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ON LARGE PRIME ACTIONS ON RIEMANN SURFACES
SEBASTI ´
AN REYES-CAROCCA AND ANITA M. ROJAS
Abstract. In this article we study compact Riemann surfaces of genus gwith an auto-
morphism of prime order g+ 1.The main result provides a classification of such surfaces.
In addition, we give a description of them as algebraic curves, determine and realise their
full automorphism groups and compute their fields of moduli. We also study some aspects
of their Jacobian varieties such as isogeny decompositions and complex multiplication.
Finally, we determine the period matrix of the Accola-Maclachlan curve of genus four.
1. Introduction and statement of the results
Let Mgdenote the moduli space of compact Riemann surfaces (smooth irreducible com-
plex algebraic curves) of genus g>2.It is classically known that Mgis endowed with a
structure of complex analytic space of dimension 3g−3,and that for g>4 its singular
locus agrees with the branch locus of the canonical projection
Tg→Mg
where Tgstands for the Teichm¨uller space of genus g. In other words, if g>4 then
Sing(Mg) = {[S]∈Mg: Aut(S)6= 1}
where Aut(S) denotes the full automorphism group of S.
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 automor-
phism group of a compact Riemann surface Sof genus g>2 is finite, and that
|Aut(S)|684(g−1).
It is well-known that there are infinitely many values of gfor which there is no compact
Riemann surfaces of genus gpossessing 84(g−1) automorphisms. Regarding this matter,
Accola [1] and Maclachlan [46] proved that, for fixed g, the largest order n0(g) of the full
automorphism group of a compact Riemann surface of genus gsatisfies
n0(g)>8(g+ 1) (1.1)
and that for infinitely many values of gthe inequality (1.1) turns into an equality.
We denote by X8the so-called Accola-Maclachlan curve, namely the compact Riemann
surface of genus gwith 8(g+ 1) automorphisms given by the algebraic curve
y2=x2(g+1) −1.
2010 Mathematics Subject Classification. 30F10, 14H37, 30F35, 14H30, 14H40.
Key words and phrases. Riemann surfaces, group actions, automorphisms, Jacobian varieties.
Partially supported by Fondecyt Grants 1180073, 11180024, 1190991 and Redes Grant 2017-170071.
2 SEBASTI ´
AN REYES-CAROCCA AND ANITA M. ROJAS
The Accola-Maclachlan 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 g6≡ 3 mod 4 then X8
is the unique compact Riemann surface of genus gwith exactly 8(g+ 1) automorphisms.
The analogous problem of finding n0(g) but for uniparametric families of compact Rie-
mann surfaces was studied in [22]. Concretely, it was proved the existence of a closed
equisymmetric complex one-dimensional family, henceforth denoted by ¯
Cg,of hyperelliptic
compact Riemann surfaces of genus gwith a group of automorphisms isomorphic to
Dg+1 ×C2acting with signature (0; 2,2,2, g + 1)
(we shall recall the precise definition of signature in §2.1 and §2.2). It was then shown that
4(g+ 1) is the largest order of the full automorphism group of complex one-dimensional
families of compact Riemann surfaces of genus gappearing for all g. These results were
recently extended to the three and four-dimensional case in [40] while the two-dimensional
case is derived from the results of [56].
It is a well-known fact that if a compact Riemann surface of genus g>2 has an auto-
morphism of prime order qsuch that q > g then either q= 2g+ 1 or q=g+ 1.The former
case corresponds to the so-called Lefschetz surfaces. This paper deals with the latter case.
Let q>5 be a prime number. Consider the singular sublocus
Mq
q−1⊂Sing(Mq−1)
consisting of all those compact Riemann surfaces of genus q−1 endowed with an automor-
phism of order q. This sublocus was studied by Urz´ua in [69] from a hyperbolic geometry
point of view, and later by Costa and Izquierdo in [21] when the existence of complex
one-dimensional isolated strata of the singular locus of the moduli space was proved.
This paper is devoted to classify and describe the surfaces lying in Mq
q−1and to study
some aspects of the corresponding Jacobians in the singular locus of the moduli space of
principally polarised abelian varieties of dimension q−1. In other words, we shall consider
all those compact Riemann surfaces of genus g>4 (and their Jacobian varieties) with a
group of automorphisms of order
λ(g+ 1) where λ>1 is an integer,
under the assumption that q:= g+ 1 is a prime number.
The classification. The first result of the paper provides a classification of these surfaces.
Theorem 1. Let q>7be a prime number. If Sis a compact Riemann surface of genus
g=q−1endowed with a group of automorphisms of order λq for some integer λ>1,then
λ∈ {1,2,3,4,8}.
Assume λ= 8.Then Sis isomorphic to the Accola-Maclachlan curve X8.
Assume λ= 4.
(1) If q≡3mod 4then Sbelongs to the closed family ¯
Cg.
ON LARGE PRIME ACTIONS ON RIEMANN SURFACES 3
(2) If q≡1mod 4then Sbelongs to the closed family ¯
Cgor Sis isomorphic to the
unique compact Riemann surface X4with full automorphism group isomorphic to
Cqo4C4acting with signature (0; 4,4, q).
Moreover, if Cgstands for the interior of ¯
Cgthen
¯
Cg−Cg={X8}.
Assume λ= 3.Then Sis isomorphic to the unique compact Riemann surface X3with
full automorphism group isomorphic to
Cq×C3acting with signature (0; 3, q, 3q).
Assume λ= 2.Then one of the following statements holds.
(1) Sis isomorphic to one of the q−3
2pairwise non-isomorphic compact Riemann sur-
faces X2,k for k∈ {1,...,q−3
2}with full automorphism group isomorphic to
Cq×C2acting with signature (0; q, 2q, 2q).
(2) Sbelongs to the closed family ¯
Kgof compact Riemann surfaces with a group of
automorphisms isomorphic to
Dqacting with signature (0; 2,2, q, q).
Moreover, the closed family ¯
Kgconsists of at most
q+3
4if q≡1mod 4
q+1
4if q≡3mod 4
equisymmetric strata; one of them being Cg.Furthermore, if Kgstands for the interior of
¯
Kgthen the full automorphism group of S∈Kg−Cgis isomorphic to Dqand
¯
Kg−Kg={X4, X8}if q≡1mod 4
{X8}if q≡3mod 4.
Remark 1. We point out some observations concerning Theorem 1.
(1) If S∈Mq
q−1then either Aut(S)∼
=Cqor Slies in one of the cases described in the
theorem. These two possible situations were considered in [21] where the focus was
put on finding isolated equisymmetric strata of Sing(Mg). We shall discuss later
the results of [21] in terms of our terminology (see Remark 2in §3).
(2) The case q= 5 is slightly different. As a matter of fact, if Shas genus 4 and is
endowed with a group of automorphisms of order 5λfor some λ>1 then, in addition
to the case Aut(S)∼
=C5and the possibilities given in the theorem, λcan equal 12
and 24.In the last two cases, Sis isomorphic to the classical Bring’s curve; see [19]
and [44].
(3) We conjecture that the upper bound given in the theorem for the number of equi-
symmetric strata of the closed family ¯
Kgis sharp. By means of computer routines
developed in [7], it can be seen the sharpness of the bound for small primes (q623).
(4) We emphasise that the equisymmetric family ¯
Cgis contained in the family ¯
Kg.
4 SEBASTI ´
AN REYES-CAROCCA AND ANITA M. ROJAS
(5) An analogous classification as in the theorem but for the compact Riemann surfaces
lying in Mq
q+1 was obtained in a series of articles due to Belolipetsky, Izquierdo,
Jones and the first author; see [8], [38], [39] and [56].
Algebraic description. Although the literature still shows few general results in this direc-
tion, 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.
Theorem 2. Let q>5be a prime number and let g=q−1.Set ωl=exp(2πi
l).
If Sbelongs to the closed family ¯
Cgthen Sis isomorphic to the normalisation of the
singular affine algebraic curve
Xt:y2= (xq−1)(xq−t)for some t∈C− {0,1}.
In addition, if S∈Cgthen the full automorphism group of S∼
=Xtis generated by
(x, y)7→ (ωqx, −y)and (x, y)7→ (q
√t1
x,√ty
xq).
Assume q≡1mod 4and choose ρ∈ {2, . . . , q −2}such that ρ4≡1mod q. Then X4is
isomorphic to the normalisation of the singular affine algebraic curve
yq= (x−1)(x−i)ρ(x+ 1)q−1(x+i)q−ρ
where i2=−1.In the previous model the full automorphism group of X4is generated by
(x, y)7→ (x, ωqy)and (x, y)7→ (ix, ϕ(x)yρ)
where ϕ(x) = −(x+i)e−ρ
(x−i)e−1(x+1)ρ−1and e=ρ2+1
q.
X3isomorphic to the normalisation of the singular affine algebraic curve
y3=xq−1
and, in this model, its full automorphism group is generated by (x, y)7→ (ωqx, ω3y).
For each k∈ {1,...,q−3
2}there exists nk∈ {1, . . . , q −1}different from q−2such that
X2,k is isomorphic to the normalisation of the singular affine algebraic curve
yq=xnk(x2−1)
and, in this model, its full automorphism group is generated by
(x, y)7→ (x, ωqy)and (x, y)7→ (−x, (−1)nky).
If Sbelongs to the closed family ¯
Kg, then Sis isomorphic to the normalisation of the
singular affine algebraic curve
Zt:yq= (x−1)(x+ 1)q−1(x−t)(x+t)q−1for some t∈C− {0,±1}
and, if S6=X4and S /∈¯
Cgthen the full automorphism group of S∼
=Ztis generated by
(x, y)7→ (x, ωqy)and (x, y)7→ (−x, φt(x)y−1)
where φt(x)=(x2−1)(x2−t2).
The theorem above overlaps results obtained in [69,§11].
ON LARGE PRIME ACTIONS ON RIEMANN SURFACES 5
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 q>7 is a prime number and Sis a compact Riemann
surface of genus q−1 with a group of automorphisms of order λq then
Sis non-hyperelliptic implies λ∈ {1,2,3,4}.
The following result lengthens the implication above; it follows from Theorem 1.
Proposition 1. Let q>7be a prime number. The compact Riemann surfaces lying in
Mq
q−1that are non-hyperelliptic are X2,k, X3, X4,the surfaces which belong to Kg−Cgand
the ones for which Aut(S)∼
=Cq.
Jacobian variety. Let Sbe a compact Riemann surface of genus g>2.We denote by JS
the Jacobian variety of S, that is, the quotient
JS =H1(S, C)∗/H1(S, Z),
where H1(S, C)∗stands for the dual of the g-dimensional complex vector space of holo-
morphic forms of Sand H1(S, Z) stands for the first integral homology group of S.
We emphasise the following two classical facts (see, for example, [10]):
(1) JS is an irreducible principally polarised abelian variety of dimension g, and
(2) up to isomorphism, the surface is determined by its Jacobian (Torelli’s theorem).
If Gis a group acting on Sthen Galso acts on JS and this action, in turn, induces the
so-called group algebra decomposition of J S. Concretely
JS ∼A1× · ·· × Ar∼Bn1
1× · ·· × Bnr
r
where the factors Ajare pairwise non-G-isogenous abelian subvarieties of JS uniquely
determined, and in correspondence with central idempotents generating the simple algebras
decomposing the rational group algebra of G. Each Ajdecomposes further as Bnj
jwhere
the abelian subvarieties Bjare no longer unique and are related to the decomposition of
each simple algebra as a product of minimal left ideals. The numbers rand njdepend only
on the algebraic structure of G. See [16] and [45].
The following result provides the group algebra decomposition of the Jacobian varieties
of the surfaces of Theorem 1, with the exception of X3and X2,k . In fact, the group
algebra decomposition of J X3is trivial whilst the one of J X2,k agrees with the classical
decomposition
JX2,k ∼J(X2,k/H)×Prym(X2,k →X2,k/H )
where Prym stands for the Prym variety and H6Aut(X2,k) is isomorphic to C2.
Theorem 3. Let q>5be a prime number and let g=q−1.
The Jacobian variety JX8decomposes, up to isogeny, as the square power
JX8∼JY 2
8
where Y8is quotient compact Riemann surface given by the action of hzion X8,where
Aut(X8)∼
=hx, y, z :x2q=y2=z2= 1,[x, y]=[z, y] = 1, zxz =x−1yi.
6 SEBASTI ´
AN REYES-CAROCCA AND ANITA M. ROJAS
The Jacobian variety JX4of X4decomposes, up to isogeny, as the fourth power
JX4∼JY 4
4
where Y4is quotient compact Riemann surface given by the action of hBion X4,where
Aut(X4)∼
=hA, B :Aq=B4= 1, BAB −1=Aρi
and ρis a primitive fourth root of unity in Zq.
The Jacobian variety JS of S∈Kgdecomposes, up to isogeny, as the square power
JS ∼JX2
where Xis quotient compact Riemann surface given by the action of hsion S, where
Aut(S)∼
=Dqif S∈Kg−Cg
Dq×C2if S∈Cg
and Dq=hr, s :rq=s2= (sr)2= 1i.
Field of moduli and fields of definition. Let Gal(C/Q) denote the group of field automor-
phisms of C.The correspondence
Gal(C/Q)×Mg→Mggiven by (σ, [S]) 7→ [Sσ]
where Sσis the Galois σ-transformed of S(considered as algebraic curve) defines an action.
The field of moduli of a compact Riemann surface Sis the fixed field M(S) of the isotropy
group of Sunder the aforementioned action, namely
M(S) = fix{σ∈Gal(C/Q) : Sσ∼
=S}.
The field of moduli of Sagrees with the intersection of all its fields of definition and, as
proved by Koizumi in [41], Scan be defined over a finite degree extension of M(S).
Necessary and sufficient conditions under which Scan be defined over its field of moduli
were provided by Weil in [70] (see also [33] for a constructive proof of Weil’s theorem); these
conditions are trivially satisfied if Shas no non-trivial automorphisms. Besides, as proved
by Wolfart in [71], if Sis quasiplatonic then Scan 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], [31], [34], [35], [42] and [54]. In this
direction, it is a known fact that if the genus of S/Aut(S) is zero then either Scan be
defined over M(S) or over a quadratic extension of it; see [23] and also [30] for recent
results.
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:
(1) As proved in Theorem 2, the surfaces X3, X2,k and X8are defined over Qand
therefore their fields of moduli are Q.
(2) As mentioned above, the fact that X4is quasiplatonic implies that it can be defined
over its field of moduli. Moreover, the uniqueness of X4implies that its field of
moduli is Q.In fact, we shall see later (Remark 3in §4) that X4is isomorphic to
yq=x(x+ 1)ρ(x−1)q−ρ
ON LARGE PRIME ACTIONS ON RIEMANN SURFACES 7
The remaining cases (that is, the surfaces lying in the family Kgsince it contains Cg) are
given in the following proposition.
Proposition 2. Let q>5be prime and let g=q−1.If Sbelongs to the family Kgand
S∼
=Zt={(x, y) : yq= (x−1)(x+ 1)q−1(x−t)(x+t)q−1}
for t∈C− {0,±1}then the field of moduli of Sis Q(t).
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 [64] and also [47].
The following result is a direct consequence of the above.
Corollary 1. The compact Riemann surfaces of Theorem 1and their Jacobian varieties
can be defined over their fields of moduli.
The sublocus of Agwith G-action. It is well-known that the moduli space Agof principally
polarised abelian varieties of dimension gis isomorphic to the quotient
π:Hg→ Ag∼
=Hg/Sp(2g, Z)
of the Siegel upper half-space Hgby the action of the symplectic group Sp(2g, Z). If the
isomorphism class of JS is represented by ZS∈Hgthen there is an isomorphism of groups
Aut(JS)∼
=ΣS:= {R∈Sp(2g, Z) : R·ZS=ZS},
where ΣSis well-defined up to conjugation in Sp(2g, Z).The subset of Hggiven by
SS:= {Z∈Hg:R·Z=Zfor all R∈ΣS}
consists of those matrices representing principally polarised abelian varieties of dimension
gadmitting an action which is equivalent to the one of Aut(JS).This subset is, indeed,
an analytic submanifold of Hgclosely related with some special subvarieties and Shimura
families of Ag.
Observe that if ¯
Ugis an equisymmetric family of compact Riemann surfaces of genus g
and if Sis any surface lying in the interior Ugof ¯
Ugthen
{JX :X∈Ug} ⊆ π(SS).
In general, those loci of Agdo not agree. Nonetheless, the uncommon cases in which these
dimensions do agree have been useful in finding Jacobians with complex multiplication.
Although a satisfactory description of the matrices in SSseems to be a difficult problem,
as we shall see in §2.7, there is a simple representation theoretic way to compute the
dimension of the (component which contains JS of) SS. We shall denote the aforementioned
dimension by NS.
Theorem 4. Let q>5be prime, let g=q−1and let S∈Kg.Then
NX8=NX3=NX2,k = 0, NX4=q−1
4and NS=q−1
2.
According to results due to Streit in [68] (and later generalised in [26] for higher di-
mension), if NSequals zero then the full automorphism group of Sdetermines the period
matrix for JS and JS admits complex multiplication. We refer to [49] and [50] for recent
applications of this result for quasiplatonic curves that are hyperelliptic and superelliptic.
As a direct consequence of the previous theorem we recover the following known result.
8 SEBASTI ´
AN REYES-CAROCCA AND ANITA M. ROJAS
Corollary 2. The Jacobian varieties of X3, X2,k and X8admit complex multiplication.
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 the Macbeath’s curve of genus seven
and of the Bring’s curve were determined in [9] and [60] respectively. A method to find
the period matrices of the Accola-Maclachlan 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 Accola-Maclachlan curve of genus two in an explicit way.
At the end of the paper we determine explicitly the period matrix of the Accola-Maclachlan
curve of genus four.
This article is organised as follows. In §2we succinctly review the basic preliminaries:
Fuchsian groups and group action on Riemann surfaces and abelian varieties. The proof
of Theorem 1is given in §3and the proofs of Theorem 2and Proposition 2are given in
§4. In §5we prove some basic algebraic lemmata needed to prove, in §6, Theorems 3and
4. Finally, we include an addendum in which the period matrix of the Accola-Maclachlan
curve of genus four is computed.
2. Preliminaries
2.1. Fuchsian groups. AFuchsian group is a discrete group of automorphisms of the
upper half-plane H.If ∆ is a Fuchsian group and the orbit space H/∆ given by the action
of ∆ on His compact, then the algebraic structure of ∆ is determined by its signature:
σ(∆) = (γ;k1, . . . , ks),(2.1)
where γis the genus of H/∆ and k1, . . . , ksare the branch indices in the universal canonical
projection H→H/∆.In this case, ∆ has a canonical presentation in terms of canonical
generators α1, . . . , αγ,β1, . . . , βγ, x1, . . . , xsand relations
xk1
1=··· =xks
s= Πγ
i=1[αi, βi]Πs
i=1xi= 1,(2.2)
where the brackets stand for the commutator. The Teichm¨uller space of ∆ is a complex
analytic manifold homeomorphic to the complex ball of dimension 3γ−3 + s.
Let ∆0be a group of automorphisms of Hsuch that ∆ 6∆0of finite index. Then ∆0is
also Fuchsian and they are related by the so-called Riemann-Hurwitz formula
2γ−2+Σs
i=1(1 −1
ki) = [∆0: ∆] ·[2γ0−2+Σr
i=1(1 −1
k0
i)].
where σ(∆0) = (γ0;k0
1, . . . , k0
r).
2.2. Group action on Riemann surfaces. Let Sbe a compact Riemann surface of genus
g>2.A finite group Gacts on Sif there is a group monomorphism :G→Aut(S).The
orbit space S/G given by the action of G∼
=(G) on Sinherits naturally a Riemann surface
structure such that the canonical projection S→S/G is holomorphic.
By the classical uniformisation theorem, there is a unique, up to conjugation, Fuchsian
group Γ of signature (g;−) such that S∼
=H/Γ.Moreover, Gacts on Sif and only if there
is a Fuchsian group ∆ containing Γ together with a group epimorphism
θ: ∆ →Gsuch that ker(θ)=Γ.(2.3)
ON LARGE PRIME ACTIONS ON RIEMANN SURFACES 9
It is said that Gacts on Swith signature σ(∆) and that the action is represented by the
surface-kernel epimorphism (2.3); henceforth, we write ske for short. Abusing notation, we
shall also identify θwith the tuple of the images of the canonical generators of ∆.
2.3. Extending actions. Assume that G0is a finite group such that G6G0.The action
of Gon Srepresented by the ske (2.3) is said to extend to an action of G0on Sif:
(1) there is a Fuchsian group ∆0containing ∆,
(2) the Teichm¨uller spaces of ∆ and ∆0have the same dimension, and
(3) there exists a ske
Θ:∆0→G0in such a way that Θ|∆=θand ker(θ) = ker(Θ).
An action is called maximal if it cannot be extended in the previous sense. Singerman
in [66] determined the complete list of pairs of signatures of Fuchsian groups ∆ and ∆0for
which it may be possible to have an extension as before. See also [62] and [67].
2.4. Equivalence of actions. Two actions i:G→Aut(S) are topologically equivalent if
there exist ω∈Aut(G) and an orientation preserving self-homeomorphism fof Ssuch that
2(g) = f1(ω(g))f−1for all g∈G. (2.4)
Each fsatisfying (2.4) yields an automorphism f∗of ∆ where H/∆∼
=S/G. If Bis the
subgroup of Aut(∆) consisting of them, then Aut(G)×Bacts on the set of skes defining
actions of Gon Swith signature σ(∆) by
((ω, f ∗), θ)7→ ω◦θ◦(f∗)−1.
Two skes θ1, θ2: ∆ →Gdefine topologically equivalent actions if and only if they belong
to the same (Aut(G)×B)-orbit; see, for example, [11]. If the genus of S/G is zero then B
is generated by the so-called braid transformations Φi, for 1 6i < l, defined by
xi7→ xi+1, xi+1 7→ x−1
i+1xixi+1 and xj7→ xjwhen j6=i, i + 1.
2.5. Equisymmetric stratification of Mg.Following [12], the singular locus of Mgad-
mits an equisymmetric stratification where each equisymmetric stratum, if nonempty, cor-
responds to one topological class of maximal actions (see also [29]). More precisely:
Sing(Mg) = ∪G,θ ¯
MG,θ
g
where the equisymmetric stratum MG,θ
gconsists of surfaces of genus gwith full automor-
phism group isomorphic to Gsuch that the action is topologically equivalent to θ. In
addition, the closure ¯
MG,θ
gof MG,θ
gis a closed irreducible algebraic subvariety of Mgand
consists of surfaces of genus gwith a group of automorphisms isomorphic to Gsuch that
the action is topologically equivalent to θ.
The subset ¯
Fg(G, σ) = ¯
Fgof Mgof all those compact Riemann Ssurfaces of genus g
with a group of automorphisms isomorphic to a given group Gacting with a given signature
σwill be called a closed family. Observe that if the signature of the action of Gon Sis
(2.1) then
dim( ¯
Fg) = 3γ−3 + s.
Assume that the action of Gis maximal. Then
(1) the interior Fgof ¯
Fgconsists of those surfaces Ssuch that G= Aut(S),
10 SEBASTI´
AN REYES-CAROCCA AND ANITA M. ROJAS
(2) Fgis formed by finitely many equisymmetric strata that are in correspondence with
the pairwise non-equivalent topological actions of G, and
(3) the set ¯
Fg− Fgis formed by those surfaces Ssuch that G < Aut(S) properly.
2.6. Abelian varieties. A complex abelian variety is a complex torus which is also a
complex projective algebraic variety. Each abelian variety X=V/Λ admits a polarisation,
that is, a non-degenerate real alternating form Θ on Vsuch that for all v, w ∈V
Θ(iv, iw) = Θ(v, w) and Θ(Λ ×Λ) ⊂Z.
If each elementary divisor of Θ|Λ×Λequals 1 then Θ is called principal and Xis called
aprincipally polarised abelian variety; from now on, we write ppav for short. In this case,
there exists a basis for Λ such that the matrix for ΘΛ×Λwith respect to it is given by
J=0Ig
−Ig0where g= dim(X); (2.5)
such a basis is called symplectic. In addition, there exist a basis for Vwith respect to which
the period matrix for Xis
Π=(IgZ) where Z∈Hg={Z∈M(g, C) : Z=Zt,Im(Z)>0},
with Ztdenoting the transpose matrix of Z. The space Hgis called Siegel upper half-space.
By an isomorphism of ppavs we mean an isomorphism of the underlying complex tori
preserving the involved polarisations. In other words, if (IgZi) is the period matrix of Xi
then an isomorphism X1→X2is given by invertible matrices
M∈GL(g, C) and R∈GL(2g, Z) such that M(IgZ1) = (IgZ2)R. (2.6)
Since Rpreserves the polarisation (2.5), it belongs to the symplectic group
Sp(2g, Z) = {R∈M(2g, Z) : RtJR =J}.
It follows from (2.6) that the correspondence Sp(2g, Z)×Hg→Hggiven by
(R=A B
C D , Z)7→ R·Z:= (A+ZC)−1(B+Z D)
defines an action that identifies period matrices representing isomorphic ppavs. Hence
Hg→ Ag:= Hg/Sp(2g, Z)
is the moduli space of isomorphism classes of ppavs of dimension g. See [51].
2.7. Abelian varieties with G-action. Let Sbe a compact Riemann surface of genus
g>2.Consider the Jacobian variety JS and its full (polarisation-preserving) automorphism
group Aut(JS).Every automorphism of Sinduces a unique automorphism of J S. In fact
[Aut(JS) : Aut(S)] ∈ {1,2}
according to whether or not Sis hyperelliptic; moreover, in the latter case
Aut(JS)/Aut(S) = {±1}.
As mentioned in the introduction, once a symplectic basis of Λ = H1(S, Z) is fixed, there
is an isomorphism
Aut(JS)∼
=ΣS:= {R∈Sp(2g, Z) : R·ZS=ZS},
ON LARGE PRIME ACTIONS ON RIEMANN SURFACES 11
where (IgZS) is the period matrix of J S. A change of basis induces a different but equivalent
choice of ZSand a conjugate subgroup ΣS.One obtains a well-defined analytic submanifold
SS:= {Z∈Hg:R·Z=Zfor all R∈ΣS}
of Hgwhose points represent ppavs admitting an action equivalent to the one of Aut(JS)
in the symplectic group. Equivalently, as −1∈ΣS, the previous submanifold represents
ppavs admitting an action equivalent to the one of Aut(S).Clearly, ZS∈SS.
According to [68] (see also [26, Lemma 3.8]), the dimension NSof (the component which
contains JS of ) SSagrees with
dim(Sym2H1,0(S, C))Aut(S)
where Sym2H1,0(S, C) stands for the symmetric square of H1,0(S, C). It follows that
NS=hχsym
ρa|1iGwhere G= Aut(S)
and χsym
ρadenotes the character of the symmetric square of the analytic representation ρa
of Gand the brackets denote the usual inner product of characters of G.
It is worth mentioning that SSis related to some special subvarieties of Ag.Indeed, as
Aut(JS) can be considered as a subgroup of
LS:= EndQ(JS) = End(J S)⊗ZQ
one sees that SScontains a complex submanifold of Hgof matrices representing ppavs
containing LSin their endomorphism algebras. This submanifold is called a Shimura domain
for Sand the corresponding ppavs form a so-called Shimura family for S; this a special
subvariety of Ag(see [48,§3] for a precise definition). We refer to [72,§3] for more details.
2.8. The group algebra decomposition. The action of a group Gon a compact Riemann
surface Sinduces a Q-algebra homomorphism from the rational group algebra of Gto LS
Ξ : Q[G]→LS.
Let W1, . . . , Wrbe the rational irreducible representations of G, and for each Wllet Vl
be a complex irreducible representation of Gassociated to it. Following [45], the equality
1 = e1+· ·· +erin Q[G],(2.7)
where elis a uniquely determined central idempotent associated to Wl,yields an isogeny
JS ∼A1× · ·· × Arwhere Al:= Ξ(αlel)(J S)
which is G-equivariant, with αl>1 chosen to satisfy αlel∈Z[G]. Additionally, there are
idempotents fl1, . . . , flnlsuch that
el=fl1+· ·· +flnl(2.8)
where nl=dl/slis the quotient of the degree dland the Schur index slof Vl. These
idempotents provide nlpairwise isogenous subvarieties of JS. If we denote by Blone of
them for each l, then (2.7) and (2.8) provide the isogeny
JS ∼Bn1
1× · ·· × Bnr
r(2.9)
known as the group algebra decomposition of JS with respect to G. See [16].
Let Hbe a subgroup of G. We denote by dH
lthe dimension of the vector subspace of Vlof
those elements which are fixed under H. Following [16, Proposition 5.2], the group algebra
12 SEBASTI´
AN REYES-CAROCCA AND ANITA M. ROJAS
decomposition (2.9) induces the following isogeny of the Jacobian J(S/H) of the quotient
S/H
J(S/H)∼BnH
1
1× · ·· × BnH
r
rwhere nH
l=dH
l/sl.(2.10)
The previous isogeny has proved to be fruitful in finding Jacobians J S isogenous to a
product of Jacobians of quotients of S. See, for example, [58] and also [59].
Assume that (γ;k1, . . . , ks) is the signature of the action of Gon Sand that this action
is represented by the ske θ: ∆ →G, with ∆ as in (2.2). Observe that if V1=W1denotes
the trivial representation of Gthen B1∼J(S/G) and therefore dim B1=γ. If l>2 then,
according to [62, Theorem 5.12], we have that
dim Bl=ml[dl(γ−1) + 1
2Σs
j=1(dl−dhθ(xj)i
l)] (2.11)
where mlis the degree of Q≤Llwith Lldenoting a minimal field of definition for Vl.
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], [55], [57] and [61].
3. Proof of Theorem 1
The proof of Theorem 1is presented as a consequence of a series of propositions proved
in this section. Hereafter, we assume q>7 to be prime and Sto be a Riemann surface of
genus g:= q−1 with a group of automorphisms Gof order λq where λ>1 is an integer.
Proposition 3.1. If λ= 3 then Gis cyclic and acts with signature (0; 3, q, 3q).Moreover,
Sis unique up to isomorphism and Gis its full automorphism group.
Proof. Let (γ;k1, . . . , kl) be the signature of the action of Gon S. The Riemann-Hurwitz
formula implies that
2(q−2) >3q(2γ−2 + 2
3l).(3.1)
Observe that if γ>1 then l= 0 and therefore q= 2,contradicting the assumption q>7.
We then assume γ= 0 and therefore (3.1) shows that l= 3.It follows that the signature of
the action of Gis
(0; k1, k2, k3) where 1
k1+1
k2+1
k3=1
3+4
3qand kj∈ {3, q, 3q}.
After a routine computation, one sees that the unique solution of the previous equation is,
up to permutation, k1= 3, k2=qand k3= 3q. The last equality implies that
G∼
=Cq×C3=hα, β :αq=β3= 1,[α, β]=1i.
Consider the Fuchsian group ∆ of signature (0; 3, q , 3q) canonically presented
∆ = hw1, w2, w3:w3
1=wq
2=w3q
3=w1w2w3= 1i
and let θ: ∆ →Gbe a ske representing an action of Gon S. It is not difficult to see that,
up to an automorphism of G, the ske θis given by
θ(w1) = β, θ(w2) = αand θ(w3) = α−1β2;
this proves the uniqueness of S. By the results of [66], if Gis strictly contained in the
full automorphism group Aut(S) of Sthen Aut(S) has order 12q, acts on Swith signature
(0; 2,3,3q) and Gis a non-normal subgroup of it. By the classical Sylow’s theorem, if a
group of order 12qwith q > 11 has a non-normal subgroup isomorphic to Gthen it is
ON LARGE PRIME ACTIONS ON RIEMANN SURFACES 13
isomorphic to Cq×A4where A4stands for the alternating group of order 12. However, the
product of an element of order two and an element of order three of Cq×A4cannot have
order 3q. The cases q611 are not realised either; see [18]. The proof is done.
Proposition 3.2. λis different from 5,6and 7.
Proof. If λequals 5,6 or 7 then Gis a large group of automorphisms (that is, |G|>4(g−1))
and therefore (see, for example, [43,§2.3]) the signature of the action is either
(1) (0; k1, k2, k3) for some 2 6k16k26k3,
(2) (0; 2,2,2, k) for some k>3,or
(3) (0; 2,2,3, k) for some 3 6k65.
If λequals 5 or 7 then Ghas no involutions; then the signature of the action Gis
(0; k1, k2, k3) for some kj∈ {5, q, 5q}or kj∈ {7, q, 7q}
respectively. The Riemann-Hurwitz formula implies that
1
k1+1
k2+1
k3=3
5+4
5qand 1
k1+1
k2+1
k3=5
7+4
7q
respectively and this, in turn, implies that qis negative; a contradiction.
We now assume that Ghas order 6q. If the signature of the action of Gis (0; 2,2,2, k)
then, by the Riemann-Hurwitz formula, we have that q+ 4 divides 6qand therefore q= 2;
contradicting the assumption q>7. The signatures (0; 2,2,3,4) and (0; 2,2,3,5) cannot be
realised either, since a group of order 6qdoes not have elements of order 4 nor 5.Besides, a
direct computation shows that the signature (0; 2,2,3,3) contradicts the Riemann-Hurwitz
formula.
It follows that the signature of the action is (0; k1, k2, k3) where kj∈ {2,3,6, q, 2q, 3q, 6q}
satisfy, by the Riemann-Hurwitz formula, the equality
1
k1+1
k2+1
k3=2
3+2
3q.
Set v= #{kj:kj= 3}.It is clear that v6= 2,3. Assume v= 1 and say k1= 3. If
k2, k3>6 then q60.Then, we can assume k2= 2 and therefore
1
k3=−1
6+2
3qshowing that q63.
Thus, v= 0.Now, let u= #{kj:kj= 2}and observe that u6= 2,3.If u= 1 then
1
k2+1
k3=2
3q−1
6and therefore q63.
All the above ensures that each kj>6.It follows that
1
k1+1
k2+1
k3=2
3+2
3q61
2
and therefore q < 0; contradicting the assumption q>7.
Proposition 3.3. If λ>8then λ= 8 and S∼
=X8.
Proof. If the order of Gis at least 8(g+ 1) then, following [2, p. 77], the signature of the
action of Gis either
(1) (0; 2,2,2,3),(5) (0; 2,6, k) where 6 6k611,
(2) (0; 2,3, k) where k>7,(6) (0; 2,7, k) where 7 6k69,
(3) (0; 2,4, k) where k>5,(7) (0; 3,3, k) where 4 6k611,or
(4) (0; 2,5, k) where 5 6k619,(8) (0; 3,4, k) where 4 6k65.
14 SEBASTI´
AN REYES-CAROCCA AND ANITA M. ROJAS
We observe that cases (1), (5) and (8) are not realised. Indeed, this fact follows from the
contradiction between the fourth and fifth columns in the following table.
case signature |G|condition Riemann-Hurwitz formula
(1) (0; 2,2,2,3) 6λ0q λ0>2λ0= 1 −2
q
(5) (0; 2,6, k) 6λ0q λ0>2λ0=k
k−3(1 −2
q)
(8.1) (0; 3,4,4) 12λ0q λ0>1λ0= (1 −2
q)
(8.2) (0; 3,4,5) 60λ0q λ0>1λ0=2
13 (1 −2
q)
We also note that cases (4), (6) and (7) are not realised. Indeed
case signature |G|condition Riemann-Hurwitz formula
(4) (0; 2,5, k) 10λ0q λ0>1λ0=2k
3k−10 (1 −2
q)
(6) (0; 2,7, k) 2λ0q λ0>4λ0=14k
5k−14 (1 −2
q)
(7) (0; 3,3, k) 3λ0q λ0>3λ0=2k
k−3(1 −2
q)
and notice that:
(a) in case (4) we have that λ0= 1 and therefore q= 4k/(10 −k) and 5 6k69.
However, for each kas before we obtain that qis not prime;
(b) in case (6) we have that λ0= 4 and therefore qequals 14, 28 and 126 for k= 7,8
and 9 respectively; and
(c) in case (7) the facts that λ>3 and q>7 imply that k68.If k= 4 then qis not
prime, if k= 5 then λ0= 3 or 4 and q= 5 or 10, and if k= 6,7,8 then λ0= 3 and
qis not prime.
We claim that case (2) is not realised either. Indeed, note that otherwise the order of G
equals 6λ0qfor some λ0>2 and the Riemann-Hurwitz formula reads
k=6qλ0
q(λ0−2)+4 and therefore k0:= 6λ0
q(λ0−2)+4 ∈Z+
(1) If k0= 1 then λ0= 2 + 8/(q−6),showing that q= 7 and λ0= 10.Consequently, the
order of Gis 420 and acts on Sof genus six with signature (0; 2,3,7).However, such
a Riemann surface does not exist because the maximal number of automorphisms
that a Riemann surface of genus six can admit is 150 (see, for instance, [18]).
(2) If k0>2 then λ062+2/(q−3) and therefore λ0= 2.It follows that Ghas order 12q
and acts on Swith signature (0; 2,3,3q).Observe that the signature of the action
shows, in particular, that Shas a cyclic subgroup H < G of automorphisms of order
3q. However, as proved in Proposition 3.1, if Shas a group of automorphisms of
order 3qthen Sdoes not have more automorphisms; a contradiction.
This proves the claim.
All the above ensures that the signature of the action is (0; 2,4, k) for some k>5.Observe
that the order of Gis 4λ0qfor some λ0>2 and the Riemann-Hurwitz formula says
λ0=2k
k−4(1 −2
q)<2k
k−4.
ON LARGE PRIME ACTIONS ON RIEMANN SURFACES 15
It follows that one of the following statements holds.
(1) k= 5 and λ0∈ {3,...,9}and therefore q= 20/(10 −λ0).
(2) k= 6 and λ0∈ {3,4,5}and therefore q= 12/(6 −λ0).
(3) k>5 and λ0= 2,and therefore q=k/2.
The first two cases must be disregarded because qis not prime; then Ghas order 8qand
acts with signature (0; 2,4,2q). By [43,§5], we obtain that S∼
=X8as desired.
We recall that, following [22], the closed family ¯
Cgconsists of all those compact Riemann
surfaces of genus gendowed with a group of automorphisms Gisomorphic to
Dq×C2∼
=D2q
acting with signature (0; 2,2,2, q).Moreover, if Sbelongs to the interior of ¯
Cgthen Gagrees
with the full automorphism group of S. It was also observed in [22] that X8∈¯
Cg−Cg.
Proposition 3.4. ¯
Cg−Cg={X8}.
Proof. Observe that if Xbelongs to ¯
Cg−Cgthen its automorphism group has order 4tq for
some t>2.It follows from Proposition 3.3 that t= 2 and that X∼
=X8.
For later and repeated use, we recall here that
Aut(X8)∼
=hx, y, z :x2q=y2=z2= 1,[x, y]=[z, y] = 1, zxz =x−1yi(3.2)
and its action on X8is represented by the ske
Θ : ∆8→Aut(X8) given by (z1, z2, z3)→(z, zx, x−1),(3.3)
where ∆8is a Fuchsian group of signature (0; 2,4,2q) presented as
∆8=hz1, z2, z3:z2
1=z4
2=z2q
3=z1z2z3= 1i.(3.4)
Proposition 3.5. If Xis a compact Riemann surface of genus gwith a group of automor-
phisms isomorphic to Cq×C2
2acting with signature (0; 2,2q, 2q)then X∼
=X8.
Proof. Let ∆2be a Fuchsian group of signature (0; 2,2q, 2q) presented as
∆2=hy1, y2, y3:y2
1=y2q
2=y2q
3=y1y2y3= 1i(3.5)
and consider the group G∼
=Cq×C2
2presented as
hA, B, C :Aq=B2=C2= (BC )2= [A, B ] = [A, C]=1i.
Let θ: ∆2→Gbe a ske representing the action of Gon X. Up to an automorphism of
Gwe can assume θ(y1) = B. Moreover, after considering the automorphism of Ggiven by
A7→ A, B 7→ B, C 7→ BC,
we can assume that θ(y2) equals either AiBor AiCfor some i∈Z∗
q.Note that the former
case is impossible, since θ(y1y2) would not have order 2q. Thus, after sending Ato an
appropriate power of it, we obtain that θis equivalent to
∆2→Cq×C2
2given by (y1, y2, y3)7→ (B, AC, (ABC )−1).(3.6)
Observe that, with the notations of (3.4), the elements
ˆy1:= z−2
2,ˆy2:= z−1
3and ˆy3=z−1
2z−1
3z2
16 SEBASTI´
AN REYES-CAROCCA AND ANITA M. ROJAS
generate a subgroup of ∆8isomorphic to ∆2and the restriction of (3.3) to it
∆2→Cq×C2
2is given by (ˆy1,ˆy2,ˆy3)7→ (y, x, x−1y).(3.7)
By letting x=AC and y=Bwe see that (3.6) and (3.7) agree; consequently X∼
=X8.
Proposition 3.6. If Yis a compact Riemann surface of genus gwith a group of automor-
phisms isomorphic to Cqo2C4acting with signature (0; 4,4, q)then Y∼
=X8.
Proof. Let ∆3be a Fuchsian group of signature (0; 4,4, q) presented as
∆3=hy1, y2, y3:y4
1=y4
2=yq
3=y1y2y3= 1i(3.8)
and consider the group G∼
=Cqo2C4presented as
hA, B :Aq=B4= 1, BAB −1=A−1i.
Let θ: ∆3→Gbe a ske representing the action of Gon Y. Then, after sending Bto
B−1if necessary, θis given by
(y1, y2, y3)7→ (AiB, AjB−1, Ak) for some i, j ∈Zqand k∈Z∗
q.
Up to conjugation, we can assume i= 0 and after sending Ato an appropriate power of it,
we can assume k= 1.It follows that j= 1 and therefore θis equivalent to
∆3→Cqo2C4given by (y1, y2, y3)7→ (B, AB−1, A).(3.9)
Now, as done in the previous proposition, with the notations of (3.4), we see that
˜y1:= z2,˜y2:= z3z2z−1
3and ˜y3=z2
3
generate a subgroup of ∆8isomorphic to ∆3and the restriction of (3.3) to it
∆3→Cqo2C4is given by (˜y1,˜y2,˜y3)7→ (zx, x−3z , x−2).(3.10)
Write A=x−2and B=zx to see that (3.9) and (3.10) agree; consequently Y∼
=X8.
Proposition 3.7. Assume q≡1mod 4.There exists a unique, up to isomorphism, compact
Riemann surface X4of genus gwith full automorphism group isomorphic to Cqo4C4acting
on it with signature (0; 4,4, q).
Proof. Consider the Fuchsian group ∆3as in (3.8), and the group
G∼
=Cqo4C4=hA, B :Aq=B4= 1, BAB −1=Aρi(3.11)
where ρis a primitive fourth root of unity in Zq.If θ: ∆3→Cqo4C4is a ske representing
the action of Gon a compact Riemann surface Zof genus gthen, by proceeding similarly
as done in the previous proposition, one sees that θis equivalent to
θ1(y1, y2, y3) = (A−1B, B −1, A) or θ2(y1, y2, y3)=(A−1B−1, B, A).
It follows that, up to isomorphism, there are at most two surfaces Zas before; namely
Zj:= H/Kjwhere Kj= ker(θj) for j= 1,2.
Observe that if the full automorphism group of Zjis different from Gthen, by Proposition
3.3, necessarily Zj∼
=X8and, in particular, Aut(X8) contains a subgroup isomorphic to
Cqo4C4.However, this is not possible. Indeed, with the notations of (3.2), if ι∈Aut(X8)
has order 4 then ι2equals the central element y. It follows that Aut(Zj)∼
=Gfor j= 1,2.
ON LARGE PRIME ACTIONS ON RIEMANN SURFACES 17
We record here that Z1and Z2are isomorphic if and only if K1and K2are conjugate in
Aut(H). As the normaliser of each Kjis ∆3,it can be seen that K1and K2are conjugate if
and only if they are conjugate in the normaliser N(∆3) of ∆3.Now, the action by conjugation
of N(∆3) on {K1, K2}has orbits of length [N(∆3):∆3] which is, by [66, Theorem 1], equal
to 2. Hence, K1and K2are conjugate and thus X4:= Z1∼
=Z2as desired.
Proposition 3.8. If λ= 4 and S /∈¯
Cgthen q≡1mod 4and S∼
=X4.
Proof. As in the proof of Proposition 3.2, the signature of the action of Gon Sis either
(1) (0; k1, k2, k3) for some 2 6k16k26k3,
(2) (0; 2,2,2, k) for some k>3,or
(3) (0; 2,2,3, k) for some 3 6k65.
The third case must be disregarded since there is no group of order 4qwith an element
of order three. If the signature is as in the second case, then the Riemann-Hurwitz formula
implies that k=q. We now assume the signature to be as in the first case. The Riemann-
Hurwitz formula says
1
k1+1
k2+1
k3=1
2+1
q.(3.12)
Note that among k1, k2, k3not two or three of them can be equal to 2. It follows that, up
to permutation, there are two cases to consider.
(1) Assume k1= 2 and k2, k3>4.Then (3.12) turns into 1
k2+1
k3=1
q.Note that if
k2= 4 then k360.It follows that k2, k3>qand therefore k2=k3= 2q.
(2) Assume k1, k2, k3>4.If the number of periods kjthat are equal to 4 is 2 then
(3.12) implies that the signature is (0; 4,4, q); otherwise q64.
Thereby, the signature of the action of Gon Sis either
(0; 2,2,2, q),(0; 2,2q, 2q) or (0; 4,4, q).
We recall that if q≡3 mod 4 then Gis isomorphic to either C4q,Cq×C2
2,D2qor Cqo2C4,
and if q≡1 mod 4 then, in addition, Gcan be isomorphic to Cqo4C4.
(1) If Gacts with signature (0; 2,2,2, q) then Gis generated by three involutions and
therefore G∼
=D2q,showing that S∈¯
Cg.
(2) If Gacts with signature (0; 2,2q, 2q) then Gis generated by two elements of order
2qwhose product is an involution; thus G∼
=Cq×C2
2.By Proposition 3.5, we see
that S∼
=X8and therefore S∈¯
Cg.
(3) If Gacts with signature (0; 4,4, q) then Gis generated by two elements of order 4
and therefore Gis isomorphic to Cqo2C4or Cqo4C4.By Proposition 3.6 the former
case implies S∼
=X8whilst by Proposition 3.7 the latter case implies S∼
=X4.
This finishes the proof.
Proposition 3.9. If λ= 2 and Gis cyclic acting with signature (0; 2,2, q, q)then S∈¯
Cg.
Proof. Let ∆4be a Fuchsian group of signature (0; 2,2, q, q) presented as
∆4=hx1, x2, x3, x4:x2
1=x2
2=xq
3=xq
4=x1x2x3x4= 1i(3.13)
18 SEBASTI´
AN REYES-CAROCCA AND ANITA M. ROJAS
and consider the cyclic group of order 2qgenerated by aof order qand bof order two. As
Ghas only one involution, it is clear that, after sending ato a suitable power of it, each
ske representing an action of Gon Sis equivalent to
θ0: ∆4→Cq×C2such that θ0(x1, x2, x3, x4)=(b, b, a, a−1).(3.14)
Then, such surfaces Sform an equisymmetric complex one-dimensional family. Let
∆1=hy1, y2, y3, y4:y2
1=y2
2=y2
3=yq
4=y1y2y3y4= 1i(3.15)
be a Fuchsian group of signature (0; 2,2,2, q) and consider the group
D2q=hR, T :R2q=T2= (T R)2= 1i.(3.16)
We recall that, following [22], the action of D2qon S0∈¯
Cgis represented by the ske
θ: ∆1→D2qgiven by (y1, y2, y3, y4)7→ (Rq, T, T R, Rq−1) (3.17)
Now, the elements of ∆1
ˆx1:= y1,ˆx2:= y2y1y2,ˆx3:= y4and ˆx4:= y1y2y4y2y1
generate a Fuchsian group isomorphic to ∆4.The restriction of (3.17) to it
∆4→ hRi∼
=Gis given by (ˆx1,ˆx2,ˆx3,ˆx4)7→ (Rq, Rq, Rq−1, R1−q).(3.18)
Set a:= Rq−1and b:= Rqto see that (3.18) agrees with (3.14) and the result follows.
Proposition 3.10. If λ= 2 and Gis a cyclic group acting with signature (0; q, 2q, 2q),
then either S∼
=X8or the full automorphism group of Sagrees with G. In the latter case,
there are exactly q−3
2pairwise non-isomorphic compact Riemann surfaces.
Proof. Let ∆5be a Fuchsian group of signature (0; q, 2q, 2q) presented as
∆5=hx1, x2, x3:xq
1=x2q
2=x2q
3=x1x2x3= 1i
and consider the cyclic group of order 2qgenerated by aof order qand bof order two.
If θ: ∆5→Gis a ske representing the action of Gon Sthen after sending ato an
appropriate power of it, we see that θis equivalent to
θj= (a, ajb, a−j−1b) for some j6=−1,0.
Let Sjbe the compact Riemann surface defined by θjand write j∗=q−1
2.
We claim that Sj∗∼
=X8.To prove that, we notice that, by Proposition 3.5, it suffices
to verify that θj∗extends to the action of Cq×C2
2with signature (0; 2,2q, 2q).With the
notations of the proof of Proposition 3.5, the elements
ˆx1:= y2
3,ˆx2:= y1y2y1and ˆx3:= y2
generate a subgroup of (3.5) isomorphic to ∆5and the restriction of (3.6) to it
∆5→ hA, Ci∼
=Gis given by (ˆx1,ˆx2,ˆx3)7→ (A−2, AC, AC ).(3.19)
By setting a:= A−2and b:= C, we see that (3.19) is equivalent to θj∗,as desired.
Let j6=j∗.If Aut(Sj)6=Gthen, by [66] and Proposition 3.3, the action θjmust extend
to the action (3.6) of Cq×C2
2with signature (0; 2,2q, 2q).Observe that an element yof
(3.5) has order 2qif and only if it is conjugate to yk
2or to yk
3for some k∈ {1,...,2q−1}
odd and different from q. As the target group is abelian, the image of yunder (3.6) is either
ON LARGE PRIME ACTIONS ON RIEMANN SURFACES 19
AC or (ABC)−1. Now, if ∆0is a subgroup of (3.5) isomorphic to ∆5then the restriction
of (3.6) to the canonical generators of ∆0must be
(A−2, AC, AC),(B, AC, (ABC)−1) or (A2,(ABC)−1,(ABC)−1).
The second case is impossible since it does not have the required signature; the other two
cases are equivalent to θj∗. We conclude that if j6=j∗then Aut(Sj) = Gand, in particular,
Sjis not isomorphic to Sj∗.
Write Kj= ker(θj) for each j∈Zq− {−1,0, j∗}.As argued in the proof of Proposition
3.7, we have that Sj1and Sj2are isomorphic if and only if Kj1and Kj2are conjugate in
the normaliser N(∆5) of ∆5.The action by conjugation of N(∆5) on
{Kj:j∈Zq− {−1,0, j∗}}
has orbits of length [N(∆5):∆5] which is, by [66, Theorem 1], equal to 2. Hence,
{Sj:j∈Zq− {−1,0, j∗}}
splits into q−3
2isomorphism classes. Finally, observe that the elements x2and x3of ∆5
are conjugate in N(∆5); thus, Sjand S−j−1are isomorphic and therefore the isomorphism
classes are represented by Sjwhere 1 6j6q−3
2.
Proposition 3.11. There exists a closed family ¯
Kgof compact Riemann surfaces with a
group of automorphisms isomorphic to the dihedral group of order 2qacting with signature
(0; 2,2, q, q).The number of equisymmetric strata of ¯
Kgis at most
q+3
4if q≡1mod 4
q+1
4if q≡3mod 4
and, independently of q, one of them equals Cg.
Proof. Let ∆4be a Fuchsian group of signature (0; 2,2, q, q) presented as in (3.13) and
consider the dihedral group of order 2q
G∼
=Dq=hr, s :rq=s2= (sr)2= 1i.
The existence of the family ¯
Kgfollows after considering the ske
Φ:∆4→Ggiven by (x1, x2, x3, x4)7→ (s, s, r−1, r).
Let us now assume that θ: ∆4→Gis a ske representing the action of Gon S∈¯
Kg. If
θ(x1) = θ(x2) then, after a conjugation and after sending rto an appropriate power of it,
we see that θis equivalent to Φ.On the other hand, if θ(x1)6=θ(x2) then, after considering
a suitable automorphism of G, we see that θis equivalent to the ske
θi:= (s, sr, ri, r−i−1) for some i∈Zq− {−1,0}.
The braid transformation Φ3(see §2.4) shows that θi∼
=θ−i−1.The rule i7→ −i−1 has
order two, restricts to a bijection of Zq− {−1,0}and has exactly one fixed point; namely
i∗=q−1
2. Observe that if ϕuis the automorphism of Ggiven by r7→ ruthen
Φ=Φ2
2◦ϕ(i∗)−1(θi∗).
All the above ensures that θis equivalent to either
Φ or θifor some i∈ {1,...,q−3
2}.
20 SEBASTI´
AN REYES-CAROCCA AND ANITA M. ROJAS
Now, for each i∈ {1,...,q−3
2}the transformation ϕi−1◦Φ2
2provides an equivalence
θi∼
=θ−i(2i+1)−1.
The rule i7→ −i(2i+ 1)−1has order two and (up to the identification i∼ −i−1) restricts
to a bijection of {1,...,q−3
2}; it has a fixed point if and only if
−i−1 = −i(2i+ 1)−1⇐⇒ 2i2+ 2i+ 1 = 0 (3.20)
and the quadratic equation above has solution in Zqif and only if q≡1 mod 4.It follows
that the number of pairwise non-equivalent skes θis at most
1 + 1
2(q−3
2) = q+1
4and 2 + 1
2(q−3
2−1) = q+3
4
if q≡3 mod 4 and q≡1 mod 4 respectively. Finally, with the notations of (3.15), define
ˆx1:= y1y2y1,ˆx2:= y2,ˆx3:= y2y1y4y1y2and ˆx4:= y4
and notice that they generate a Fuchsian group isomorphic to ∆4.The restriction of (3.17)
to it is given by
(ˆx1,ˆx2,ˆx3,ˆx4)7→ (T , T, R1−q, Rq−1).(3.21)
If we write s:= Tand r:= Rq−1we see that (3.21) agrees with Φ.Hence, the action of Φ
extends to (3.17) and therefore the stratum defined by Φ agrees with Cg.
Proposition 3.12. If Kgstands for the interior of the closed family ¯
Kgthen the full
automorphism group of S∈Kgis isomorphic to either Dqor D2q.In addition
¯
Kg−Kg={X8, X4}if q≡1mod 4
{X8}if q≡3mod 4.
Proof. We keep the notations of the proof of Proposition 3.11. The first statement is clear
since the full automorphism group of Sis isomorphic to D2qor Dqaccording to whether
or not θiis equivalent to Φ.
Let Ki
gdenote the equisymmetric stratum defined by θi.
(1) If θiis equivalent to Φ then Propositions 3.11 and 3.4 imply that ¯
Ki
g−Ki
g={X8}.
(2) If θiis non-equivalent to Φ then, by Proposition 3.8, we see that:
(a) if q≡3 mod 4 then ¯
Ki
g−Ki
gis empty, and
(b) if q