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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 5 Riemann surface with automorphism group (160,234) 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 (2,4,5).
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.
Magma V2.21-7 Thu May 26 2016 14:48:33 on Davids-MacBook-Pro-2 [Seed =
3302282922]
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Type ? for help. Type -D to quit.
> load "autcv10e.txt";
Loading "autcv10e.txt"
> G:=SmallGroup(160,234);
> MatrixGens,MatrixSKG,Q,C:=RunExample(G,5,[2,4,5]);
Set seed to 0.
Character Table of Group G
--------------------------
------------------------------------------
Class | 1 2 3 4 5 6 7 8 9 10
Size | 1 5 5 5 20 20 20 20 32 32
Order | 1 2 2 2 2 4 4 4 5 5
------------------------------------------
p = 2 1 1 1 1 1 3 4 2 10 9
p = 5 1 2 3 4 5 6 7 8 1 1
------------------------------------------
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 + 2 2 2 2 0 0 0 0 Z1 Z1#2
X.4 + 2 2 2 2 0 0 0 0 Z1#2 Z1
X.5 + 5 -3 1 1 -1 1 1 -1 0 0
X.6 + 5 -3 1 1 1 -1 -1 1 0 0
X.7 + 5 1 -3 1 1 1 -1 -1 0 0
X.8 + 5 1 -3 1 -1 -1 1 1 0 0
X.9 + 5 1 1 -3 1 -1 1 -1 0 0
X.10 + 5 1 1 -3 -1 1 -1 1 0 0
Explanation of Character Value Symbols
--------------------------------------
# denotes algebraic conjugation, that is,
#k indicates replacing the root of unity w by w^k
Z1 = (CyclotomicField(5: Sparse := true)) ! [ RationalField() | -1, 0, -1,
-1 ]
Conjugacy Classes of group G
----------------------------
[1] Order 1 Length 1
Rep Id(G)
[2] Order 2 Length 5
Rep G.3
[3] Order 2 Length 5
Rep G.4
[4] Order 2 Length 5
Rep G.5
[5] Order 2 Length 20
Rep G.1
[6] Order 4 Length 20
Rep G.1 * G.4 * G.5 * G.6
[7] Order 4 Length 20
Rep G.1 * G.3 * G.5
[8] Order 4 Length 20
Rep G.1 * G.3 * G.4 * G.6
[9] Order 5 Length 32
Rep G.2
[10] Order 5 Length 32
Rep G.2^2
Surface kernel generators: [ G.1, G.1 * G.2^4 * G.6, G.2 * G.3 ]
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, 0, 0, 0, 0, 0, 1 ]
H^0(C,K) =[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ]
I_2 =[ 1, 0, 1, 0, 0, 0, 0, 0, 0, 0 ]
S_2 =[ 1, 0, 1, 1, 0, 1, 1, 0, 0, 0 ]
H^0(C,2K)=[ 0, 0, 0, 1, 0, 1, 1, 0, 0, 0 ]
I_3 =[ 0, 0, 0, 0, 0, 0, 0, 0, 1, 2 ]
S_3 =[ 0, 0, 0, 0, 1, 0, 0, 1, 2, 3 ]
H^0(C,3K)=[ 0, 0, 0, 0, 1, 0, 0, 1, 1, 1 ]
I2timesS1=[ 0, 0, 0, 0, 0, 0, 0, 0, 1, 2 ]
Is clearly not generated by quadrics? false
Matrix generators for action on H^0(C,K):
Field K Cyclotomic Field of order 160 and degree 64
[
[-1 0 0 0 0]
[-1 -1 -1 0 -1]
[ 2 0 1 -1 1]
[ 0 0 0 0 1]
[ 0 0 0 1 0],
[-1 -1 -1 0 -1]
[-1 0 0 0 0]
[ 2 0 1 -1 1]
[ 0 0 0 0 -1]
[ 0 1 0 0 0],
[-1 0 -1 1 0]
[ 0 -1 -1 -1 0]
[ 0 0 1 0 0]
[ 0 0 0 1 0]
[ 2 0 1 -1 1],
[-1 0 0 0 0]
[ 0 1 0 0 0]
[ 0 0 0 1 0]
[ 0 0 1 0 0]
[ 2 0 1 -1 1],
[-1 0 0 0 0]
[ 0 -1 -1 -1 0]
[ 0 0 0 1 0]
[ 0 0 1 0 0]
[ 0 0 0 0 -1],
[-1 0 -1 1 0]
[ 0 1 1 1 0]
[ 0 0 0 -1 0]
[ 0 0 -1 0 0]
[ 2 0 1 -1 1]
]
Matrix Surface Kernel Generators:
[
[-1 0 0 0 0]
[-1 -1 -1 0 -1]
[ 2 0 1 -1 1]
[ 0 0 0 0 1]
[ 0 0 0 1 0],
[ 0 1 1 1 0]
[-1 0 -1 1 0]
[ 0 0 0 -1 0]
[ 0 0 1 0 0]
[-1 -1 -1 0 -1],
[-1 1 0 1 -1]
[ 1 0 1 -1 0]
[ 0 0 0 0 1]
[-2 0 -1 1 -1]
[ 0 -1 -1 -1 0]
]
Finding quadrics:
I2 contains a 1-dimensional subspace of CharacterRow 1
Dimension 1
Multiplicity 1
[
x_0^2 - x_0*x_2 + x_0*x_3 - x_0*x_4 + x_1^2 - x_1*x_2 - x_1*x_3 + x_2^2 +
x_3^2 + x_4^2
]
I2 contains a 2-dimensional subspace of CharacterRow 3
Dimension 2
Multiplicity 1
[
x_0^2 - 2*x_0*x_4 + (-z^48 + z^32)*x_1^2 + 2*x_1*x_2 + 2*x_1*x_3 - x_2^2 -
2*x_2*x_3 - x_3^2 + (z^48 - z^32)*x_4^2,
x_0*x_2 - x_0*x_3 - x_0*x_4 + 1/2*(-z^48 + z^32)*x_1^2 + (z^48 -
z^32)*x_1*x_2 + (z^48 - z^32)*x_1*x_3 + 1/2*(-z^48 + z^32 - 1)*x_2^2 +
(-z^48 + z^32 + 1)*x_2*x_3 + 1/2*(-z^48 + z^32 - 1)*x_3^2 + 1/2*(z^48 -
z^32 + 1)*x_4^2
]
The ideal contains quadrics from two isotypical subspaces of \(S_2\). The polynomials shown above are exactly those we need. Note that the exponent of z \( = \zeta_{160}\) is always a power of 16; therefore we reduce these to z_10 \(= \zeta_{10}.\)
The first isotypical subspace corresponds to the character \( \chi_{1}\) in the character table shown above. It yields the polynomial \[ \begin{array}{l} x_0^2 - x_0 x_2 + x_0 x_3 - x_0 x_4 + x_1^2 - x_1 x_2 - x_1 x_3 + x_2^2 + x_3^2 + x_4^2. \end{array} \]
The second isotypical subspace corresponds to the character \( \chi_{3}\) in the character table shown above. It yields the polynomials
\[ \begin{array}{l} x_0^2 - 2 x_0 x_4 + (-\zeta_{10}^3 + \zeta_{10}^2) x_1^2 + 2 x_1 x_2 + 2 x_1 x_3 - x_2^2 - 2 x_2 x_3 - x_3^2 + (\zeta_{10}^3 - \zeta_{10}^2) x_4^2,\\ x_0 x_2 - x_0 x_3 - x_0 x_4 + 1/2 (-\zeta_{10}^3 + \zeta_{10}^2) x_1^2 + (\zeta_{10}^3 - \zeta_{10}^2) x_1 x_2 + (\zeta_{10}^3 - \zeta_{10}^2) x_1 x_3 + 1/2 (-\zeta_{10}^3 + \zeta_{10}^2 - 1) x_2^2 + (-\zeta_{10}^3 + \zeta_{10}^2 + 1) x_2 x_3 + 1/2 (-\zeta_{10}^3 + \zeta_{10}^2 - 1) x_3^2 + 1/2 (\zeta_{10}^3 - \zeta_{10}^2 + 1) x_4^2 \end{array} \]We check that these equations yield a smooth genus 5 curve with the desired automorphisms.
> K<z_10>:=CyclotomicField(10);
> P4<x_0,x_1,x_2,x_3,x_4>:=ProjectiveSpace(K,4);
> X:=Scheme(P4,[
> x_0^2 - x_0*x_2 + x_0*x_3 - x_0*x_4 + x_1^2 - x_1*x_2 - x_1*x_3 + x_2^2 + x_\
3^2 + x_4^2,
> x_0^2 - 2*x_0*x_4 + (-z_10^3 + z_10^2)*x_1^2 + 2*x_1*x_2 + 2*x_1*x_3 - x_2^2\
- 2*x_2*x_3 - x_3^2 + (z_10^3 - z_10^2)*x_4^2,
> x_0*x_2 - x_0*x_3 - x_0*x_4 + 1/2*(-z_10^3 + z_10^2)*x_1^2 + (z_10^3 - z_10^\
2)*x_1*x_2 + (z_10^3 - z_10^2)*x_1*x_3 + 1/2*(-z_10^3 + z_10^2 - 1)*x_2^2 + (-\
z_10^3 + z_10^2 + 1)*x_2*x_3 + 1/2*(-z_10^3 + z_10^2 - 1)*x_3^2 + 1/2*(z_10^3 \
- z_10^2 + 1)*x_4^2
> ]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
8*$.1 - 4
2
> A:=Matrix([
> [-1,0,0,0,0],
> [-1,-1,-1,0,-1],
> [2,0,1,-1,1],
> [0,0,0,0,1],
> [0,0,0,1,0]
> ]);
> B:=Matrix([
> [0,1,1,1,0],
> [-1,0,-1,1,0],
> [0,0,0,-1,0],
> [0,0,1,0,0],
> [-1,-1,-1,0,-1]
> ]);
> Order(A);
2
> Order(B);
4
> Order( (A*B)^(-1));
5
> GL5K:=GeneralLinearGroup(5,K);
> IdentifyGroup(sub<GL5K | A,B>);
<160, 234>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations :
-x_0 - x_1 + 2*x_2
-x_1
-x_1 + x_2
-x_2 + x_4
-x_1 + x_2 + x_3
and inverse
-x_0 - x_1 + 2*x_2
-x_1
-x_1 + x_2
-x_2 + x_4
-x_1 + x_2 + x_3
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations :
-x_1 - x_4
x_0 - x_4
x_0 - x_1 + x_3 - x_4
x_0 + x_1 - x_2
-x_4
and inverse
x_1 - x_4
-x_0 + x_4
-x_0 + x_1 - x_3
-x_0 - x_1 + x_2 + x_4
-x_4