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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 (15,1) 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 (3,5,15).
We use Magma to compute equations of this Riemann surface.
Magma V2.21-7 Mon Apr 25 2016 15:33:22 on Davids-MacBook-Pro-2 [Seed =
2597567684]
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Type ? for help. Type -D to quit.
> load "autcv10e.txt";
Loading "autcv10e.txt"
> G:=SmallGroup(15,1);
> MatrixGens,MatrixSKG,Q,C:=RunExample(G,4,[3,5,15]);
Set seed to 0.
Character Table of Group G
--------------------------
---------------------------------------------------------------------------
Class | 1 2 3 4 5 6 7 8 9 10 11 12 13
Size | 1 1 1 1 1 1 1 1 1 1 1 1 1
Order | 1 3 3 5 5 5 5 15 15 15 15 15 15
---------------------------------------------------------------------------
p = 3 1 1 1 5 7 4 6 7 5 7 4 4 6
p = 5 1 3 2 1 1 1 1 3 2 2 2 3 3
---------------------------------------------------------------------------
X.1 + 1 1 1 1 1 1 1 1 1 1 1 1 1
X.2 0 1-1-J J 1 1 1 1 -1-J J J J -1-J -1-J
X.3 0 1 J-1-J 1 1 1 1 J -1-J -1-J -1-J J J
X.4 0 1 1 1 Z1 Z1#3 Z1#2 Z1#4 Z1#3 Z1 Z1#3 Z1#2 Z1#2 Z1#4
X.5 0 1-1-J J Z1 Z1#3 Z1#2 Z1#4 Z2 Z2#2 Z2#11 Z2#14 Z2#4 Z2#13
X.6 0 1 J-1-J Z1 Z1#3 Z1#2 Z1#4 Z2#11 Z2#7 Z2 Z2#4 Z2#14 Z2#8
X.7 0 1 1 1 Z1#2 Z1 Z1#4 Z1#3 Z1 Z1#2 Z1 Z1#4 Z1#4 Z1#3
X.8 0 1-1-J J Z1#2 Z1 Z1#4 Z1#3 Z2#7 Z2#14 Z2#2 Z2#8 Z2#13 Z2
X.9 0 1 J-1-J Z1#2 Z1 Z1#4 Z1#3 Z2#2 Z2#4 Z2#7 Z2#13 Z2#8 Z2#11
X.10 0 1 1 1 Z1#3 Z1#4 Z1 Z1#2 Z1#4 Z1#3 Z1#4 Z1 Z1 Z1#2
X.11 0 1-1-J J Z1#3 Z1#4 Z1 Z1#2 Z2#13 Z2#11 Z2#8 Z2#2 Z2#7 Z2#4
X.12 0 1 J-1-J Z1#3 Z1#4 Z1 Z1#2 Z2#8 Z2 Z2#13 Z2#7 Z2#2 Z2#14
X.13 0 1 1 1 Z1#4 Z1#2 Z1#3 Z1 Z1#2 Z1#4 Z1#2 Z1#3 Z1#3 Z1
X.14 0 1-1-J J Z1#4 Z1#2 Z1#3 Z1 Z2#4 Z2#8 Z2#14 Z2#11 Z2 Z2#7
X.15 0 1 J-1-J Z1#4 Z1#2 Z1#3 Z1 Z2#14 Z2#13 Z2#4 Z2 Z2#11 Z2#2
---------------------
Class | 14 15
Size | 1 1
Order | 15 15
---------------------
p = 3 5 6
p = 5 3 2
---------------------
X.1 + 1 1
X.2 0 -1-J J
X.3 0 J -1-J
X.4 0 Z1 Z1#4
X.5 0 Z2#7 Z2#8
X.6 0 Z2#2 Z2#13
X.7 0 Z1#2 Z1#3
X.8 0 Z2#4 Z2#11
X.9 0 Z2#14 Z2
X.10 0 Z1#3 Z1#2
X.11 0 Z2 Z2#14
X.12 0 Z2#11 Z2#4
X.13 0 Z1#4 Z1
X.14 0 Z2#13 Z2#2
X.15 0 Z2#8 Z2#7
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(5: Sparse := true)) ! [ RationalField() | 0, 0, 0, 1 ]
Z2 = (CyclotomicField(15: Sparse := true)) ! [ RationalField() | 1, 1, 1, 1,
1, 1, 1, 1 ]
Conjugacy Classes of group G
----------------------------
[1] Order 1 Length 1
Rep Id(G)
[2] Order 3 Length 1
Rep G.1^2
[3] Order 3 Length 1
Rep G.1
[4] Order 5 Length 1
Rep G.2^3
[5] Order 5 Length 1
Rep G.2^4
[6] Order 5 Length 1
Rep G.2
[7] Order 5 Length 1
Rep G.2^2
[8] Order 15 Length 1
Rep G.1^2 * G.2^4
[9] Order 15 Length 1
Rep G.1 * G.2^3
[10] Order 15 Length 1
Rep G.1 * G.2^4
[11] Order 15 Length 1
Rep G.1 * G.2
[12] Order 15 Length 1
Rep G.1^2 * G.2
[13] Order 15 Length 1
Rep G.1^2 * G.2^2
[14] Order 15 Length 1
Rep G.1^2 * G.2^3
[15] Order 15 Length 1
Rep G.1 * G.2^2
Surface kernel generators: [ G.1^2, G.2, G.1 * G.2^4 ]
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, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0 ]
H^0(C,K) =[ 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0 ]
I_2 =[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ]
S_2 =[ 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 2, 0 ]
H^0(C,2K)=[ 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0 ]
I_3 =[ 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0 ]
S_3 =[ 2, 1, 2, 2, 0, 1, 2, 0, 1, 3, 1, 1, 2, 1, 1 ]
H^0(C,3K)=[ 1, 1, 1, 1, 0, 1, 1, 0, 1, 2, 1, 1, 2, 1, 1 ]
I2timesS1=[ 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 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 15 and degree 8
[
[ z^5 0 0 0]
[ 0 -z^5 - 1 0 0]
[ 0 0 -z^5 - 1 0]
[ 0 0 0 -z^5 - 1],
[z^3 0 0 0]
[0 z^3 0 0]
[0 0 z^6 0]
[0 0 0 z^7 - z^6 - z^3 + z^2 - 1]
]
Matrix Surface Kernel Generators:
[
[-z^5 - 1 0 0 0]
[ 0 z^5 0 0]
[ 0 0 z^5 0]
[ 0 0 0 z^5],
[z^3 0 0 0]
[0 z^3 0 0]
[0 0 z^6 0]
[0 0 0 z^7 - z^6 - z^3 + z^2 - 1],
[z^2 0 0 0]
[ 0 z^7 0 0]
[ 0 0 z^4 0]
[ 0 0 0 z]
]
Finding quadrics:
I2 contains a 1-dimensional subspace of CharacterRow 14
Dimension 2
Multiplicity 2
[
x_1*x_3,
x_2^2
]
Finding cubics:
I3 contains a 1-dimensional subspace of CharacterRow 1
Dimension 2
Multiplicity 2
[
x_1^2*x_3,
x_1*x_2^2
]
I3 contains a 1-dimensional subspace of CharacterRow 3
Dimension 2
Multiplicity 2
[
x_0*x_1*x_3,
x_0*x_2^2
]
I3 contains a 1-dimensional subspace of CharacterRow 4
Dimension 2
Multiplicity 2
[
x_1*x_2*x_3,
x_2^3
]
I3 contains a 1-dimensional subspace of CharacterRow 7
Dimension 2
Multiplicity 2
[
x_1*x_3^2,
x_2^2*x_3
]
I3 contains a 1-dimensional subspace of CharacterRow 10
Dimension 3
Multiplicity 3
[
x_0^3,
x_1^3,
x_2*x_3^2
]
From the output, we see that this curve is cyclic trigonal. The canonical ideal is of the form \[ c_1 x_1 x_3 +c_2 x_2^2, c_3 x_0^3 + c_4 x_1^3 + c_5 x_2 x_3^2 \] Assume that these coefficients are nonzero. Then after scaling the variables and dividing, we may assume that the canonical ideal is \[ x_1x_3-x_2^2, x_0^3-x_2 x_3^2 +x_1^3. \]
> K<z_15>:=CyclotomicField(15);
> z_5:=z_15^3;
> z_3:=z_15^5;
> P3<x_0,x_1,x_2,x_3>:=ProjectiveSpace(K,3);
> X:=Scheme(P3,[ x_1*x_3-x_2^2,x_0^3-x_2*x_3^2+x_1^3]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
6*$.1 - 3
2
> A:=DiagonalMatrix([z_3^2,z_3,z_3,z_3]);
> B:=DiagonalMatrix([z_5,z_5,z_5^2,z_5^-2]);
> Order(A);
3
> Order(B);
5
> Order( (A*B)^-1);
15
> GL4K:=GeneralLinearGroup(4,K);
> IdentifyGroup(sub<GL4K | A,B>);
<15, 1>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations :
(-z_15^5 - 1)*x_0
z_15^5*x_1
z_15^5*x_2
z_15^5*x_3
and inverse
z_15^5*x_0
(-z_15^5 - 1)*x_1
(-z_15^5 - 1)*x_2
(-z_15^5 - 1)*x_3
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations :
z_15^3*x_0
z_15^3*x_1
z_15^6*x_2
(z_15^7 - z_15^6 - z_15^3 + z_15^2 - 1)*x_3
and inverse
(-z_15^7 - z_15^2)*x_0
(-z_15^7 - z_15^2)*x_1
(z_15^7 - z_15^6 - z_15^3 + z_15^2 - 1)*x_2
z_15^6*x_3