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Abstract

In this article, we extend results of Zomorrodian to determine upper bounds for the order of a nilpotent group of automorphisms of a complex d-dimensional family of compact Riemann surfaces, where d1.d \geqslant 1. We provide conditions under which these bounds are sharp. In addition, for the one-dimensional case, we construct and describe an explicit family attaining the bound for infinitely many genera. We obtain similar results for the case of p-groups of automorphisms.
arXiv:2004.06506v2 [math.AG] 19 May 2021
NILPOTENT GROUPS OF AUTOMORPHISMS
OF FAMILIES OF RIEMANN SURFACES
SEBASTI ´
AN REYES-CAROCCA
Abstract. In this article we extend results of Zomorrodian to determine upper bounds
for the order of a nilpotent group of automorphisms of a complex d-dimensional family
of compact Riemann surfaces, where d>1.We provide conditions under which these
bounds are sharp. In addition, for the one-dimensional case we construct and describe
an explicit family attaining the bound for infinitely many genera. We obtain similar
results for the case of p-groups of automorphisms.
1. Introduction and statement of the results
The classification of groups of automorphisms of compact Riemann surfaces is a classi-
cal subject of study which has attracted considerable interest ever since Hurwitz proved
that the full automorphism group of a compact Riemann surface of genus g>2 is finite
and that its order is at most 84(g1).Later, this problem acquired a new relevance
when its relationship with Teichm¨uller and moduli spaces was developed.
It is classically known that there are infinitely many values of gfor which there exists
a compact Riemann surface of genus gwith automorphism group of maximal order; they
are called Hurwitz curves and correspond to branched regular covers of the projective
line with three branch values, marked with 2, 3 and 7.
We recall the known fact that each finite group can be realized as a group of auto-
morphisms of a compact Riemann surface of a suitable genus. In part due to the above,
an interesting problem is to study those compact Riemann surfaces whose automor-
phism groups share a common property and, after that, to describe among them those
possessing the maximal possible number of automorphisms.
Perhaps, the most noteworthy examples concerning that are the abelian and cyclic
cases. In fact, in the late nineteenth century, Wiman showed that the largest cyclic
group of automorphisms of a compact Riemann surface of genus g>2 has order at most
4g+ 2.Moreover, the compact Riemann surface given by the algebraic curve
y2=x2g+1 1
shows that this upper bound is attained for each g. See [53] and also [20] and [29].
2010 Mathematics Subject Classification. 14H30, 30F35, 14H37, 14H40.
Key words and phrases. Riemann surfaces, Fuchsian groups, Group actions, Jacobian varieties.
Partially supported by Fondecyt Grants 11180024, 1190991 and Redes Grant 2017-170071.
2 SEBASTI ´
AN REYES-CAROCCA
Meanwhile, as a consequence of a result due to Maclachlan, the order of an abelian
group of automorphisms of a compact Riemann surface of genus g>2 is at most 4g+ 4;
see [33]. In addition, the fact that for each gthere exists a compact Riemann surface of
genus gwith a group of automorphisms isomorphic to C2×C2g+2,shows that that this
upper bound is attained for each value of g.
Similar bounds for special classes of groups can be found in the literature in plentiful
supply. For instance, the solvable case can be found in [10] and [16], the supersolvable
case in [17] and [50], the metabelian case in [11] and [15], the metacyclic case in [44] and
several special cases of solvable groups in [46]. We also refer to the survey article [14].
By contrast, it seems that not much is known in this respect when considering complex
d-dimensional families of compact Riemann surfaces, for d>1. Very recently, Costa
and Izquierdo in [12] proved that the maximal possible order of the automorphism group
of the form ag +b, where a, b Z,of a complex one-dimensional family of compact
Riemann surfaces of genus g>2,appearing for all genera, is 4(g+ 1).In addition, they
went even further by exhibiting an explicit equisymmetric family of non-hyperelliptic
compact Riemann surfaces attaining this bound for all g(c.f. [5]). Later, the analogous
problem for complex low dimensional families (d64) was addressed in [27] and [37].
The aim of this article is to deal with nilpotent groups and p-groups of automorphisms
of complex d-dimensional families of compact Riemann surfaces, where d>1.
We recall that the Jacobian variety JC of a compact Riemann surface Cof genus gis
an irreducible principally polarized abelian variety of dimension g. The relevance of the
Jacobian variety lies, in part, in the classical Torelli’s theorem, which establishes that
C1
=C2if and only if JC1
=JC2.
In this paper we shall also consider isogenous decompositions of Jacobian varieties of
certain compact Riemann surfaces with a nilpotent group of automorphisms.
Nilpotent groups acting on families of Riemann surfaces
In [32] Macbeath considered homomorphisms from co-compact Fuchsian groups onto
finite nilpotent groups. Since every finite nilpotent group is isomorphic to the direct
product of its Sylow subgroups, after introducing the concept of p-localization of groups,
he succeeded in providing necessary and sufficient conditions under which a given sig-
nature appears as the signature of the action of a nilpotent group of automorphisms on
a compact Riemann surface.
Soon after and based on the aforementioned Macbeath’s result, Zomorrodian in [52]
proved that the order of a nilpotent group of automorphisms of a compact Riemann
surface of genus g>2 is at most 16(g1).Moreover, he noticed that if the previous
bound is sharp then g1 is a power of two and the signature of the action is (0; 2,4,8).
Here we extend the previous result from (zero-dimensional families of) compact Rie-
mann surfaces to d-dimensional families of compact Riemann surfaces, where d>1.
NILPOTENT GROUP OF AUTOMORPHISMS OF RIEMANN SURFACES 3
Theorem 1. Let d>1, g >2be integers. Let Gbe a nilpotent group of automorphisms
of a complex d-dimensional family of compact Riemann surfaces Cof genus g.
(1) The order Gis at most
M2,d =8(g1) if d= 1
4
d1(g1) if d>2.
(2) The order of Gis M2,d if and only if the signature of the action of Gon Cis
σ2,d =(0; 2,2,2,4) if d= 1
(0; 2,d+3
. . ., 2) if d>2.
(3) If the order of Gis M2,d then Gis a 2-group. In particular if, in addition, d= 1
or d1a power of two then g1is a power of two.
If g1 is a power of two, in [52] it was also proved the existence of at least one
compact Riemann surface of genus gwith a nilpotent group of automorphisms of order
16(g1), showing that this upper bound is attained for infinitely many values of g.
Note that for d= 2 the previous theorem guarantees that, if the order of Gis maximal
then g1 is a power of two. We notice that the converse is also true. Indeed, following
[37], for each g>2,there exist a complex two-dimensional family of compact Riemann
surfaces of genus gwith a dihedral group of automorphisms of order
M2,2= 4(g1) acting with signature σ2,2= (0; 2,2,2,2,2).
Thus, in particular, if g1 is a power of two then the involved dihedral group is nilpotent
and therefore the upper bound M2,2is attained.
It is worth pointing out here that Zomorrodian’s method to prove the existence of a
compact Riemann surface of genus gwith a nilpotent group of automorphisms of order
16(g1) is based on an inductive argument and does not provide neither the Riemann
surface nor the nilpotent group in an explicit manner; see [52, p. 254].
The following theorem shows that the upper bound Md,1is sharp for infinitely many
values of g. In contrast with the zero-dimensional case, our strategy is to construct a
complex one-dimensional family in an explicit enough way in order to provide a detailed
description of the family. We include an isogeny decomposition of the associated family
of Jacobian varieties.
Theorem 2. For each integer n>5there is a complex one-dimensional closed family of
compact Riemann surfaces Cof genus 1 + 2n3with a nilpotent group of automorphisms
Gof order 2nisomorphic to the semidirect product
(C2×D2n3)C2
presented in terms of generators a, b, r, s and relations
r2n3=s2= (sr)2=a2=b2= 1,[s, b] = [r, b] = 1, ara =r1, asa =sr, aba =br2n4
acting on Cwith signature (0; 2,2,2,4).Furthermore:
4 SEBASTI ´
AN REYES-CAROCCA
(1) the family consists of at most 22n6equisymmetric strata,
(2) up to possibly finitely many exceptions, Cis non-hyperelliptic and its automor-
phism group agrees with G, and
(3) the Jacobian variety JC of Ccontains an elliptic curve isogenous to JChriand
decomposes, up to isogeny, as
JC JChsi×J Chbi,
where the dimensions of JChsiand J Chbiare 2n4and 2n4+ 1 respectively.
Remarks.
(1) The cases n= 3 and n= 4 are exceptional in the sense that the upper bound is
attained by a group with a different algebraic structure. Concretely
(a) for n= 3 (g= 2) the bound is attained by D4,and
(b) for n= 4 (g= 3) the bound is attained by C2×D4and by (C2×C4)C2.
See [1] and [3].
(2) We announce that for each odd integer d>3,the bound M2,d is attained for
infinitely many genera. We shall deal with this problem in a forthcoming paper.
p-groups acting on families of Riemann surfaces.
The fact that nilpotent groups of automorphisms of compact Riemann surfaces of
maximal order turn out to be 2-groups led Zomorrodian to ask for similar bounds for
the class of p-groups. Indeed, he proved in [51] that if Gis a p-group of automorphisms
of a compact Riemann surface of genus g>2 then
|G|6ǫ(g1) where ǫ=
16 if p= 2
9 if p= 3
2p
p3if p>5,
(1.1)
and that (1.1) turns into an equality if and only if the signature of the action is
(0; 2,4,8),(0; 3,3,9) and (0; p, p, p)
respectively. Furthermore, in the same paper it was also proved the existence of a p-
group of order pnacting on a compact Riemann surface of genus 1+ pn for each n>4,
showing that the bounds (1.1) are sharp for infinitely many values of g.
The following result is a direct consequence of Theorems 1and 2.
Corollary 1. Let d>1and g>2be integers. If Gis 2-group of automorphisms of a
complex d-dimensional family of compact Riemann surfaces Cof genus gthen:
(1) the order Gis at most M2,d,
(2) the order of Gis M2,d if and only if the signature of the action is σ2,d, and
(3) the upper bound M2,1is attained for infinitely many values of g.
The following theorem extends both the previous corollary from p= 2 to odd prime
numbers p>3 and the results in [51] from the zero-dimensional situation to complex
d-dimensional families. For each rational number t>0 we denote its integer part by [t].
NILPOTENT GROUP OF AUTOMORPHISMS OF RIEMANN SURFACES 5
Theorem 3. Let d>1and g>2be integers and let p>3be a prime number. Let G
be a p-group of automorphisms of a complex d-dimensional family of compact Riemann
surfaces Cof genus g.
Assume p= 3.
(1) The order of Gis at most
M3,d =3
d(g1).
(2) The order of Gis M3,d if and only if the signature of the action of Gon Cis
σ3,d,h = (h; 3,d+33h
. . . , 3) for some h {0,...,[d
3+ 1]}.
Assume p>5.
(3) Let λdbe the smallest non-negative representative of dmodulo 3. The order of
Gis at most
Mp,d =2
N(g1) where N=2
3d+λd(1
31
p).
(4) The order of Gis Mp,d if and only if the signature of the action of Gon Cis
σp,d = (ˆ
h;p, d+33ˆ
h
... ,p)where ˆ
h= [d
3+ 1].
The previous theorem applied to d= 1 says that if p>3 is a prime number and if G
is a p-group of automorphism of a complex one-dimensional family of compact Riemann
surfaces of genus gthen
|G|6Mp,1=2p
p1(g1) (1.2)
and the that equality holds if and only if the signature of the action is (1; p) for p>5,
and (1; 3) or (0; 3,3,3,3) for p= 3.
The following theorem provides a detailed description of a complex one-dimensional
family of compact Riemann surfaces whose existence shows that the bound (1.2) is sharp
for each prime p>3 and for infinitely many values of g.
Theorem 4. Let p>3be a prime number. For each integer n>3there is a complex
one-dimensional closed family of compact Riemann surfaces Cof genus
1 + (p1)pn1
2
with a p-group of automorphisms Gof order pnisomorphic to the semidirect product
Cpn1pCp=ha, b :apn1=bp= 1, bab1=ari,
where r=pn2+ 1,acting on Cwith signature (1; p).In addition,
(1) the family consists of p1equisymmetric strata,
(2) Cis elliptic-p-gonal,
(3) up to possibly finitely many exceptions, the automorphism group of Cagrees with
G, and
6 SEBASTI ´
AN REYES-CAROCCA
(4) the Jacobian variety JC of Cdecomposes, up to isogeny, as
JC E×Ap,
where Eis an elliptic curve isogenous to JCGand Ais an abelian subvariety of
JC of dimension (p1)pn2
2.
Remark. The groups involved in this paper have order of the form ρ(g1) where ρQ
and g1 is a power of a prime number. We remark that this situation differs radically
from the case in which ρZand g1 is prime; see [2], [25], [26] and [37].
This paper is organized as follows. Section §2will be devoted to briefly review the basic
background: Fuchsian groups, group actions on Riemann surfaces, the equisymmetric
stratification of the moduli space and the decomposition of Jacobian varieties with group
action. The proofs of the theorems will be given in Sections §3,§4,§5and §6.
2. Preliminaries
2.1. Fuchsian groups. AFuchsian group is a discrete group of automorphisms of
H={zC: Im(z)>0}.
If is a Fuchsian group and the orbit space Hgiven by the action of on His
compact, then the algebraic structure of is determined by its signature:
σ(∆) = (h;m1,...,ml),(2.1)
where his the genus of the quotient Hand m1,...,mlare the branch indices in the
universal canonical projection HH.The signature (2.1) is called degenerate if
h= 0 and l= 1 or h= 0 and l= 2 with m16=m2.
Let be a Fuchsian group of signature (2.1). Then
(1) has a canonical presentation with generators α1,...,αh,β1,...,βh, γ1,...,γl
and relations
γm1
1=···=γml
l= Πh
i=1[αi, βil
i=1γi= 1,(2.2)
where [u, v] stands for the commutator uvu1v1.
(2) The elements of of finite order are conjugate to powers of γ1,...,γl.
(3) The Teichm¨uller space of is a complex analytic manifold homeomorphic to the
complex ball of dimension 3h3 + l.
(4) The hyperbolic area of each fundamental region of is given by
µ(∆) = 2π[2h2 + Σl
i=1(1 1
mi)].
(5) The Euler characteristic of the signature σ(∆) is the rational number
χ(σ(∆)) = 1
2πµ(∆).
NILPOTENT GROUP OF AUTOMORPHISMS OF RIEMANN SURFACES 7
We refer to the classical articles [21] and [49] for further details.
Let Γ be a group of automorphisms of H.If is a subgroup of Γ of finite index then Γ
is also Fuchsian and their hyperbolic areas are related by the Riemann-Hurwitz formula
µ(∆) = : ∆] ·µ(Γ).
2.2. Group actions on Riemann surfaces and localization. Let Cbe a compact
Riemann surface of genus g>2 and let Aut(C) denote its automorphism group. A finite
group Gacts on Cif there is a group monomorphism GAut(C).The space of orbits
CGof the action of Gon Cis naturally endowed with a Riemann surface structure such
that the canonical projection CCGis holomorphic.
By the classical uniformization theorem, there is a unique, up to conjugation, Fuchsian
group Γ of signature (g;) such that C
=HΓ.Moreover, Gacts on Cif and only if
there is a Fuchsian group containing Γ together with a group epimorphism
θ: Gsuch that ker(θ) = Γ.
In such a case, the group Gis said to act on Cwith signature σ(∆) and the action is
said to be represented by the surface-kernel epimorphism θ. See [21], [43] and [49]
If Gis a subgroup of Gthen the action of Gon Cis said to extend to an action of
Gon Cif:
(1) there is a Fuchsian group containing ,
(2) the Teichm¨uller spaces of and have the same dimension, and
(3) there exists an epimorphism
Θ : Gin such a way that Θ|=θand ker(θ) = ker(Θ).
An action is called maximal if it cannot be extended in the afore introduced sense.
A complete list of signatures of pairs of Fuchsian groups and for which it may be
possible to have an extension as before was provided by Singerman in [48].
Let be a Fuchsian group of signature (2.1) and let pbe a prime number. Define ei
as the largest integer such that peiis a divisor of mi.Following [32], the signature
σp:= (h;pe1,...,pel),
where the (i+ 1)-entry is dropped if ei= 0,is called the p-localization of σ=σ(∆).The
signature σis called nilpotent-admissible if σpis non-degenerate for each prime p.
Macbeath proved that if σis a nilpotent-admissible signature then there exists at least
one nilpotent group acting as a group of automorphisms of a compact Riemann surface
with signature σ. Furthermore, if in addition the signature satisfies that χ(σp)60 for
al least one prime p, then there are infinitely many nilpotent groups with the same
property. See [32, Theorem (8.1)] and [32, Theorem (8.2)].
8 SEBASTI ´
AN REYES-CAROCCA
2.3. Equisymmetric stratification. Let Hom+(C) denote the group of orientation
preserving self-homeomorphisms of C. Two actions ψi:GAut(C) of Gon Care
topologically equivalent if there exist ωAut(G) and fHom+(C) such that
ψ2(g) = fψ1(ω(g))f1for all gG. (2.3)
Each homeomorphism fsatisfying (2.3) yields an automorphism fof where H
=
CG. If Bis the subgroup of Aut(∆) consisting of them, then Aut(G)×Bacts on the
set of epimorphisms defining actions of Gon Cwith signature σ(∆) by
((ω, f ), θ)7→ ωθ(f)1.
Two epimorphisms θ1, θ2: Gdefine topologically equivalent actions if and only
if they belong to the same (Aut(G)×B)-orbit (see [1], [3], [21] and [31]).
We remark that if the genus of CGis one then Bcontains the transformations
A1,n :α17→ α1, β17→ β1αn
1, γjγj,and A2,n :α17→ α1βn
1, β17→ β1, γjγj
for each nZ.See [3, Proposition 2.5].
Let Mgdenote the moduli space of compact Riemann surfaces of genus g>2.It is
well-known that Mgis endowed with a structure of complex analytic space of dimension
3g3,and that for g>4 its singular locus Sing(Mg) agrees with the set of points
representing compact Riemann surfaces with non-trivial automorphisms.
Following [4], the singular locus of Mgadmits an equisymmetric stratification
Sing(Mg) = G,θ ¯
MG,θ
g
where each equisymmetric stratum MG,θ
g, if nonempty, corresponds to one topological
class of maximal actions (see also [21]). More precisely:
(1) the equisymmetric stratum MG,θ
gconsists of those Riemann surfaces Cof genus
gwith (full) automorphism group isomorphic to Gsuch that the action is topo-
logically equivalent to θ,
(2) the closure ¯
MG,θ
gof MG,θ
gis a closed irreducible algebraic subvariety of Mgand
consists of those Riemann surfaces Cof genus gwith a group of automorphisms
isomorphic to Gsuch that the action is topologically equivalent to θ, and
(3) if the equisymmetric stratum MG,θ
gis nonempty then it is a smooth, connected,
locally closed algebraic subvariety of Mgwhich is Zariski dense in ¯
MG,θ
g.
In this article we employ use the following terminology.
Definition. The subset of Mgconsisting of those compact Riemann surfaces Cof genus
gwith action of a given group Gwith a given signature will be called a (closed) family.
The complex dimension of the family is the complex dimension of the Teichm¨uller
space associated to a Fuchsian group such that CG
=H.Note that the interior of a
family consists of those Riemann surfaces whose full automorphism group is isomorphic
to Gand is formed by finitely many equisymmetric strata which are in correspondence
NILPOTENT GROUP OF AUTOMORPHISMS OF RIEMANN SURFACES 9
with the pairwise non-equivalent topological actions of G. Besides, the members of the
family that do not belong to the interior is formed by those Riemann surfaces that have
strictly more automorphisms than G.
2.4. Decomposition of Jacobians with group action. It is well-known that if Gacts
on a compact Riemann surface Cthen this action induces a Q-algebra homomorphism
Φ : Q[G]EndQ(JC) = End(J C)ZQ,
from the rational group algebra of Gto the rational endomorphism algebra of J C.
For each αQ[G] we define the abelian subvariety
Aα:= Im(α) = Φ()(JC)J C
where nis some positive integer chosen in such a way that Z[G].
Let W1,...,Wrbe the rational irreducible representations of G. For each Wjwe denote
by Vja complex irreducible representation of Gassociated to it. The decomposition of
1 as the sum e1+···+er,where ejQ[G] is a uniquely determined central idempotent
computed explicitly from Wj, yields an isogeny
JC Ae1× · · · × Aer
which is G-equivariant; see [30]. Additionally, there are idempotents fj1,...,fjnjsuch
that ej=fj1+···+fj njwhere nj=dVj/sVjis the quotient of the degree dVjof Vjand
its Schur index sVj. These idempotents provide njsubvarieties of JC which are pairwise
isogenous; let Bjbe one of them, for every j. Thus, we obtain the following isogeny
JC GBn1
1× · · · × Bnr
r(2.4)
called the group algebra decomposition of JC with respect to G. See [9] and also [41].
If W1(= V1) denotes the trivial representation of Gthen n1= 1 and B1JCG.
Let Hbe a subgroup of Gand consider the associated regular covering map πH:C
CH.It was proved in [9] that (2.4) induces the isogeny
JCHBnH
1
1× · · · × BnH
r
rwith nH
j=dH
Vj/sVj(2.5)
where dH
Vjis the dimension of the vector subspace VH
jof Vjof elements fixed under H.
Assume that (2.1) is the signature of the action of Gon Cand that this action is
represented by θ: G, with as in (2.2). Following [43, Theorem 5.12]
dim(Bj) = kVj[dVj(γ1) + 1
2Σl
k=1(dVjdhθ(γk)i
Vj)] for 2 6j6r(2.6)
where kVjis the degree of the extension Q6LVjwith LVjdenoting a minimal field of
definition for Vj.
The decomposition of Jacobian varieties with group actions has been extensively stud-
ied, going back to contributions of Wirtinger, Schottky and Jung. For decompositions
of Jacobians with respect to special groups, we refer to [6], [7], [8], [13], [22], [24], [26],
[28], [34], [35], [38], [39], [40] and [42].
10 SEBASTI ´
AN REYES-CAROCCA
Notation. We denote the cyclic group of order nby Cnand the dihedral group of order
2nby Dn.
3. Proof of Theorem 1
Let d>1 and g>2 be integers. We assume that Gis a nilpotent group acting as a
group of automorphisms of a complex d-dimensional family of compact Riemann surfaces
Cof genus g, and that the signature of the action of Gon Cis σ= (h;m1,...,ml).
Assume that d>2.Note that, as each mi>2,the hyperbolic area µof a fundamental
domain of a Fuchsian group of signature σsatisfies
µ= 2π(2h2 + Σl
i=1 2
mi)>2π(h
2+d1
2)>2πd1
2.
Thus, by the Riemann-Hurwitz formula, one easily obtains that
2(g1) = µ
2π|G|>d1
2|G| |G|64
d1(g1)
as claimed. Now, if we assume that
|G|=4
d1(g1) then Σl
i=1 1
mi=d+3
2h,
which is at most l
2.It follows that
h= 0 and Σd+3
i=1 1
mi=d+3
2.
The unique solution of the equation above is mi= 2 for each i, and then σ= (0; 2,d+3
. . ., 2).
Assume that d= 1.We have only two cases to consider; namely, (h, l) = (1,1) and
(h, l) = (0,4).In the former case it is clear that µ>π. Assume σ= (0; m1, m2, m3, m4)
and denote by vthe number of periods mithat are equal to 2. Note that v63 because
if v= 4 then µ= 0.
(a) If v= 0 then each mi>3 and therefore µ>4π
3.
(b) If v= 1 then σ= (0; 2, m2, m3, m4) where mi>3.Note that if m2, m3, m4were
equal to 3 then the 2-localization of σwould be degenerate. Then, we can assume
m4>4 and therefore µ>7π
6.
(c) If v= 2 then σ= (0; 2,2, m3, m4) where m3, m4>3 and µ>2π
3.
(d) If v= 3 then σ= (0; 2,2,2, m4) where m4>3.Note that m4must be a power of
two, since otherwise the p-localization of σwould be degenerate for some prime
p>3.Thus µ>π
2.
All the above ensures that µ>π
2and therefore, by the Riemann-Hurwitz formula,
2(g1) = |G|µ
2π>|G|
4 |G|68(g1)
as claimed. Now, if |G|= 8(g1) then
Σl
i=1 1
mi=7
4h643h
2and therefore h= 0 and Σ4
j=1 1
mj=7
4,
showing that m1=m2=m3= 2 and m4= 4.Thus, σ= (0; 2,2,2,4) as desired.
NILPOTENT GROUP OF AUTOMORPHISMS OF RIEMANN SURFACES 11
Finally, as the group Gis assumed to be nilpotent and as, in each case, the genus of
the corresponding quotient is zero, we can apply [52, Theorem 2.11] to ensure that the
prime factors of |G|are necessarily contained in the set of prime factors of the periods of
σ. Thus, the group Gis a 2-group. Consequently, if we assume that, in addition, d= 1
or d1 is a power of two, then we can conclude that g1 is a power of two as well.
4. Proof of Theorem 2
Let be a Fuchsian group of signature (0; 2,2,2,4) with canonical presentation
= hγ1, γ2, γ3, γ4:γ2
1=γ2
2=γ2
3=γ4
4=γ1γ2γ3γ4= 1i
and, for each n>5,consider the group G
=(C2×D2n3)C2of order 2nwith
presentation in terms of generators a, b, r, s and relations
r2n3=s2= (sr)2=a2=b2= [s, b] = [r, b] = 1, ara =r1, asa =sr, aba =br2n4.
Note that the Riemann-Hurwitz formula is satisfied for a branched 2n-fold regular
covering map from a compact Riemann surface of genus 1 + 2n3onto the projective
line, ramified over three values marked with 2 and one value marked with 4. Thus, by
virtue of Riemann’s existence theorem, the existence of the desired family follows after
noticing that the correspondence
Gdefined by (γ1, γ2, γ3, γ4)7→ (s, bs, a, ab)
is a surface-kernel epimorphism. Henceforth, we denote this family by F.
In order to determine an upper bound for the number of equisymmetric strata of
F, we have to determine an upper bound for the number of pairwise non-equivalent
surface-kernel epimorphisms θ: G. For each such epimorphism θ, we write
gi:= θ(γi) for each i= 1,2,3,4,
and, for the sake of simplicity, we identify θwith the 4-uple θ= (g1, g2, g3, g4).
We notice that:
(1) the elements of order four of Gare abrland w:= r2n5,and
(2) the involutions of Gare b, arl, z := r2n4, bz, srland bsrl,
where 1 6l62n3.
Claim. The central element zis different from g1, g2and g3.
Clearly, not three or two among g1, g2, g3can be equal to z. In addition, if one of them
equals z, say g1=z, then, as g1g2g3must have order four, either
g2, g3 {srl1, bsrl2: 1 6lj62n3}or g2, g3/ {srl1, bsrl2: 1 6lj62n3}.
In the former case g4does not have order four, whilst in the latter one sdoes not
belong to the image of θ, contradicting its surjectivity.
12 SEBASTI ´
AN REYES-CAROCCA
Similarly as argued before, we can see that the number of g
isthat are of the form srl
or bsrlis exactly two. In addition, if g4equals wthen hg1, g2, g4iis a proper subgroup
of G, showing that θis not surjective. Thus, θis of one of the following forms:
(srl1, srl2, g3, abrl3),(srl1, bsrl2, g3, abrl3) or (bsrl1, bsrl2, g3, abrl3)
for some 1 6l1, l2, l362n3.The fact that g1g2g3g4= 1 implies that necessarily θis
(srl1, bsrl2, arl2+l3l1, abrl3) for some 1 6l1, l2, l362n3.
Note that, after applying an appropriate conjugation, we can assume l1= 0 or l1= 1.
Furthermore, by considering the action of the automorphism of Ggiven by
r7→ r1, s 7→ sr, a 7→ a, b 7→ b
we obtain that θis equivalent to
θu,v := (s, bsru, arv, abrvu) where 1 6u, v, 62n3.
Thereby, the number of topologically non-equivalent actions of Gon Cis as most 22n6.
Following [48, Theorem 1], the signature (0; 2,2,2,4) is maximal; thus, if Clies in the
interior of the family then its automorphism group agrees with G. It is easy to verify that
Ghas exactly five conjugacy classes of subgroups of order two, and that among them
only K=hziis a normal subgroup. Consider the associated two-fold regular covering
map given by the action of K
π:CCK,
and notice that, independently of the equisymmetric stratum to which Cbelongs (or,
in other words, independently of the surface-kernel epimorphism θu,v representing the
corresponding action), the covering πramifies over exactly 2n2values marked with 2.
Thus, the Riemann-Hurwitz formula implies that CKis an elliptic curve and therefore
Cis non-hyperelliptic.
If a compact Riemann surface Xbelongs to Fbut does not belong to its interior then
Gis strictly contained in the full automorphism group of X(this is a general result that
can be found, for instance, in [3]). Now, as the complex dimension of the family Fis
one, it follows that the signature of the action of Aut(X) on Xmust be triangle; namely,
of the form (0; t1, t2, t3).Note that there are finitely many possibilities for t1, t2, t3and,
in turn, to each of these possible signatures correspond at most finitely many Riemann
surfaces. Thus, the family contains at most finitely many surfaces that do not belong to
its interior.
We now proceed to prove the announced isogeny decomposition of JC for each Cin
the family F. Let us consider the normal subgroup Nof Ggiven by
hr, s, b :r2n3=s2= (sr)2=b2= 1,[b, s] = [b, r] = 1i
and the complex irreducible representation of Ngiven by
r7→ (ω0
0 ¯ω), s 7→ (0 1
1 0 ), b 7→ (1 0
0 1 ) where ωis a 2n3-th primitive root of unity.
NILPOTENT GROUP OF AUTOMORPHISMS OF RIEMANN SURFACES 13
This representation induces the complex representation Vof Ggiven by
r7→ ω000
0 ¯ω0 0
0 0 ¯ω0
0 0 0 ω, s 7→ 0 1 0 0
1 0 0 0
0 0 0 ω
0 0 ¯ω0, b 7→ 1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1, a 7→ 0 0 1 0
0 0 0 1
1 0 0 0
0 1 0 0
which is, by [23, Theorem 6.11], irreducible. In addition, as Vis constructed from a
complex irreducible representation of a dihedral group, it is easy to infer that its Schur
index is 1. Note the character field of Vis Q(ω+ ¯ω); this is an extension of Qof degree
1
2ϕ(2n3) = 2n5
where ϕis the Euler function. We denote by W2the rational irreducible representation
of Gassociated to Vand by W1the rational irreducible representation Ggiven by
r7→ 1, s 7→ 1, b 7→ 1, a 7→ 1.
Then, as explained in §2.4, there is an abelian subvariety Pof JC such that
JC BW1×B4
W2×P, (4.1)
where BWjis the factor associated to Wjin the group algebra decomposition of JC with
respect to G. As the action of Gon Cis determined by θu,v for some u, v {1,...,2n3},
we can apply the equation (2.6) to notice that, independently of the choice of uand v,
the following equalities hold:
dim(BW1) = 1 and dim(BW2) = 2n5.
Then, by considering dimensions is the relation (4.1), one sees that
dim(JC) = 1 + 2n3= 1 + 4(2n5) + dim Pand therefore P= 0.
Now, we consider the induced isogeny (2.5) (with H=hbiand H=hsi) to obtain
JChbiBW1×B2
W2and JChsiB2
W2
The previous two isogenies together with isogeny (4.1) permits us to conclude that
JC JChbi×J Chsi
as claimed. Finally, is a similar way, we consider the induced isogeny (2.5) with H=hri
to obtain that JChriand BW1are isogenous and, consequently, JC contains an elliptic
curve isogenous to JChri.
5. Proof of Theorem 3
Let d>1, g >2 be integers and let p>3 be a prime number. Let Gbe a p-group
of automorphisms of a complex d-dimensional family of compact Riemann surfaces Cof
genus gand assume the signature of the action of Gon Cto be σ= (h;m1,...,ml).
The hyperbolic area µof a fundamental region of a Fuchsian group of signature σ
satisfies
µ>2π[d+ 1 d+3
p+h(3
p1)] >4
3 if p= 3
2π[d+ 1 d+3
p+ˆ
h(3
p1)] if p>5
14 SEBASTI ´
AN REYES-CAROCCA
where ˆ
his the largest possible genus of the quotient CG.Note that ˆ
h= [d
3+ 1].
Assume p= 3.The Riemann-Hurwitz formula ensures that
2(g1) = |G|µ
2π>2
3d|G| |G|6M3,d
as claimed in (1). Now, if we suppose that the order of Gequals M3,d then, by the
Riemann-Hurwitz formula, we easily obtain that
Σl
i=1 1
mi=l
3and, consequently, each mi= 3.
Note that there is no restriction on l. Thus, σ=σ3,d,h for some h {0,...,ˆ
h}.The only
if part of (2) is a direct computation.
Assume p>5.Then
µ>
4
3πd if d0 mod 3
4
3πd + 2π(1
31
p) if d1 mod 3
4
3πd + 4π(1
31
p) if d2 mod 3.
In other words, if λdis the smallest non-negative representative of dmodulo 3 then
2(g1) = |G|µ
2π>|G|(2
3d+λd(1
31
p)) |G|6Mp,d,
as claimed in (3). If we now assume that the order of Gequals Mp,d then
Σl
i=1 1
mi=l
3λd(1
31
p).(5.1)
(1) If d0 mod 3 then (5.1) turns into Σl
i=1 1
mi=l
3and l= 0.Thus,
σ= (d+3
3;) = σp,d
(2) If d1 mod 3 then (5.1) turns into Σl
i=1 1
mi=l
31
3+1
pand l= 1.Thus,
σ= (d+2
3;p) = σp,d
(3) If d2 mod 3 then (5.1) turns into Σl
i=1 1
mi=l
32
3+2
pand l= 2.Thus,
σ= (d+1
3;p, p) = σp,d
The only if part of (4) is a direct computation.
6. Proof of Theorem 4
Let p>3 and let be a Fuchsian group of signature (1; p) with canonical presentation
= hα1, β1, γ1:α1β1α1
1β1
1γ1=γp
1= 1i
and, for each n>5,consider the group G
=Cpn1pCpof order pnwith presentation
ha, b :apn1=bp= 1, bab1=ari,
where r=pn2+1.Observe that rp1 mod pn1and rk6≡ 1 mod pn1for 1 6k6p1.
Note that the Riemann-Hurwitz formula is satisfied for a branched pn-fold regular
covering map from a compact Riemann surface of genus 1 + (p1)pn1
2onto a Riemann
NILPOTENT GROUP OF AUTOMORPHISMS OF RIEMANN SURFACES 15
surface of genus one, ramified over one value marked with p. Thus, by virtue Riemann’s
existence theorem, the existence of the family follows after noticing that the rule
Gdefined by (α1, β1, γ1)7→ (a, b, apn2)
is a surface-kernel epimorphism. Henceforth, we denote this family by G.
We now proceed to prove that there are exactly p1 pairwise non-equivalent surface-
kernel epimorphisms θ: G. For each such epimorphism θ, we write
x:= θ(α1), y =θ(β1) and z=θ(γ1)
and, for the sake of simplicity, we identify θwith the 3-uple θ= (x, y, z).Note that
x=albkand y=asbmfor some 1 6l, s 6pn1and 1 6k, m 6p.
If k6= 0 and u=mk,where kis the inverse of kin the field of pelements, then
the transformation A1,u (see §2.3) shows that we can assume, up to equivalence, that
x=albkand y=as.(6.1)
On the other hand, if k= 0 then
x=aland y=asbm,(6.2)
and the transformation A2,1A1,1shows that (6.1) and (6.2) are equivalent. Now, in
(6.2) one sees that if land pn1are not coprime then θis not surjective. Thus, after
sending ato an appropriate power of it, we can be assume l= 1.Then
x=aand b=asbmwhere m6= 0.(6.3)
Now, if we set v=srmthen we apply A1,v to (6.3) to ensure that θis equivalent to
θm= (a, bm, arm1) for some 1 6m6p1.
The result follows after noticing that θmand θmare non-equivalent if m6=m.
Note that K=hapn2iis a cyclic group of order pand that, independently of the
equisymmetric stratum to which Cbelongs, the associated regular covering map
CCK
ramifies over pn1values marked with p. It follows that the quotient Riemann surface CK
has genus one; thus, Cis an elliptic-p-gonal Riemann surface. Due to the explicitness
of the family, one can easily see that Kis the unique group of automorphisms of C
providing the elliptic-p-gonal structure (c.f. [45, Theorem 1.3] and also [18] and [19]).
According to [48, Theorem 1], the action of Gon each Cin Gmight be extended to
only an action of a group of order 2pnacting on Cwith signature σ= (0; 2,2,2,2p).
Claim. Such extension is not possible in our case.
To prove the claim we shall proceed by contradiction; namely, we assume that:
(1) there is a group Gof order 2qnwith a subgroup isomorphic to G, and that
16 SEBASTI ´
AN REYES-CAROCCA
(2) there is a surface-kernel epimorphism G,where is a Fuchsian group of
signature σ.
By the classical Schur-Zassenhaus theorem, we can ensure that
G
=GC2with C2=ht:t2= 1i.
Observe that C2must act on Gwith order 2, because of the direct product G×C2
cannot be generated by three involutions. Thus, by considering an automorphism of G
that sends ato an appropriate power of it and after some routine computations, one can
see that, up to an isomorphism of G, the action of C2on Gis given by
tat =a1and tbt =b.
In particular, the involutions of Gare of the form takfor 0 6k6p1.However, three
of them cannot generate G,contradicting the surjectivity of θ. This proves the claim.
As observed in the proof of Theorem 2, the surface Cbelongs to the interior of the
family G(and therefore for all up to possibly finitely many exceptions) if and only if G
is the full automorphism group of it (see, for instance, [3]).
We now proceed to decompose the Jacobian variety JC of each Cin the family G.
We apply the method of little groups of Wigner–Mackey (see, for example, [47, p. 62]),
to guarantee the irreducibility of the complex representation Vof Ggiven by
a7→ diag(ω, ωr,...,ωrp1) and b7→
0 1 0 ... 0
0 0 1 ··· 0
...
0 0 0 ... 1
1 0 0 ··· 0
where ωis a pn1-th primitive root of unity. We notice that the character field of Vis
Q(ωp), which is an extension of Qof degree
ϕ(pn2) = pn3(p1),
where ϕis the Euler function. We recall that p-groups with p>3 only possess repre-
sentations with Schur index 1 (see, for example, [36, Theorem 41.9]).
We denote by Wthe rational irreducible representation of Gassociated to V. Then,
as explained in §2.4, there is an abelian subvariety Qof JC such that
JC E×Ap×Q, (6.4)
where Eis an elliptic curve isogenous to JCGand Ais the factor associated to Win
the group algebra decomposition of J C with respect to G. Now, as the action of Gon
Cis determined by θmfor some 1 6m6p1,we apply the equation (2.6) to notice
that, independently of the choice of m, the following equality holds:
dim(A) = pn3(p1) ·1
2(p0) = (p1)pn2
2.
Finally, by considering dimensions in the relation (6.4), one concludes that Q= 0 and
the desired decomposition of J C is obtained.
NILPOTENT GROUP OF AUTOMORPHISMS OF RIEMANN SURFACES 17
Acknowledgements. The author is very grateful to the referee for his/her valuable
comments and suggestions.
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We prove that the maximal number ag+b of automorphisms of equisymmetric and complex-uniparametric families of Riemann surfaces appearing in all genera is 4g+4. For each integer g2g\ge 2 we find an equisymmetric complex-uniparametric family Ag{\mathcal {A}}_{g} of Riemann surfaces of genus g having automorphism group of order 4g+4. For g1mod4g\equiv -1\,{\mathrm {mod}}\,4 we present another uniparametric family Kg{\mathcal {K}}_{g} with automorphism group of order 4g+4. The family Ag{\mathcal {A}}_{g} contains the Accola–Maclachlan surface and the family Kg{\mathcal {K}}_{g} contains the Kulkarni surface.