Advances in Mathematical
Sciences and Applications
Vol. 27, No. 1 (2018), pp. 45–63

GAKKOTOSHOTOKYO JAPANGALOIS THEORY AND THE UNIFORMIZATION OF
RIEMANN SURFACES
Simon DAVIS
Research Foundation of Southern California
8861 Villa La Jolla Drive #13595
La Jolla, CA 92037(sbdavis@resfdnsca.org)

Abstract. The solvability of an algebraic equation is represented by the vanishing of anirreducible polynomial f (x0 , y) over Q and it is based on the properties of the Galois groupof the splitting ﬁeld. The function of two variables f (x, y) which generates this equation
deﬁnes a Riemann surface, derived from the Puiseux expansion, with a covering surface
that is equivalent to a punctured sphere. The relation between the conformal group of
the punctured sphere and the Galois group of the function ﬁeld on the Riemann surface
yields a description of the entire set of symmetries of the integrals over compactiﬁed
moduli spaces of surfaces with a higher number of punctures.

————————————————————
Communicated by Nobuyuki KENMOCHI; Received August 9, 2017
AMS Subject Classification: 12A55, 14G32, 14H05, 14H30
Keywords: Galois group, function field, Riemann surface, symmetric permutation, group,punctured spheres, moduli spaces

45

46

1

Introduction

Riemann surfaces can be constructed by the sewing of thrice-punctured spheres, and theconformal group of these punctured spheres contains a symmetric permutation group
S3 that interchanges the punctures. More generally, the Riemann surface constructedfrom an irreducible polynomial of two variables can be shown to have a group of decktransformations represented by the Galois group of the function ﬁeld. Since all of thesmooth ﬁnite-genus Riemann surfaces may be described as algebraic curves, solvability ofthe equation with the Galois group equal to Sn , resulting from the ﬁxing of the value of
one of the variables of a two-variable polynomial, implies that the uniformizing group is
projected to the symmetric group on the solution set.
The uniformization of spheres with n punctures is known to introduce functions withspecial modular properties. For the thrice-punctured sphere, it is the elliptic modular
function, and the additional invariance under the permutations of the punctures yields
a group generated by two transformations of order 2 adjoined to Γ(2). These transformations also generate the mapping class group Γ0,[3] for three unmarked points, which isshown to be isomorphic to the quotient of the braid group by equivalence relations forthe two generators.
The isomorphism between Γ(1)/Γ(2) is examined in light of the matrices with negative
determinant, and it is found that consistency is restored modulo 2. Once the transformations representing S3 modulo 2 are allowed to act on punctures on C∗ , a sets of threepoints, which are being permuted, would diﬀer for each generator. Consequently, the inclusion of matrices of negative determinant in the presentation of the permutation grouphas been found to be necessary.
The punctures of a collection of thrice-punctured spheres can be sewed together toconstruct a higher-genus surface with fewer punctures. The integrals over these highergenus surfaces therefore should reﬂect some of the symmetries of the thrice-punctured
sphere including invariance of under the permutation group. This result is found to hold,and the implications for the vestiges of this symmetry are elaborated.
Amongst the surfaces that may be formed by the sewing of the punctures of thricepunctured spheres are the n-punctured spheres. Translating the problem of deﬁning theuniformizing parameters of higher-genus surfaces to the punctured sphere, a paracompact
model of the universal moduli space of punctured spheres is deﬁned. The connection withthe absolute Galois group is described.

2

Galois Groups and Irreducible Polynomials

If f (x, y) is an irreducible polynomial of ﬁnite degree, suppose that g(x, y) ∈ C(x)[y] is
the monic irreducible polynomial over Q with Y as a root, where Y is a primitive element

47of the splitting ﬁeld Kf . Fixing y, the equation in x resulting from f (x, y) has a solution
in the algebraic numbers. The coeﬃcients, which may be expressed as combinations of
the roots, also must belong to a ﬁeld of algebraic numbers. Therefore, f (x, y) ∈ Q(x)[y],where Q(x) is a ﬁeld extension of the rational numbers. By the Hilbert irreducibility
theorem [13], there exist inﬁnitely many values x0 such that the polynomial in Q(x0 )[y]is irreducible over the rational numbers. It is known that Y can be chosen such that the
N roots of g(x, y) in Kf are polynomials of degree less than or equal to N − 1 in Y [18],Yj = c0j + c1j Y + ... + cN −1,j Y N −1
cij ∈ Q(α1 , ..., αn )(x).

(1)

where α1 , ..., αn are introduced through a set of conditions on the nonlinear combinationsof the coeﬃcients. The algebraicity of the roots is determined therefore by Y . The rolesof the roots Y = Y1 , Y2 ,..., YN can be permuted when the Galois group is SN , and theroot which is an algebraic number in Kf may be selected. From the relations between theroots, it follows that all roots of g(x0 , y) are algebraic numbers given by ﬁnite radicals if
x0 , α1 , ..., αn and one of the roots are similarly characterized.
The ﬁnite-degree algebraic equations deﬁne surfaces with a ﬁnite number of branchcuts and curves with a ﬁnite number of covering sheets. The conclusions regarding algebraic numbers can be restricted to ﬁnite algebraic expressions. If one root is a ﬁnite radical
expression, and g(x0 , y) ∈ Q(x0 )[y] where Q(x0 ) is an extension of Q by ﬁnite radicals,
then all roots of g(x0 , y) are ﬁnite radical expressions if the Galois grouup of the equation
g(x0 , y) = 0 is SN . It is necessary to establish the connection between the nature of thepolynomial g(x, y) and the roots Y = Y1 , ..., YN . Given a choice of x0 , Y is a ﬁnite radical
expression, it may be viewed as a root of an irreducible polynomial g(x0 , y) that belongsto Q(α1 , ..., αn )(x0 ), where α1 , ..., αn are Ruﬃni radicals, and cij ∈ Q(α1 , ..., αn )(x). For
ﬁxed x0 , cij ∈ Q(α1 , ..., αn )(x0 ). If x0 is an algebraic number that is expressed in terms ofa ﬁnite number of radicals, cij ∈ Q(α1 , ..., αn , x0 ) is a soluble extension of Q. Therefore,
if one root is a Ruﬃni radical, all of the roots would be Ruﬃni radicals. The basis for theconclusion stems from the arrangement of the decomposition of the polynomial g(x, y)
into linear factors in y. Dividing g(x, Y ) by y − Y yields a polynomial of degree N − 1 in
Y . It follows that(Y − Y2 )...(Y − YN ) = Y

N −1

−

N
−1
∑

Yj + ... + (−1)N −1 Y2 ...YN

(2)

j=1

∑
∑
where j Yj = j (c0j + c1j Y + ... + cN −1,j Y N −1 ). The powers of Y which must have avanishing coeﬃcient are Y N , ..., Y N (N −1) , yielding N + N (N − 2) equations for N (N − 1)
coeﬃcients with values in Q(x). There exists a nontrivial solution to these equations
consisting of nonlinear combinations of the coeﬃcients cij , since N (N − 2) coeﬃcientscan be eliminated systematically without introducing any new radicals. The remaining
N coeﬃcients can be expressed in terms of Ruﬃni radicals α1 , ..., αn , because it requires
only of ﬁnite number of arithmetical operations to transform the nonlinear combinations
into linear expressions in the coeﬃcients, Y has been given by a ﬁnite set of radicals, andthe polynomial ring generated by Ruﬃni radicals is closed. This polynomial ring diﬀers

48
from the ﬁeld extensions Kj of K0 = Q(s1 , ..., sN ), with s1 , ..., sN being the symmetricnpolynomials of the roots Y1 , ..., YN , where Kj = Kj−1 (αj ), αj j ∈ Kj , j = 0, 1, ..., r and
Kr = Q(Y1 , ..., YN ), since a sequential nesting of the radicals {αj } is not required in a
general polynomial of {αj }. Therefore, interchanging Y with Yj , j = 2, ..., N , it would
not be possible to express Y as a polynomial of degree N − 1 in Yj unless Yj is given by(j)a ﬁnite arithmetical combination of Ruﬃni radicals, and ck,1 ∈ Q(α1 , ..., αn , x0 ), whereY = c0,1 + c1,1 Yj + ... + cN −1,1 YjN −1 . The nature of the root Y may be deduced from Galois
theory. The Galois group of the splitting ﬁeld of an algebraic equation is Sn generally,
although polynomials with discriminants that are perfect squares have this group An[13][21]. If there is one proper normal subgroup of Sn with an abelian quotient, this canbe used to generate a root that is a Ruﬃni radical of the equation. Therefore, when thisproperty holds for the Galois group of the equation, all of the roots have ﬁnite radical
expressions.
(j)

(j)

(j)

The deﬁnition of the Riemann surface as an algebraic curve and the uniformization ofthese surfaces can be reformulated in terms of parameters describing the covering groupof the surface constructed from the Puiseux elements of the irreducible polynomial. The
permutation group of three elements can be interpreted as a Galois group of a function
ﬁeld on a Riemann surface related to an equation of degree three. The algebraicity of theroots of the equation, derived after the ﬁxing of one of the variables, is conﬁrmed by theﬁniteness of the number of uniformizing parameters for ﬁnite-genus surfaces.

3

Galois Group of the Function Field on a RiemannSurface and the Conformal Group of the Punctured Sphere

It has been demonstrated that the Riemann surface deﬁned by the function f (x, y) may
be constructed from the Puiseux elements of the irreducible polynomial f (x, y), which are
the triples (a, r, u) with(i) r is an integer larger than 0∗(ii) u ∈ C((t))∑(iii) u =am tm , supp u = {m|am ̸= 0}, d(u) = gcd supp(u), gcd(r, d(u)) = 1(iv) f (a + tr , u) = 0 in C((t)), a ∈ C.
The set of Puiseux elements for the function at a shall be denoted by P(a, f ) and let Pf
be the set of points at which Puiseux elements are deﬁned.
If f (x, y) = c0 y n + c1 (x)y n−1 + ... + cn (x), let S+ be the set of zeros of c0 (x), S− bethe set of poles of ci (x) and D be the union of the set of zeros of D(f ) and ∞. Then the
Riemann surface R is equated toR∗ × In ∪ P × In × (ℓ × In × {L.R})

(3)

49
where S = S+ ∪ S− ∪ D, where In is an index set {1, ..., n},ℓ = ∪p ℓq,p , q ∈ C∗ , q ̸∈ S
ℓq,p = πq,p (0, 1); πq,p is a piecewise linear map of [0, 1] into C∗(4)

and R∗ = C ∗ /({Pf ∪ ℓ}).
Each (z, i) ∈ R∗ × In will be contained in an open set (D(z, r), i), and the union ofsuch sets would form D(z, r) ⊂ R∗ . Since there are permutations of these sets given byequivalences (z, i, L) ∼ (z, σz (i), R), this union also can expressed as (∆1 , i)∪(∆2 , σp−1
(i))∪
1
−1
−1 −1
...(∆r+1 , σpr σpr−1 ...σ1 (i)) ∪ ..., where σ1 ...σpn = 1 [21][22].
Since the Riemann surface Rg is constructed by this method, the group of conformal transformations contains G(Kf /Q(x)) [21]. The identiﬁcation of a subgroup of theconformal group with the Galois group of permutations results from the interchange ofthe sector {∆i }, provided that the maps are the identity for identical domains. However, a more general set of maps from ∆i to ∆j and pi to pj can be described, such thatthe continuous transformations are not identity mappings. There is a group of conformaltransformations of Riemann spheres with n punctures, with the locations of the punctures
permuted under the mappings, that includes these transformations.
Since the covering space of a Riemann surface of genus greater than one is H, a presentation of the conformal group may be found. For C ∗ \{0, 1, ∞}, the three generators
satisfy g1 g2 g3 = 1. If these transformations{ ( are)required to ﬁx only one of the }punca btures, the group is conjugate to Γ(2) =∈ P SL(2, Z) b ≡ c ≡ 0 mod 2 . Asc d
Γ(1)/Γ(2) ≃ S3 , the permutation group of the three points, the use of the larger groupcan be considered.

Theorem 3.1. A union of the sets of generators of S3 ⋊ Γ(2) permuting the points{0, 1, ∞} in the thrice-punctured sphere and ﬁxing separately one puncture has four independent basis elements. Factorization by those matrices that do not leave invariant the setof three punctures yields the permutation group S3 , which is isomorphic to the subgroupof the conformal group of the covering space that ﬁxes the set {0, 1, ∞}. The normalizerof S3 within GL(2; R) is S3 and it is included in the modular group SL(2; Z). The subgroup of the conformal group is isomorphic to the Galois group of the function ﬁeld of thethrice-punctured sphere.

Proof. The presentation of the generators satisfying the conditions

50

g1 (1) = ∞g2 (1) = 0
g3 (1) = ∞g4 (1) = 1

g1 (0) = 1
g2 (0) = ∞g3 (0) = 0
g4 (0) = ∞g1 (∞) = 0
g2 (∞) = 1
g3 (∞) = 1
g4 (∞) = 0(5)

and tr g1 = tr g2 = 2 would be
(
g1 =
(
g2 =

0 2
−2 2
2 −2
2 0

)
(6)
)

)
a′ 0
g3 =
a′ −a′
(
)
0 b′′
.
g4 =
b′′ 0
(

Neither g3 or g4 can be parabolic elements, and the requirement of a parabolic conformalgroup must be relaxed to include these generators. Alternatively, since 12 g1 is an elliptic
element representing( a fractionaltransformation of order 3 that permutes the three punc)2
1
tures, even though 2 g1 = −I and g3 at a′ = 1 is an element of order 2, it is suﬃcientto add these matrices to the group conjugate to Γ(2), yielding the group
)⟩
) (
) (
) (
⟨(
1 0
1 2
1 0
0 −1
/⟨±I⟩.
(7)
,
,
,
−2 1
0 1
1 −1
1 −1
which represent a basis of generators of S3 ⋊ Γ(2). Since the latter two transformations
do not leave invariant the set {0, 1, ∞}, factorization by the group generated by theseelements leaves S3 .
When n = 3, the normalizer NΓ(2) of the group Γ(2), which is conjugate to a group witheach generator ﬁxing only one puncture, is a proper subset of SL(2; Z) and NΓ(2) /Γ(2) ⊂Γ(1)/Γ(2). The Galois group Gal(Kf /Q(x)) is not contained in NΓ(2) /Γ(2) [7] but it is a
subgroup of the conformal group of the surface.
It is necessary to consider the subgroup of the conformal group of the covering space
which leaves invariant the set of three punctures. Consequently, the generators of thisconformal group would only diﬀer by a scale factor from the matrices gi , i = 1, 2, 3, 4 thatis established by the determinant condition. Within multiplication by ±I, the matrices
in the group
⟨(

)⟩
) (
1
0
0 −1
, 1 −1
1 −1

51
are

(
I,

0 1
−1 1

) (
) (
) (
) (
)
1 0
−1 1
1 −1
0 1
,
,
,
,
1 −1
−1 0
0 −1
1 0

(8)

and the matrices 12 g2 and g4 at b′′ = 1 belong to this set.
These matrices will belong to the normalizer of S3 . The maximal subgroup of GL(2; R),
which is a normalizer of S3 and ﬁxes the three punctures, will be
Consider,( determined.
)
0 1
for example, a similarity transformation with determinant 1 of
.
−1 1
(

a b
c d

)(

0 1
−1 1

)(

d −b
−c a

)
=
(

If the transformed matrix is set equal to(−bd − (a + b)c b2 + (a + b)a−d2 − (c + d)c bd + (c + d)a

1 0
−1 1

)
(9)

)
,

−bd − (a + b)c = 1
b 2 + (a + b)a = 0− d2 − (c + d)c = −1
bd + (c + d)a = 1(10)

The second condition is satisﬁed by a = b = 0 for real matrices. The only
( group)element−1 1
in Eq.(5) which could equal a nontrivial conjugation of this matrix is.
−1 0
Further suppose) determinant of the GL(2; R) matrix is allowed to be ±1.
( that the−1 1
Equality with ±yields four relations for a GL(2; R) matrix are−1 0−bd − (a + b)c = −1
b 2 + (a + b)a = 1− d2 − (c + d)c = −1
bd + (c + d)a = 0(11)

From the second and third relations,()√
b = 1/2 −a ± 4 − 3a2()√
2
d = 1/2 −c ± 4 − 3c

(12)

52
The fourth condition then is equivalent to
√
√
√
(
)(
) (
)
−a ± 4 − 3a2−c ± 4 − 3c2−c ± 4 − 3c2
+ c+
a=0
2
2
2

(13)

or
(

Dividing by

3a ∓

√

)
√
√
4 − 3a2 c ± a 4 − 3a2 + (4 − 3a2 )(4 − 3c2 ) = 0(14)√
4 − 3c2 gives√∓a − 4 − 3a2c√√=
4 − 3c2
3a ∓ 4 − 3a2

(15)

√
)
±a − 4 − 3a2√∓a − 4 − 3a2

(16)

Then
(
d=c

Finally, it follows from the ﬁrst condition
(
c

−a ±

√
√
√
)(
) (
)
4 − 3a2±a − 4 − 3a2
a ± 4 − 3a2
√
+
c=1
2
2
∓a − 4 − 3a2
√
√
(
)
(
)
a ∓ 4 − 3a2
a ± 4 − 3a2
c
+c
=1
2
2

(17)

which implies that c = a1 . By Eq.(8),√∓a − 4 − 3a2√√=
3a ∓ 4 − 3a2
4 − 32(18)√
1
4 − 2a2 ± 2a 4 − 3a2√=
4a2 − 34 + 6a2 ∓ 6a 4 − 3a2(19)√
4 + 6a2 − (4a2 − 3)(4 − 2a2 ) = ±8a3 4 − 3a2

(20)

1
a

a

Squaring this relation gives

andor(a 4 − 2a 2 + 2 )2 = a6 (4 − 3a2 )(a4 − a2 + 1)2 = a4

(21)

53
Therefore, either a4 − 2a2 + 1 = 0 or a4 + 1 = 0. The ﬁrst set of solutions is a = ±1
while the second equation has complex solutions only, and therefore c = ±1. From theconditions on the matrix elements, b = 0 if a = 1, − 1 and d = −1 if c = 1 or d = 1 if
c = −1.
The matrices to consider are
(

1 0
1 −1

) (
)
−1 0
,
.
−1 1

(22)

The matrices, with determinants equal to −1, produce the required matrix within afactor of ±I, since the inverse must include a negative sign with respect to the matrix
multiplication (5).
The eﬀect of the fractional linear transformation represented by the ﬁrst matrix in
2 −iy
x+iy
Eq.(13) is x + iy → (x−1)+iy= x(x−1)−y, which has a negative imaginary part. The(x−1)2 +y 2∗
conformal group of C \{0, 1, ∞} will be required to allow transformations from the upper
half-plane to the lower half-plane that leave the real line invariant. Because three ofthe matrices) have determinant −1, the group SL(2; R) should be enlarged to( in Eq.(4)1 0SL(2; R). While the covering space of C∗ \{p1 , ..., pn } is generally
SL(2; R) ∪1 −1chosen to be H or the unit disk U [14][19], the covering group can be selected such thatits action on H has a range which includes H̄ for some of the elements.
The ﬁnal permutation of two elements( ′′′ ′′′ )a bg5 =would satisfy
c′′′ d′′′g5 (0) = 1

g5 (∞) = ±∞g5 (1) = 0

(23)

or

b′′′
a′′′ + d′′′ = 0c′′′ = 0=1′′′dwith ∞ and −∞ identiﬁed. Setting b′′′ = 1 gives the matrix
(

−1 0
1 1

)

⟨(
̸∈

0 1
−1 1

) (
)⟩
1 0
,
/⟨±I⟩
1 −1

which, again, has a determinant equal to −1.
Multiplication of the matrices in Eq.(4) by
(

−1 0
1 1

(

−1 0
1 1

(24)

(25)

)
yields

) (
) (
) (
) (
)
0 −1
−1 0
1 −1
−1 1
,
,
,
,
,
−1 2
2 −1
−2 1
1 −2
(
)
0 −1
.
1 1

(26)

54
These matrices, however, represent fractional transformations that map {0, 1, ∞} tothe set {−1, − 12 , 0}. Furthermore, multiplication of the elements
would() close( only after
)
0 1
−1 0
generating a much larger group. Finally, the conjugation of
by
is
−1 1
1 1
(
)
−1 −1
, which does not belong to the set of matrices in Eq.(4).
3
2
((
)
)
1 0
SL(2; R) is S3 , andTherefore, the normalizer of S3 in SL(2; R) ∪1 −1
NS3 /S3 would not contain the symmetric permutation group. It is suﬃcient, however, toconsider that subgroup of this conformal group that preserves the set {0, 1, ∞} withina permutation, which is isomorphic with the Galois group of the function ﬁeld of thethrice-punctured sphere.
Since three matrices in Eq.(4) have a determinant equal to −1, it may be investigated whether all presentations of the symmetric permutation group as fractional linear
transformations on C∗ are represented by matrices with a determinant equal to −1.

Theorem 3.2. The presentation of generators of S3 , which are fractional linear transformations on C∗ preserving a set of three points in the ﬁnite complex plane, requires
matrices with determinants equal to −1 for the permutation of two elements.
Proof. Because the action of fractional linear transformations on C∗ \{0, 1, ∞} can becomplicated by the point at ∞, consider instead the locationsof the three{} punctures at−1 0 1{−1, 0, 1}. The action of the permutation of order 3 is, the conditions
0 1 −1on the matrix elements are

a1 = b1

a1 + b1 = −(c1 + d1 )b1 = d1

c1 = −3a1(

and a unit determinant results if a1 = ± 21 .

Let a1 = 21 .

Then the matrix is

Within multiplication by ±I, this matrix generates a cyclic group of order 3.

1
2

− 23

1
2
1
2

)
.

The permutation of two points is given by{−1 0 10 −1 1} {} {}−1 0 1−1 0 1,,−1 1 01 0 −1

The conditions on the matrix representing the ﬁrst permutation are
a2 = b2

b2 = −d2

a2 + b2 = c2 + d2

c2 = 3a2(27)

55
such that[(

det a2

1 1
3 −1

)]
= −4a22 = −1

(28)

if a2 = ± 21 . When a2 = 12 , the matrix is( 1 1 )
2
2
with a determinant equal to −1. The relations for the second permutation
3
1
−
2
2
are
a3 = −b3b3 = d3

a3 − b3 = −(c3 − d3 )c3 = −3a3

(29)

and
[

(

det a3(
if a3 = ± 12 , yielding

1
2

− 23

− 12
− 12

1 −1
−3 −1

)]
= −4a23 = −1

(30))
. The constraints for the third permutation are

a4 − b4 = c4 − d4b4 = 0c4 = 0a4 = −d4

a4 + b4 = −(c4 + d4 )

(31)

(

)
1 0
and, again, the matrix
has a determinant equal to −1. Each of the permu0 −1
tations of order 2, combined with the permutations of order 3, produces a group of six
elements represented by the matrices)( 1 1 ) ( 1 1 ) ( 1 1 ) ( 1 1 ) (
−
−
1
0
2
2
2
2
2
2
2
2
,
,
,
,
(32)
I,
3
3
1
0 −1
− 23 12− 32 − 12
− 12
2
2
2
within multiplication by ±I.
By contrast with the points {0, 1, ∞}, the interchange of any two of the punctures inthe set {−1, 0, 1} would not alter the presentation of the permutation group. For any threeﬁnite real values, the application of an SL(2; R) matrix would be suﬃcient to transform{−1, 0, 1} to these points. Then multiplication of these generators by this matrix again
would give a group of order 6, with three elements represented by matrices of determinant−1.
The diﬀerential equation for the universal covering map of C∗ \{p1 , ..., pn } has been
given [12]. A presentation for the subgroup of the conformal group of transformations
on the n-punctured sphere leaving invariant p1 , ..., pn can be found by an embedding

56of the matrices in SO(2, 1). Although there are continuous transformations from one
sector, ∆i , to another, ∆j , these analytic mappings may be retracted to the identity,
After identiﬁcation of the domains, combined with permutation of the points {pi }. The
discrete group G(Kf /Q(x)) results from a retraction of the group of transformations of
the conformal equivalence class of the Riemann surface Rg . Permutation of the ﬁxed
points is suﬃcient for this retraction.

4

The Moduli Spaces of Punctured Spheres and theSymmetric Permutation GroupThe elliptic modular function is the conformal universal covering map for the thricepunctured sphere [1]. The relation between Γ(1) = SL(2; Z) and the congruence subgroup
2 −λ+1)3
Γ(2) is evident in the modular invariance of J(ω) = (λλ2 (λ−1)
2 under modular transforma(w )(w )()
e1 −e31
2
2
2
tions, where λ = e2 −e3 , e1 = J 2 , e2 = J 2 and e3 = J w1 +w,ω= w, invariance2
w1of λ with respect to Γ(2) and (λλ2 −λ+1)
under the transformations λ → λ1 and λ → 1 − λ,(λ−1)2
which generates a group of order 6.
2

3

Theorem 4.1. The group S3 may be generated by two permutations of order 2, which
are represented by matrices with determinants equal to −1. The elements also generate
the fundamental group of the moduli space of the sphere with three unmarked points which
is a quotient of the braid group B3 by equivalence relations for the two generators.
Proof. This presentation of the symmetric group is determined by the transformationsof order 21
λσ2 : λ → 1 − λσ1 : λ →

(33)

since
1
1−λ
λ−1
σ2 σ1 : λ →λσ1 σ2 : λ →σ1 σ2 σ1 = σ2 σ1 σ2 : λ →

(34)

λ
λ−1

57
The relations for the two generators areσ2 σ1 σ2 = σ1 σ2 σ1

σ12 = 1σ22 = 1(35)

For the braid group Bn with generators σ1 , ..., σn−1 , the relations areσi σi+1 σi = σi+1 σ1 σi+1
σi σj = σj σi
if |i − j| ≥ 2(36)

For the braid group B3 , |i − j| ≤ 1, and the the ﬁrst relation isσ2 σ1 σ2 = σ1 σ2 σ1(37)

iteSchneps03.
Given thatyi = σi−1 ...σ1 · σ1 ...σi−1

2≤i≤n

(38)

zi = (σ1 ...σi−1 )iit follows thaty3 = σ2 σ12 σ2z3 = (σ1 σ2 )3(39)

If y3 = 1, it follows from Eq.(24) thatz3 = σ1 (σ2 σ1 σ2 )σ1 σ2 = σ1 (σ1 σ2 σ1 )σ1 σ2 = σ12 (σ2 σ12 σ2 ) = σ12(40)The equality z3 = 1 is equivalent to σ12 = 1. Substituting this relation into y3 gives
σ22 = 1. Therefore, S3 ≃ B3 /⟨y3 = 1, z3 = 1⟩. This space can be identiﬁed with Γ0,[3] ,the mapping class group of the sphere with three unmarked points [11][20]. From thetransformations representing σ1 , σ2 , σ1 σ2 , σ2 σ1 and σ1 σ2 σ1 , it follows that I, σ1 σ2 and σ2 σ1
correspond to matrices with a determinant equal to 1, while σ1 , σ2 and σ1 σ2 σ1 = σ2 σ1 σ2
are permutations of two points that are represented by matrices of determinant -1.
The principal congruence group of level 2 is the kernel of the homomorphism µ2 :P SL(2; Z) → P SL(2;Z2)). The( only matrices
in P SL(2; Z2 ) that have a determinant
(
)
1 1
1 0
equal to 1 are I,and, and the six elements modulo 2 are
0 1
1 1
(
) (
) (
) (
) (
)
1 1
1 0
0 1
1 1
0 1
I,
,
,
,
,
(41)
0 1
1 1
1 1
1 0
1 0
Again, the determinants of three of the matrices equals −1, although, modulo 2, it canbe equated to 1. However, it can be veriﬁed that there is no set of three points which are

58
permuted by the second matrix, a coalescence of the three punctures at 0 is required forthe third matrix, and the locations of the three punctures for the remaining matrices are
(
(
(

√
√
−3 ± 52−1 ± 3i: x=, y= √
, z=
2
2
± 5−1
√
√
√
)
1 ± 3i3 ± 3i1± 5
1√ , z=
, y=
: x=
0
2
2
1 ± 3i)1: x = 0, y = ∞, z = ±1
0

0 1
1 1
1
1
0
1

)

(42)

Consequently, the fractional linear transformations cannot ﬁx the same set of points inC∗ , and the conclusions of Theorem 3.2 are conﬁrmed.
It follows that for a thrice-punctured sphere, an S3 permutation symmetry, representing the interchange of the punctures, exists. This group also acts freely on any ﬁnite
complex homotopic to S 3 , and by restriction, S 2 [4]. Furthermore, this symmetry will be
included in the group of deck transformations of the covering of the surface.
String theory amplitudes have been given formally by a sum over genus of moduli
space integrals deﬁned over all quasiconformal deformations of the metric not related by
the identity component of the diﬀeomorphism group and factored by Weyl rescalings of thetwo-dimensional metrics on the Riemann surfaces. The action of the S3 transformation
may considered for string theory amplitudes. Since the thrice-punctured spheres are used
initially for surfaces of genus g ≥ 2, it is necessary to introduce vertex operators at zeroand one loops. Furthermore, it can be concluded that the invariance under an appropriate
S3 action on the modular parameter integration region is reﬂected at lower energies in auniversality principle for trilinear couplings in the ﬁeld theory limit.
The genus-g surface can be constructed by the sewing of spheres with three punctures.
Consider 2g − 2 spheres with three punctures with g − 1 components in two separate
(1)
(g−1)
groups. The ﬁrst set will be labelled as {Σ0,3 , ..., Σ0,3 }, while the second group is(g)(2g−2){Σ0,3 , ..., Σ0,3 }. Suppose that a handle is placed between two of the punctures on(1)(2)
Σ0,3 and the third puncture is joined to a puncture of Σ0,3 . The process of creating ahandle between two punctures is the reverse of that of the pinching of a handle thatyields a singular surface in the compactiﬁcation divisor of moduli space. By connecting
(2)
(3)
(g+1)
(2)
the two remaining punctures of Σ0,3 to punctures on Σ0,3 and Σ0,3 , Σ0,3 is sewed to two
other components of the Riemann surface. This procedure can be iterated with the two
(i)
(i+1)
(g+i−1)remaining punctures of Σ0,3 being joined to Σ0,3 and Σ0,3
, i = 3, ..., g −1. Then all of
the punctures of the ﬁrst set of components have been sewed, and there are two available
punctures on the each of the components in the second set. Since it is not necessary to
sew any of these components to other thrice-punctured spheres, handles can be placed on(g)(2g−2)
Σ0,3 , ..., Σ0,3 . It follows that the sewed surface has g handles and no punctures.
This technique can be extended to N -point, g-loop amplitudes represented by thesewing of 2g − 2 + N thrice-punctured spheres. The integration over moduli space would

59
be described by the nature of the handles and the propagators between diﬀerent components The interchange of two punctures which are replaced by a handle does not aﬀect theamplitude, whereas the interchange of the punctures that are connected to diﬀerent components would transform the integration to a separate part of moduli space. The action of
the symmetric permutation group may be applied to each component Σ0,3 of the decomposition on which no handles are attached, it must be possible to partition the integration
region of the modular parameters at genus g according to an S3 × S3 × ... × S3 = S3g−2
symmetry.
The method of sewing thrice-punctured spheres to construct a genus-g surface begins
at g = 2. Summing the amplitudes from genus 3, there would be S3 , S3 × S3 , S33 , S34 ,... actions on the integration regions for the path integral. The total amplitude would
contain an invariance only under the intersection of these symmetries, which would ben∩∞n=3 (S3 ) = S3 .

(43)

At genus 1, when the handle is placed between two punctures, the inclusion an additional vertex operators would require a thrice-punctured sphere. The genus 2 surfaces
are constructed by connecting two once-punctured tori. Likewise, at tree level, thricepunctured spheres again occur when N ≥ 3. The eﬀect of an S3 transformation on the
integral for the one-point, one-loop diagram would be the interchange of the location of
the vertex operator with that of the handle in the corner of moduli space deﬁned by thethinning of the handle until a puncture appears. For the higher-point one-loop diagrams,
the interchange of vertex operators arises as well. The action of the symmetric permutation group therefore implies a comparison of the coincidences of vertex and handle
operators with the coincidences of vertex operators [2]. This possibility, which potentiallycomplicated the analysis of divergences in superstring theory, can be shown to have aregular solution.
It may be recalled that one-point functions determine the stability of the vacuum,
two-point amplitudes are related to mass or wavefunction renormalization and three-point
amplitudes yield predictions of the couplings in the low-energy ﬁeld theory. Consequently,
the three-point tree-level amplitude would govern the leading-order contribution to the
Yukawa couplings in the ﬁeld theory [25].
It is known that N -point superstring amplitudes vanish for N < 4, and therefore, it
might appear that the three-point amplitude cannot be used. However, it may be conjectured that the supersymmetry breaking in a phenomenological Lagrangian will render thethree-point functions nonvanishing. Therefore, if the formal S3 invariance in the integral
for the superstring amplitude is preserved at lower energies, its eﬀect would be evident
in the ﬁeld-theory limit. This result is conﬁrmed by the universality of the couplings ininteractions between photons, intermediate vector bosons and gluons and the fermions in
the three diﬀerent generations.
The symmetric permutation group of three elements also arises in the automorphismgroup of the standard model. It has been demonstrated that the spinor space of thestandard model may be expressed as ⊕3i=1 (Ci ⊗ Hi ⊗ Oi ). The automorphism group [3]60then equalsAut(⊕3i=1 (Ci ⊗ Hi ⊗ Oi )) = Aut(C ⊗ H ⊗ O) wr S3 = G2 × SU (2) × U (1) wr S3 .
Symmetry breaking generates diﬀerent masses for the quarks and leptons in each of thegenerations. There are relations between the masses of the charged leptons, however,which reveal an underlying S3 invariance [16] . It is known that there is an extended
model with a larger S3 × S3 invariance [8] which can be used as a theoretical basis for
the Koide relation. Again, it can be concluded that this eﬀected may be connected toinvariances of amplitudes in string theory.
Thrice-punctured spheres may be used construct the moduli spaces of higher-genus
surfaces with punctures and the rays in the Hilbert space representing the universal Grassmannian [23]. The energy-momentum tensor of the string can be identiﬁed with the actionof meromorphic vector ﬁelds on the Riemann surface, which generate a Krichever-Novikov
algebra at higher genus [15].

Theorem 4.2. A paracompact model of the universal moduli space of surfaces with n
1 3g+n−3punctures is diﬀeomorphic to the union (CP1 )n−3 ∪ (CP1 )n ∪ ∪∞/(S3g−1 ) for
g=2 ((CP )n ≥ 4.
Proof. The moduli space of the four-punctured sphere, CP1 − {0, 1, ∞}, is diﬀeomorphic
to the thrice-punctured sphere and its closed compactiﬁcation is M̄0,4 = CP1 . Because
the moduli space of the ﬁve-punctured sphere is M0,5 = (CP1 − {0, 1, ∞})2 − ⟨x = y⟩,with x, y ∈ CP1 , and the divisor includes M0,4 [20], the closed compactiﬁcation could
be identiﬁed with (CP1 )2 after ﬁlling in the remaining punctures and abbreviating theprocess of adding the boundary by excluding three extra lines at the divisor. Similarly,
since M0,n = (CP1 − {0, 1, ∞})n−3 − ∆ when n ≥ 5, where ∆ is a set of codimension
greater than or equal to one. there would be a transformation from M̄0,n to ((CP1 )n−3 ).
The methods for evaluating accessory parameters for the torus with one puncture andthe four-punctured sphere, which have hyperbolic metrics, are known to be equivalent,since there is a covering of T1,1 ≃ U/G over Σ0,4 ≃ U/G̃, with punctures at ∞, e1 , e2 ande3 [17], with the Weierstrass function invariant under the interchange of e1 , e2 and e3 ,and (M1,1 )/Z4 ≃ M0,[4] ≃ M0,4 /S4 . Similarly, the transformation from M1,2 to M0,5 /S4[20] conﬁrms the factorization by S4 . The two-valued uniformizing function from thefour-punctured sphere to the complex plane yields a cancellation of the Z2 arising fromﬁxing the ﬁfth puncture in Σ0,5 at one of the two punctures in M1,2 . Consequently, theaddition of a handle is equivalent to that of three punctures, such that the compactiﬁed
moduli space is described by a tensor product with ((CP1 )3 × Z4 )/S4 . A relation between
g−1
M̄g,n and ((CP1 )3g+n−3 × Zg−1
would follow from a sewing of disks around the
4 )/(S4 )punctures to create the handles and the additional modular parameters resulting from theuse of two punctures, with 3g − 3 moduli for each genus, when g ≥ 2. The quotients by

61
the powers of the symmetric groups may deﬁned through the action of S4 on (CP1 )4 , by
the permutation of indices in the cross-ratio function which is a holomorphic mapping ofordered four-tuples on CP1 to C\{0, 1}, that can be restricted to the action of S3 on CP3[9]. There would exist a transformation from the universal moduli space of n-punctured
1 3g+n−3
surfaces with n ≥ 4 to (CP1 )n−3 ∪ (CP1 )n ∪ ∪∞/(S3g−1 ).
g=2 ((CP )The total derivative ambiguities in superstring amplitudes [24] can be eliminated at genus
one and two [6]. At higher genus, the integral of a total derivative over the covering spaceof the union of closed compactiﬁcations of the moduli spaces would vanish.
The group of quasi-special, symmetric outer automorphisms of the proﬁnite completion Γ̂0,4 [21] of the fundamental group Γ0,4 = F2 = ⟨x, y, z|xyz = 1⟩, the free group of
two elements consisting of squares of Dehn twists [20], is deﬁned by the action of automorphisms on the two elements x and y. It is known similarly that the automorphismgroups Out# (Γ̂0,n ) for n ≥ 5 are isomorphic, with M̄0,5 ≃ (CP1 )2 ≃ S 2 × S 2 , and theouter automorphisms of the proﬁnite completions of the fundamental groups of the moduli
spaces Mg,n include the absolute Galois group GQ [10]. At higher genus, stability of theouter automorphism group Out# (Γ̂g,n ) can be valid for n larger than a genus-dependent
lower bound related to a generalization of the ﬁve-value theorem [5].

5Conclusion

The relation between the roots of an monic irreducible polynomial in two variables g(x, y)
at a value x0 that is an algebraic function of the coeﬃcients, which yields an irreducible
polynomial in one variable, g(x0 , y) is suﬃcient to establish the algebraicity of the rootsof the g(x0 , y) given the algebraicity of one of the roots. A group theoretical equivalent of
this result has been given. Furthermore, there is a connection between the Galois groupof the quotient of the splitting ﬁeld of an irreducible polynomial f (x, y) and Q̄(x) andthe deck transformations of a Riemann surface that can be constructed with puncturesat the zeros of the function in a neighbourhood of a point x0 . The conformal group ofthrice-punctured sphere is examined and shown to coincide with the Galois group, thepermutation group S3 , if the matrix representation of the group of covering transformationis enlarged from SL(2; R) to SL(2; R)∪T ·SL(2; R), where det T = −1. More generally, thepresentation of matrices representing the action of the fractional linear transformations
on C∗ which preserve the set of three punctures is shown to always contain three matrices
of determinant −1 within multiplication by I.
The necessity of the matrices of determinant −1 for the presentation of S3 has implications for the uniformization theorem of Riemann surfaces. While surfaces of genus
2 are known to have the upper half plane H or the unit disk U as the covering surface,
the thrice-punctured sphere, which is conventionally viewed as belonging to the class of
surfaces with H as a covering surface, must be uniformized with an enlargement of thecovering group.

62
The study of moduli spaces of n-punctured spheres becomes nontrivial only whenn ≥ 4. Therefore, the description of Mg,n would be expected to require the moduli spacesof m-punctured spheres, with 4 ≤ m ≤ n, even though the thrice-punctured spheres may
be sewn together to form the higher-genus surfaces. Based on work relating the moduli
spaces of the punctured torus and the four-punctured sphere, the moduli and the accessory
parameters related to the addition of handles, in principle, can be evaluated. A method
for compactifying the moduli spaces of punctured surfaces at higher genus follows fromthe description of M̄1,1 , the addition of new punctures and the sewing rules. The unionof each of the compactiﬁed moduli spaces over the genus then furnishes a paracompact
model of a universal moduli space.

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