On large prime actions on Riemann surfaces
-
SebastiĂĄn Reyes-Carocca
und
Anita M. Rojas
Abstract
In this article, we study compact Riemann surfaces of genus đ with an automorphism of prime order
1 Introduction and statement of the results
Let
where
where
The classification of groups of automorphisms of compact Riemann surfaces is a classical problem which has attracted broad interest ever since it was proved that the full automorphism group of a compact Riemann surface đ of genus
It is well known that there are infinitely many values of đ for which there is no compact Riemann surfaces of genus đ possessing
and that, for infinitely many values of đ, the inequality (1.1) turns into an equality.
We denote by
The AccolaâMaclachlan curve is a remarkable example of a compact Riemann surface determined by the order of its full automorphism group.
More precisely, Kulkarni in [43] succeeded in proving that, up to finitely many values of the genus, if
The analogous problem of finding
(we shall recall the precise definition of signature in 変2.1 and 変2.2).
It was then shown that
It is a well-known fact that if a compact Riemann surface of genus
Let
consisting of all those compact Riemann surfaces of genus
This paper is devoted to classifying and describing the surfaces lying in
under the assumption that
The classification
The first result of the paper provides a classification of these surfaces.
Let
Assume
Assume
If
If
Assume
Assume
đ is isomorphic to one of the
đ belongs to the closed family
equisymmetric strata; one of them being
We point out some observations concerning Theorem 1.
If
The case
We conjecture that the upper bound given in the theorem for the number of equisymmetric strata of the closed family
We emphasise that the equisymmetric family
Algebraic description
Although the literature still shows few general results in this direction, there is a great interest in providing descriptions of compact Riemann surfaces as algebraic curves in an explicit manner. The following result gives such a description for the surfaces appearing in Theorem 1, as well as a realisation of their full automorphism groups.
Let
If đ belongs to the closed family
In addition, if
Assume
where
where
The surface
and in this model, its full automorphism group is generated by
For each
and, in this model, its full automorphism group is generated by
If đ belongs to the closed family
and, if
where
The theorem above overlaps results obtained in [68, 変11].
Hyperelliptic surfaces
Arakelian and Speziali in [3] studied groups of automorphisms of large prime order of (non-necessarily smooth) projective absolutely irreducible algebraic curves over algebraically closed fields of any characteristic.
In terms of our terminology, in [3, Theorem 4.7], they proved that if
The following result lengthens the implication above; it follows from Theorem 1.
Let
Jacobian variety
Let đ be a compact Riemann surface of genus
where
We emphasise the following two classical facts (see, for example, [10]):
up to isomorphism, the surface is determined by its Jacobian (Torelliâs theorem).
If đș is a group acting on đ, then đș also acts on
where the factors
The following result provides the group algebra decomposition of the Jacobian varieties of the surfaces of Theorem 1, with the exception of
where
Let
The Jacobian variety
The Jacobian variety
and đ is a primitive fourth root of unity in
The Jacobian variety
and
Field of moduli and fields of definition
Let
where
The field of moduli of a compact Riemann surface đ is the fixed field
The field of moduli of đ agrees with the intersection of all its fields of definition and, as proved by Koizumi in [41], đ can be defined over a finite degree extension of
Necessary and sufficient conditions under which đ can be defined over its field of moduli were provided by Weil in [69] (see also [34] for a constructive proof of Weilâs theorem); these conditions are trivially satisfied if đ has no non-trivial automorphisms. Besides, as proved by Wolfart in [71], if đ is quasiplatonic, then đ can be defined over its field of moduli.
The general question of deciding whether or not the field of moduli is a field of definition is a challenging problem; see, for example, [4, 27, 30, 33, 35, 42, 54].
In this direction, it is a known fact that if the genus of
We now study the aforementioned problem for the compact Riemann surfaces of Theorem 1. First, note that, for the quasiplatonic ones, the problem is trivial. Indeed,
as proved in Theorem 2, the surfaces
As mentioned above, the fact that
The remaining cases (that is, the surfaces lying in the family
Let
for
It it worth mentioning that a compact Riemann surface and its Jacobian variety can be defined over the same fields and that their fields of moduli agree; see [63] and also [47].
The following result is a direct consequence of the above.
The compact Riemann surfaces of Theorem 1 and their Jacobian varieties can be defined over their fields of moduli.
The sublocus of
A
g
with đș-action
It is well known that the moduli space
of the Siegel upper half-space
where
consists of those matrices representing principally polarised abelian varieties of dimension đ admitting an action which is equivalent to that of
Observe that if
In general, those loci of
Although a satisfactory description of the matrices in
Let
According to results due to Streit in [67] (and later generalised in [26] for higher dimension), if
As a direct consequence of the previous theorem, we recover the following known result.
The Jacobian varieties of
In spite of the fact that the problem of determining the period matrix of a given Jacobian variety is, in general, intractable, interesting results have been obtained for some famous Riemann surfaces. For instance, the period matrices of Macbeathâs curve of genus seven and of Bringâs curve were determined in [9] and [60] respectively. A method to find the period matrices of the AccolaâMaclachlan and Kulkarni surfaces was given in [13]. In addition, in [13, Example 3.7], the authors went even further and employed their method to provide the period matrix of the AccolaâMaclachlan curve of genus two in an explicit way.
At the end of the paper, we determine explicitly the period matrix of the AccolaâMaclachlan curve of genus four.
This article is organised as follows. In §â2, we succinctly review the basic preliminaries: Fuchsian groups and group action on Riemann surfaces and abelian varieties. The proof of Theorem 1 is given in §â3, and the proofs of Theorem 2 and Proposition 2 are given in §â4. In §â5, we prove some basic algebraic lemmata needed to prove, in §â6, Theorems 3 and 4. Finally, we include an addendum in which the period matrix of the AccolaâMaclachlan curve of genus four is computed.
2 Preliminaries
2.1 Fuchsian groups
A Fuchsian group is a discrete group of automorphisms of the upper half-plane â.
If Î is a Fuchsian group and the orbit space
where đŸ is the genus of
where the brackets stand for the commutator.
The TeichmĂŒller space of Î is a complex analytic manifold homeomorphic to the complex ball of dimension
Let
where
2.2 Group action on Riemann surfaces
Let đ be a compact Riemann surface of genus
By the classical uniformisation theorem, there is a unique, up to conjugation, Fuchsian group Î of signature
It is said that đș acts on đ with signature
2.3 Extending actions
Assume that
there is a Fuchsian group
the TeichmĂŒller spaces of Î and
there exists an ske
2.4 Equivalence of actions
Two actions
Each đ satisfying (2.4) yields an automorphism
Two skes
2.5 Equisymmetric stratification of
M
g
Following [11], the singular locus of
where the equisymmetric stratum
The subset
Assume that the action of đș is maximal. Then
the interior
the set
2.6 Abelian varieties
A complex abelian variety is a complex torus which is also a complex projective algebraic variety.
Each abelian variety
If each elementary divisor of
such a basis is called symplectic. In addition, there exists a basis for đ with respect to which the period matrix for đ is
with
By an isomorphism of ppavs, we mean an isomorphism of the underlying complex tori preserving the involved polarisations.
In other words, if
Since đ preserves the polarisation (2.5), it belongs to the symplectic group
It follows from (2.6) that the correspondence
defines an action that identifies period matrices representing isomorphic ppavs. Hence
is the moduli space of isomorphism classes of ppavs of dimension đ. See [51].
2.7 Abelian varieties with đș-action
Let đ be a compact Riemann surface of genus
according to whether or not đ is hyperelliptic; moreover, in the latter case,
As mentioned in the introduction, once a symplectic basis of
where
of
According to [67] (see also [26, Lemma 3.8]), the dimension
where
and
It is worth mentioning that
one sees that
2.8 The group algebra decomposition
The action of a group đș on a compact Riemann surface đ induces a â-algebra homomorphism from the rational group algebra of đș to
Let
where
which is đș-equivariant, with
where
known as the group algebra decomposition of
Let đ» be a subgroup of đș.
We denote by
The previous isogeny has proved to be fruitful in finding Jacobians
Assume that
where
For decompositions of Jacobians and families of Jacobians with respect to special groups, we refer to the articles [5, 14, 15, 24, 25, 32, 52, 53, 57, 55, 61].
3 Proof of Theorem 1
The proof of Theorem 1 is presented as a consequence of a series of propositions proved in this section.
Hereafter, we assume
If
Proof
Let
Observe that if
After a routine computation, one sees that the unique solution of the previous equation is, up to permutation,
Consider the Fuchsian group Î of signature
and let
this proves the uniqueness of đ.
By the results of [66], if đș is strictly contained in the full automorphism group
đ is different from 5, 6 and 7.
Proof
If đ is equal to 5, 6 or 7, then đș is a large group of automorphisms (that is,
If đ is equal to 5 or 7, then đș has no involutions; then the signature of the action đș is
respectively. The RiemannâHurwitz formula implies that
respectively, and this, in turn, implies that đ is negative, a contradiction.
We now assume that đș has order
It follows that the signature of the action is
satisfy, by the RiemannâHurwitz formula, the equality
Set
Thus
All the above ensures that each
and therefore
If
Proof
If the order of đș is at least
| Case | Signature |
|
Condition | RiemannâHurwitz formula |
|---|---|---|---|---|
| (1) |
|
|
|
|
| (5) |
|
|
|
|
| (8.1) |
|
|
|
|
| (8.2) |
|
|
|
|
We also note that cases (4), (6) and (7) are not realised.
| Case | Signature |
|
Condition | RiemannâHurwitz formula |
|---|---|---|---|---|
| (4) |
|
|
|
|
| (6) |
|
|
|
|
| (7) |
|
|
|
|
Note that,
in case (4), we have
in case (6), we have
in case (7), the facts that
We claim that case (2) is not realised either.
Indeed, note that otherwise the order of đș is equal to
If
If
All the above ensures that the signature of the action is
It follows that one of the following statements holds.
The first two cases must be disregarded because đ is not prime; then đș has order
We recall that, following [22], the closed family
acting with signature
Proof
Observe that if đ belongs to
For later and repeated use, we recall here that
and its action on
where
If đ is a compact Riemann surface of genus đ with a group of automorphisms isomorphic to
Proof
Let
and consider the group
Let
we can assume that
Observe that, with the notation of (3.4), the elements
generate a subgroup of
By letting
If đ is a compact Riemann surface of genus đ with a group of automorphisms isomorphic to
Proof
Let
and consider the group
Let
Up to conjugation, we can assume
Now, as in the previous proposition, with the notation of (3.4), we see that
generate a subgroup of
Write
to see that (3.9) and (3.10) agree; consequently
Assume
Proof
Consider the Fuchsian group
where đ is a primitive fourth root of unity in
It follows that, up to isomorphism, there are at most two surfaces đ as before, namely
Observe that if the full automorphism group of
We record here that
If
Proof
As in the proof of Proposition 3.2, the signature of the action of đș on đ is either
Note that, among
Assume
Assume
Thereby, the signature of the action of đș on đ is either
We recall that if
If đș acts with signature
If đș acts with signature
If
Proof
Let
and consider the cyclic group of order
Then such surfaces đ form an equisymmetric complex one-dimensional family. Let
be a Fuchsian group of signature
We recall that, following [22], the action of
Now, the elements of
generate a Fuchsian group isomorphic to
Set
Let
Proof
Let
and consider the cyclic group of order
If
Let
generate a subgroup of (3.5) isomorphic to
By setting
Let
The second case is impossible since it does not have the required signature; the other two cases are equivalent to
Write
has orbits of length
splits into
There exists a closed family
and, independently of đ, one of them is equal to
Proof
Let
The existence of the family
Let us now assume that
The braid transformation
has order two, restricts to a bijection of
All the above ensures that đ is equivalent to either
Now, for each
The rule
and the quadratic equation above has solution in
if
and note that they generate a Fuchsian group isomorphic to
If we write
If
Proof
We keep the notation of the proof of Proposition 3.11.
The first statement is clear since the full automorphism group of đ is isomorphic to
Let
If
if
if
Assume
where
generate a Fuchsian group isomorphic to
By letting
If
Proof
Assume that the signature of the action of đș on đ is
In both cases, we see that
As each
Let
If
showing that
If
showing that
If
(3.24)It is easy to verify that the unique solution of (3.24) is
We record here the simple fact that a group of order
If the signature is
Let
where
In terms of our terminology, the results of [21] allow us to claim that the stratum determined by
4 Proof of Theorem 2 and Proposition 2
Let
The family
C
ÂŻ
g
We recall that if
on đ has signature
Let
and observe that ÎŁ has genus zero and đ ramifies over
then it is classically known that đ is isomorphic to the normalisation of
Note that
and đ is a nonzero complex number such that
for some
restrict to automorphisms of
Note that
The surface
X
4
Following the proof of Proposition 3.7, the action of
on
ramifies over four values marked with đ. As ÎŁ admits the action of
and
Since đ has order 4 in
and note that they restrict to automorphisms of (4.2). If đ is as in the statement of the theorem, then routine computations show that
This implies that đ has order four.
Now, it is direct to see that
The Möbius transformation
and lifts to obtain an isomorphism between
This provides the explicit model for
The surface
X
3
Similarly to before, the normalisation of
The surfaces
X
2
,
k
Following the proof of Proposition 3.10, the action of
on
If
ramifies over four values marked with đ.
As
and
for some
are automorphisms of (4.3) and that
The family
K
ÂŻ
g
Following the proof of Proposition 3.11, the action of
on
If
ramifies over four values marked with đ. As ÎŁ admits the action of
and
It is straightforward to verify that the transformations
restrict to automorphisms of
Proof of Proposition 2
Assume that
and therefore, if
Let
đș is the unique group of automorphisms of đ isomorphic to
We have two possible cases.
If
If
Observe that the latter case in (1) and the former case in (2) imply
as desired.
5 Some algebraic lemmata
In this section, we collect some facts related to the representations of the groups appearing in Theorem 1; these results will be needed to prove Theorems 3 and 4.
Set
Rational and complex irreducible representations
Let
has four complex irreducible representations of degree one given by
and
| Representation | đ„ | đŠ | đ§ |
|---|---|---|---|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Here
Proof
The proof is an application of the method of Wigner and Mackey to build the complex irreducible representations of certain semidirect products. See, for example, [64, §â8.2]. â
Let
has four complex irreducible representations of degree one given by
and
for
Proof
The construction of the representations follows from [64, 変8.2], and we only need to prove the last statement.
As
Symmetric square character formula
Let đ» be a finite group, and let
According to [64, Proposition 2.3], the character
where
Let đ be a character of đ», and let
Proof
According to [64, Exercise 2.1], for any pair of characters
If we write
Since
the conclusion follows after noting that
The analytic representation
Let đ be a compact Riemann surface of genus đ, and let đ» be a group of automorphisms of đ. The action of đ» induces a complex representation
called the analytic representation of đ».
Let
then
Assume that the action of đ» on đ has signature
where
Computation of the dimension
N
S
,
G
Let đ be a compact Riemann surface of genus đ.
As mentioned in 変2.7, following [67] and [26, Lemma 3.8] together with formula (5.1), the dimension of (the component which contains
where
As a direct consequence of Lemma 3, the previous equality can be rewritten as follows.
Assume that
Note the following.
The computation of
Observe that if
6 Proof of Theorems 3 and 4
The surface
X
8
With the notation of Lemma 1, the representations
of
As explained in 変2.8, the group algebra decomposition of
We recall that the action of
do not have 1 as an eigenvalue.
In addition, it is easy to see that the character field of
Since the dimension of
Finally, we consider the subgroup
as claimed in Theorem 3.
We now proceed to prove that
We claim that the analytic representation
To prove that, observe that
then according to Lemma 4, we have
for each
In this way, we obtain if
We then can construct the following table.
| â | Order |
|
|
|---|---|---|---|
| 1 | 1 |
|
|
|
|
2 | 0 |
|
|
|
đ |
|
â2 |
|
|
|
0 | â2 |
Then we obtain, by (5.4), that
Now, the fact that
and therefore we obtain
The surface
X
4
With the notation of Lemma 2, the group algebra decomposition of
We recall that the action of
The dimension of the fixed subspace of
The previous equality shows, in addition, that
Finally, we consider the subgroup
as claimed in Theorem 3.
We now proceed to prove that
where đ« is as in Lemma 2. It is not difficult to see that
and therefore (5.4) allows us to write that
Thus
and consequently, Lemma 5 says that
The surface
X
3
The complex irreducible representations of
We denote by
In fact, according to [62, Theorem 5.10], the multiplicity of
Since
In order to determine the character
and consider the subset
Clearly,
For the elements of order 3 (that is,
Analogously, one sees that
For the elements of order đ (that is,
for every
showing that
For the elements of order
and this corresponds to the sum of all primitive
We summarise all the above in the third column of the following table; the fourth column follows from all the above and (5.4).
| Order | đ |
|
|
|---|---|---|---|
| 1 | 1 |
|
|
| 3 |
|
|
|
| đ |
|
â2 |
|
|
|
|
1 |
|
It follows that
and the desired result
The surfaces
X
2
,
k
Let
given by
are rational irreducible representations of
Let
and if
where
and therefore we can construct the following table.
| Order | đ |
|
|
|---|---|---|---|
| 1 | 1 |
|
|
| 2 | đ | 0 |
|
| đ |
|
â2 |
|
|
|
|
0 |
|
The desired conclusion
The family
C
g
The complex representation
has Schur index 1 and field of characters of degree
where đ” is the abelian subvariety of
and therefore
A routine application of (5.2) permits us to see that the analytic representation
where đ runs over the Galois group associated to character field of đ.
The following table (taken from [37, Proposition 6.1]) collects the character of
| đ |
|
|
|---|---|---|
| 1 |
|
|
|
|
1 |
|
|
|
â1 |
|
|
|
|
|
| đ | 0 |
|
|
|
0 |
|
Thus, by equation (5.3), we obtain that
The family
K
g
It is well known that
all of them are Galois conjugate. Clearly, their direct sum
is a rational irreducible representation of
where đ” is the abelian subvariety of
the dimension of the fixed subspace of
the dimension of the fixed subspace of
It follows that, independently of the equisymmetric stratum to which đ belongs (see the ske
and therefore
By Lemma 4, one sees that the analytic representation
| đ |
|
|
|---|---|---|
| 1 |
|
|
|
|
0 |
|
|
|
â1 |
|
Finally, by equation (5.3), we obtain that
Funding source: Fondo Nacional de Desarrollo CientĂfico y TecnolĂłgico
Award Identifier / Grant number: 1180073
Award Identifier / Grant number: 11180024
Award Identifier / Grant number: 1190991
Funding statement: Partially supported by Fondecyt Grants 1180073, 11180024, 1190991 and Redes Grant 2017-170071.
A Addendum
We recall here the fact that the full automorphism group of the AccolaâMaclachlan curve
Now, if we write
then with the notation of 変2.7, the fact that
implies that the coefficients of đ satisfy the relations
The solutions of (A.1) are
The values
do not have imaginary part positive, and therefore the corresponding matrices do not belong to
With the help of numerical approximations of [73] one sees that đż is positive only for
We replace the value of
with
Acknowledgements
The authors are very grateful to the referee for his/her valuable comments and suggestions.
-
Communicated by: Robert Guralnick
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© 2022 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Lower bound on growth of non-elementary subgroups in relatively hyperbolic groups
- Primitivity rank for random elements in free groups
- On the Tits alternative for cyclically presented groups with length-four positive relators
- Lie groups as permutation groups: Ulamâs problem in the nilpotent case
- A pro-2 group with full normal Hausdorff spectra
- On large prime actions on Riemann surfaces
- Powers in wreath products of finite groups
- On powers of conjugacy classes in finite groups
- Conjectural invariance with respect to the fusion system of an almost-source algebra
Artikel in diesem Heft
- Frontmatter
- Lower bound on growth of non-elementary subgroups in relatively hyperbolic groups
- Primitivity rank for random elements in free groups
- On the Tits alternative for cyclically presented groups with length-four positive relators
- Lie groups as permutation groups: Ulamâs problem in the nilpotent case
- A pro-2 group with full normal Hausdorff spectra
- On large prime actions on Riemann surfaces
- Powers in wreath products of finite groups
- On powers of conjugacy classes in finite groups
- Conjectural invariance with respect to the fusion system of an almost-source algebra
