z2#4
X.16 0 2 0-2-2*J 2*J 1+J -J -1 0 0 Z1#5 Z2#2 Z2#4 Z1 Z1#7 Z1#4
X.17 0 2 0 2*J-2-2*J -J 1+J -1 0 0 Z1#4 Z2#2 Z2#4 Z1#8 Z1#2 Z1#5
X.18 0 2 0-2-2*J 2*J 1+J -J -1 0 0 Z1#2 Z2 Z2#2 Z1#4 Z1 Z1#7
------------------------
Class | 16 17 18
Size | 2 2 2
Order | 9 9 9
------------------------
p = 2 11 14 15
p = 3 7 7 7
------------------------
X.1 + 1 1 1
X.2 + 1 1 1
X.3 0 1 J J
X.4 0 1 J J
X.5 0 1 -1-J -1-J
X.6 0 1 -1-J -1-J
X.7 + -1 -1 -1
X.8 0 -1 -J -J
X.9 0 -1 1+J 1+J
X.10 0 Z2#2 Z1#7 Z1#4
X.11 + Z2#4 Z2#2 Z2#4
X.12 0 Z2#4 Z1#4 Z1
X.13 0 Z2#2 Z1#2 Z1#5
X.14 + Z2 Z2#4 Z2
X.15 + Z2#2 Z2 Z2#2
X.16 0 Z2 Z1#8 Z1#2
X.17 0 Z2 Z1 Z1#7
X.18 0 Z2#4 Z1#5 Z1#8
Explanation of Character Value Symbols
--------------------------------------
# denotes algebraic conjugation, that is,
#k indicates replacing the root of unity w by w^k
J = RootOfUnity(3)
Z1 = (CyclotomicField(9: Sparse := true)) ! [ RationalField() | 0, 1, 0, 0,
0, 1 ]
Z2 = (CyclotomicField(9: Sparse := true)) ! [ RationalField() | 0, 0, 0, 0,
1, 1 ]
Conjugacy Classes of group G
----------------------------
[1] Order 1 Length 1
Rep Id(G)
[2] Order 2 Length 9
Rep G.1
[3] Order 3 Length 1
Rep G.2
[4] Order 3 Length 1
Rep G.2^2
[5] Order 3 Length 2
Rep G.2 * G.4
[6] Order 3 Length 2
Rep G.2^2 * G.4
[7] Order 3 Length 2
Rep G.4
[8] Order 6 Length 9
Rep G.1 * G.2^2
[9] Order 6 Length 9
Rep G.1 * G.2
[10] Order 9 Length 2
Rep G.2 * G.3 * G.4^2
[11] Order 9 Length 2
Rep G.3 * G.4^2
[12] Order 9 Length 2
Rep G.3
[13] Order 9 Length 2
Rep G.2^2 * G.3
[14] Order 9 Length 2
Rep G.2^2 * G.3 * G.4
[15] Order 9 Length 2
Rep G.2^2 * G.3 * G.4^2
[16] Order 9 Length 2
Rep G.3 * G.4
[17] Order 9 Length 2
Rep G.2 * G.3
[18] Order 9 Length 2
Rep G.2 * G.3 * G.4
Surface kernel generators: [ G.1 * G.3 * G.4, G.1 * G.2 * G.4^2, G.2^2 * G.3 *
G.4^2 ]
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, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
S_1 =[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0 ]
H^0(C,K) =[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0 ]
I_2 =[ 0, 0, 0, 2, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0 ]
S_2 =[ 0, 0, 0, 3, 1, 0, 2, 1, 0, 2, 2, 1, 0, 0, 1, 1, 2, 0 ]
H^0(C,2K)=[ 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 0 ]
I_3 =[ 1, 3, 1, 1, 0, 0, 3, 0, 3, 1, 2, 0, 2, 3, 3, 1, 2, 4 ]
S_3 =[ 1, 4, 2, 1, 1, 1, 5, 1, 4, 2, 4, 0, 3, 4, 4, 2, 3, 5 ]
H^0(C,3K)=[ 0, 1, 1, 0, 1, 1, 2, 1, 1, 1, 2, 0, 1, 1, 1, 1, 1, 1 ]
I2timesS1=[ 2, 4, 1, 1, 0, 0, 2, 0, 5, 1, 1, 0, 3, 5, 4, 1, 2, 7 ]
Is clearly not generated by quadrics? true
Matrix generators for action on H^0(C,K):
Field K Cyclotomic Field of order 54 and degree 18
[
[-1 0 0 0 0 0 0]
[0 1 0 0 0 0 0]
[0 -z^12 + z^6 + z^3 -1 0 0 0 0]
[0 0 0 z^12 - z^6 - z^3 + 1 -z^15 - z^12 + z^9 + z^6 - 1 0 0]
[0 0 0 -z^15 + z^12 + z^9 + z^6 - z^3 -z^12 + z^6 + z^3 - 1 0 0]
[0 0 0 0 0 1 0]
[0 0 0 0 0 z^15 + z^12 - z^6 -1],
[ -z^9 0 0 0 0 0 0]
[ 0 z^9 - 1 0 0 0 0 0]
[ 0 0 z^9 - 1 0 0 0 0]
[ 0 0 0 -z^9 0 0 0]
[ 0 0 0 0 -z^9 0 0]
[ 0 0 0 0 0 -z^9 0]
[ 0 0 0 0 0 0 -z^9],
[1 0 0 0 0 0 0]
[0 0 1 0 0 0 0]
[0 -1 -z^12 + z^6 + z^3 0 0 0 0]
[0 0 0 z^15 - z^12 - 1 z^15 + z^12 - z^9 - z^6 + 1 0 0]
[0 0 0 z^15 - z^9 + z^3 z^12 - z^6 - z^3 + 1 0 0]
[0 0 0 0 0 -1 -z^15 - z^3]
[0 0 0 0 0 -z^15 - z^12 + z^6 -z^15 + z^12 + 1],
[1 0 0 0 0 0 0]
[0 -z^12 + z^6 + z^3 - 1 z^15 - z^12 - 1 0 0 0 0]
[0 -z^15 + z^12 + 1 z^12 - z^6 - z^3 0 0 0 0]
[0 0 0 -z^12 + z^6 + z^3 - 1 -z^9 - z^6 + z^3 + 1 0 0]
[0 0 0 z^15 - z^12 - z^9 - z^6 + z^3 z^12 - z^6 - z^3 0 0]
[0 0 0 0 0 z^12 - z^6 - z^3 z^15 - z^12 - z^9 - z^6 + z^3]
[0 0 0 0 0 -z^9 - z^6 + z^3 + 1 -z^12 + z^6 + z^3 - 1]
]
Matrix Surface Kernel Generators:
[
[-1 0 0 0 0 0 0]
[0 -z^15 + z^12 + 1 z^12 - z^6 - z^3 0 0 0 0]
[0 -z^12 + z^6 + z^3 - 1 z^15 - z^12 - 1 0 0 0 0]
[0 0 0 -z^12 + z^6 + z^3 - 1 -z^9 - z^6 + z^3 + 1 0 0]
[0 0 0 z^15 - z^9 + z^3 z^12 - z^6 - z^3 + 1 0 0]
[0 0 0 0 0 z^12 - z^6 - z^3 + 1 z^15 - z^9 + z^3]
[0 0 0 0 0 -z^9 - z^6 + z^3 + 1 -z^12 + z^6 + z^3 - 1],
[z^9 0 0 0 0 0 0]
[0 -z^15 - z^12 + z^6 z^9 + z^6 - z^3 - 1 0 0 0 0]
[0 -z^9 + 1 z^15 + z^12 - z^6 0 0 0 0]
[0 0 0 -z^15 - z^3 -1 0 0]
[0 0 0 z^9 + z^6 - z^3 - 1 z^15 + z^3 0 0]
[0 0 0 0 0 -z^15 + z^9 - z^3 -z^9 - z^6 + z^3 + 1]
[0 0 0 0 0 z^12 - z^6 - z^3 + 1 z^15 - z^9 + z^3],
[z^9 - 1 0 0 0 0 0 0]
[0 -z^15 + z^12 + z^9 + z^6 - z^3 -z^15 + z^9 - z^3 0 0 0 0]
[0 z^15 - z^9 + z^3 z^15 - z^9 + z^3 0 0 0 0]
[0 0 0 z^9 + z^6 - z^3 - 1 z^15 + z^3 0 0]
[0 0 0 -z^12 + z^6 + z^3 -z^9 + 1 0 0]
[0 0 0 0 0 z^15 + z^12 - z^6 -1]
[0 0 0 0 0 -z^9 0]
]
The output above shows that this surface is cyclic trigonal.
We look for a trigonal morphism. The output above shows that the degree three elements belong to conjugacy classes 3-7. We compute the number of fixed points of these group elements:
> CCL:=Classes(G);
> SKG:=[ G.1 * G.3 * G.4, G.1 * G.2 * G.4^2, G.2^2 * G.3 *
> G.4^2 ];
> NumberOfFixedPoints(G,SKG,CCL[3][3]);
9
> NumberOfFixedPoints(G,SKG,CCL[4][3]);
9
> NumberOfFixedPoints(G,SKG,CCL[5][3]);
0
> NumberOfFixedPoints(G,SKG,CCL[6][3]);
0
> NumberOfFixedPoints(G,SKG,CCL[7][3]);
6
> IdentifyGroup(quo<G | CCL[3][3]>);
<18, 1>
> IdentifyGroup(quo<G | CCL[4][3]>);
<18, 1>
> IdentifyGroup(DihedralGroup(9));
<18, 1>
> Eigenvalues(MatrixGens[2]);
{
<-z^9, 5>,
}
> Eigenvalues(MatrixGens[2]^2);
{
<-z^9, 2>,
}
The eigenvalues of a representative of conjugacy class 3 are are \( \zeta_3 \) with multiplicity 5 and \( \zeta_3^2 \) with multiplicity 2. Thus, in the notation of [AchterPries2007] we have \(r = 5, s=2\), so \(d_1 = 9, d_2 = 0\). The cyclic trigonal equation therefore has the form \(y^3 = f(x)\) for some degree 9 polynomial \(f(x)\). A configuration of 9 points on the Riemann sphere with symmetry \(D_9\) is given by the ninth roots of unity. Therefore we conjecture that the cyclic trigonal equation is \(y^3 = x^9-1\), and check that this yields a smooth curve with the correct automorphism group. A basis of holomorphic differentials for this Riemann surface is given by \[ \{y \frac{dx}{y^2}, yx\frac{dx}{y^2}, \frac{dx}{y^2}, x \frac{dx}{y^2},x^2 \frac{dx}{y^2}, x^3 \frac{dx}{y^2} , x^4 \frac{dx}{y^2} \} \] The canonical ideal associated to this trigonal equation is given by the \(2 \times 2\) minors of the matrix \[ \left[ \begin{array}{rrrrrr} x_0 & x_2 & x_3 & x_4 & x_5 \\ x_1 & x_3 & x_4 & x_5 & x_6 \end{array} \right] \] together with the cubics \[ \begin{array}{c} x_0^3-x_6^2 x_3+x_2^3,\\ x_0^2 x_1-x_6^2 x_4+x_2^2 x_3,\\ x_0 x_1^2-x_6^2 x_5+x_2^2 x_4,\\ x_1^3-x_6^3+x_2^2 x_5 \end{array} \] Next we compute the action of the automorphisms on the differentials.
The rotation of the sphere \( x \mapsto \zeta_9 x \) preserves \( y^3 = x^9-1 \) if \( y \mapsto y\). This automorphism acts on the basis of differentials shown above by a diagonal matrix whose diagonal entries are \[ (\zeta_9, \zeta_9^2, \zeta_9, \zeta_9^2, \zeta_9^3,\zeta_9^4, \zeta_9^5). \]
The automorphism \( x \mapsto \frac{1}{x} \) interchanging the north and south poles preserves \( y^3 = x^9-1 \) if \(y \mapsto -y\). This automorphism acts on the basis of differentials above by the matrix \[ \left[ \begin{array}{rrrrrrr} 0 & -1 & 0 & 0 & 0 & 0 & 0\\ -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0\\ \end{array} \right]. \] Finally, the trigonal automorphism \( y \mapsto \zeta_3 y\) acts on the basis of differentials shown above by a diagonal matrix whose diagonal entries are \[ (\zeta_3^2, \zeta_3^2, \zeta_3, \zeta_3, \zeta_3, \zeta_3, \zeta_3) \] We check that these matrices generate the group (54,3):
> K<z_9>:=CyclotomicField(9);
> z_3:=z_9^3;
> A:=DiagonalMatrix([z_9,z_9^2,z_9,z_9^2,z_9^3,z_9^4,z_9^5]);
> B:=DiagonalMatrix([z_3^2,z_3^2,z_3,z_3,z_3,z_3,z_3]);
> C:=Matrix([
> [0,-1,0,0,0,0,0],
> [-1,0,0,0,0,0,0],
> [0,0,0,0,0,0,1],
> [0,0,0,0,0,1,0],
> [0,0,0,0,1,0,0],
> [0,0,0,1,0,0,0],
> [0,0,1,0,0,0,0]
> ]);
> GL7K:=GeneralLinearGroup(7,K);
> G:=sub<GL7K | A,B,C>;
> IdentifyGroup(G);
<54, 3>
> K<z_9>:=CyclotomicField(9);
> z_3:=z_9^3;
> P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
> X:=Scheme(P6,[
> -x_1*x_2+x_0*x_3,
> -x_1*x_3+x_0*x_4,
> -x_3^2+x_2*x_4,
> -x_1*x_4+x_0*x_5,
> -x_3*x_4+x_2*x_5,
> -x_4^2+x_3*x_5,
> -x_1*x_5+x_0*x_6,
> -x_3*x_5+x_2*x_6,
> -x_4*x_5+x_3*x_6,
> -x_5^2+x_4*x_6,
> x_0^3-x_6^2*x_3+x_2^3,
> x_0^2*x_1-x_6^2*x_4+x_2^2*x_3,
> x_0*x_1^2-x_6^2*x_5+x_2^2*x_4,
> x_1^3-x_6^2*x_6+x_2^2*x_5
> ]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
12*$.1 - 6
2
> A:=Matrix([
> [0,1,0,0,0,0,0],
> [1,0,0,0,0,0,0],
> [0,0,0,0,0,0,-1],
> [0,0,0,0,0,-1,0],
> [0,0,0,0,-1,0,0],
> [0,0,0,-1,0,0,0],
> [0,0,-1,0,0,0,0]
> ]);
> B:=Matrix([
> [0,z_9^2,0,0,0,0,0],
> [z_9,0,0,0,0,0,0],
> [0,0,0,0,0,0,-z_9^5],
> [0,0,0,0,0,-z_9^4,0],
> [0,0,0,0,-z_9^3,0,0],
> [0,0,0,-z_9^2,0,0,0],
> [0,0,-z_9,0,0,0,0]
> ]);
> Order(A);
2
> Order(B);
6
> Order( (A*B)^-1);
9
> GL7K:=GeneralLinearGroup(7,K);
> IdentifyGroup(sub<GL7K | A,B>);
<54, 3>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations :
x_1
x_0
-x_6
-x_5
-x_4
-x_3
-x_2
and inverse
x_1
x_0
-x_6
-x_5
-x_4
-x_3
-x_2
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations :
z_9*x_1
z_9^2*x_0
-z_9*x_6
-z_9^2*x_5
-z_9^3*x_4
-z_9^4*x_3
-z_9^5*x_2
and inverse
(-z_9^4 - z_9)*x_1
(-z_9^5 - z_9^2)*x_0
-z_9^4*x_6
-z_9^5*x_5
(z_9^3 + 1)*x_4
(z_9^4 + z_9)*x_3
(z_9^5 + z_9^2)*x_2