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A note on Jacobians of quasiplatonic Riemann surfaces with complex multiplication

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Abstract

Let m≥6m6m \ge 6 be an even integer. In this short note we prove that the Jacobian variety of a quasiplatonic Riemann surface with associated group of automorphisms isomorphic to C22⋊2CmC222CmC_2^2 \rtimes _2 C_m admits complex multiplication. We then extend this result to provide a criterion under which the Jacobian variety of a quasiplatonic Riemann surface admits complex multiplication.
A NOTE ON JACOBIANS OF QUASIPLATONIC RIEMANN
SURFACES WITH COMPLEX MULTIPLICATION
SEBASTI ´
AN REYES-CAROCCA
Abstract. Let m>6 be an even integer. In this short note we prove that
the Jacobian variety of a quasiplatonic Riemann surface with associated group
of automorphisms isomorphic to C2
2o2Cmadmits complex multiplication. We
then extend this result to provide a criterion under which the Jacobian variety
of a quasiplatonic Riemann surface admits complex multiplication.
1. Introduction
A simple complex polarized abelian variety Aof dimension gis said to admit
complex multiplication if its rational endomorphism algebra
E= End(A)ZQ
is a number field of degree 2g. In this case Eis a CM field; namely, a totally
imaginary quadratic extension of a totally real field of degree g. If Ais not simple
then, by Poincar´e Reducibility theorem, there exist pairwise non isogenous simple
abelian varieties A1, . . . , Asand positive integers n1, . . . , nsin such a way that
AAn1
1× · · · × Ans
s
where stands for isogeny. By definition, Aadmits complex multiplication if each
simple factor Ajdoes.
Let Xbe a compact Riemann surface (or, equivalently, a complex algebraic
curve) and let JX denote its Jacobian variety. Classical examples of compact Rie-
mann surfaces with Jacobian variety admitting complex multiplication are Fermat
curves and their quotients. However, in general, it is a difficult task to decide
whether or not the Jacobian variety of a given compact Riemann surface admits
complex multiplication, and much less is known about their distribution in the
moduli space Agof principally polarized abelian varieties; see, for example, [8].
As a matter of fact, a well-known conjecture due to Coleman predicts that if g>4
then the number of isomorphism classes of compact Riemann surfaces of genus g
with Jacobian variety admitting complex multiplication is finite. In spite of the
fact that this conjecture has been proved to be false for g67,currently it still
remains as an open problem for g>8.This conjecture is closely related to impor-
tant open problems of Shimura varieties, special points in the Torelli locus of Ag
and the theory of unlikely intersections.
2010 Mathematics Subject Classification. 11G15, 11G10, 14K22, 14H37.
Key words and phrases. Riemann surfaces, Jacobian varieties, complex multiplication.
Partially supported by Fondecyt Grants 11180024, 1190991 and Redes Grant 170071.
1
2 SEBASTI ´
AN REYES-CAROCCA
If the Jacobian variety of a compact Riemann surface Xadmits complex multi-
plication then Xcan be defined, as complex algebraic curve, over a number field;
see [12]. In part due to this fact, there has been an increase in the interest of these
compact Riemann surfaces, particularly in their applications to number theory and
arithmetic geometry.
By the classical Belyi’s theorem [1], a compact Riemann surface Xcan be defined
over a number field if and only if there exists a holomorphic map
β:XP1
with at most three critical values; the pair (X, β) is called a Belyi pair. Possibly
the more interesting examples of Belyi pairs are the regular ones: namely, those for
which βis given by the action of a group of automorphisms of X. In this case, Xis
known to be quasiplatonic (or to have many automorphisms); that is, it cannot be
deformed non-trivially in the moduli space together with its automorphism group.
Oort in [9, p.18] considered quasiplatonic Riemann surfaces and discussed the
problem of determining which among them have Jacobian variety admitting com-
plex multiplication. For genus at most four, this problem was completely solved
by Wolfart in [14]. Later, uller and Pink in [6] and Obus and Shaska in [7] con-
sidered the hyperelliptic and superelliptic situations respectively, and succeeded in
determining which among them have Jacobian variety admitting complex multi-
plication.
A different approach can be done by considering regular Belyi pairs
β:XP1
=X/H
whose covering groups Hshare a common property or have the same algebraic
structure. In this direction, it was proved in [14] that if His abelian then JX
admits complex multiplication, and in [2] the same conclusion was obtained for two
infinite series of compact Riemann surfaces arising as quotients of regular Belyi
curves with a metacyclic group of automorphisms.
In this short note we consider a infinite series of quasiplatonic Riemann surfaces
with associated covering group isomorphic to the semidirect product
Gm:= ha, b, t :a2=b2= (ab)2=tm= 1, tat1=a, tbt1=abi
=C2
2o2Cm
where m>6 is even integer. Based on the classification and description obtained
in [3] for quasiplatonic Riemann surfaces with action of semidirect products of the
form C2
2oCm, we prove the following result.
Theorem. Let m>6 be an even integer. If (X, β ) is a regular Belyi pair with
associated covering group isomorphic to Gmthen the Jacobian variety JX admits
complex multiplication.
The proof of the theorem –which is rather simple and based on the classical
theory of covering of Riemann surfaces– is done in Section §2. Then, in Section §3
we extend the arguments used to prove the theorem to provide a criterion under
which the Jacobian variety of a quasiplatonic Riemann surface admits complex
QUASIPLATONIC RIEMANN SURFACES WITH COMPLEX MULTIPLICATION 3
multiplication. Finally, we end this short note by recalling a couple of observations
in Section §4.
2. Proof of the theorem
Let (X, β) be a regular Belyi pair with associated covering group isomorphic to
Gmwhere m>6 is even.
Case A. Assume m2 mod 4.Following [3, Theorem 1(1)], the regular cov-
ering map
XX/Gm
=P1
ramifies over three values marked with 2, m and 2m, and the genus of Xis
gX=m2.
Moreover, as observed in [3, Subsection §4.3.1], its Jacobian variety decomposes
isogenously as the product
JX JY 2(2.1)
where Y=X/haiis the quotient Riemann surface represented by the affine alge-
braic curve
y2=xm1.
Note that haiis a normal subgroup of Gmand the quotient H=Gm/haiis an
abelian group of order 2macting as a group of automorphisms of Y. Clearly, the
corresponding orbit space Y /H has genus zero and the associated regular covering
map
βH:YY/H
ramifies over at most three values. It follows that (Y, βH) is a regular Belyi pair
with abelian covering group; thus, by [14, Theorem 4], we obtain that JY admits
complex multiplication. The result follows from the isogeny (2.1).
Case B. Assume m0 mod 4. Following [3, Theorem 1(2b)], the regular
covering map
XX/Gm
=P1
ramifies over three values marked with 2, m and mand the genus of Xis
gX=m3.
Moreover, as observed in [3, Subsection §4.3.3], its Jacobian variety decomposes
isogenously as the product
JX JY ×J Z2(2.2)
where Y=X/haiand Z=X/hbiare the quotient Riemann surfaces represented
by the affine algebraic curves
y2=xm1 and y2=xm
21
respectively; their genera are gY=m
21 and gZ=m
41.
We argue analogously as done in the case Ato ensure that J Y admits complex
multiplication. Besides, in order to prove that JZ also does, define
ι(x, y)=(x, y) and τ(x, y) = (exp(4πi
m)x, y)
4 SEBASTI ´
AN REYES-CAROCCA
and let K=hι, τ i
=C2×Cm/2.We observe that the abelian group Ksatisfies
|K|>4(gZ1) for all m>8.
Then, by the classification of large abelian groups of automorphisms of compact
Riemann surfaces given in [5, Theorem 3.1], we see that the branched regular
covering map
βK:ZZ/K
=P1
ramifies over three values, marked with 2, m
2and m
2. Thus, again by [14, Theorem
4], we conclude that JZ admits complex multiplication and the result follows from
the isogeny (2.2).
3. A generalization
Let (X, β) be a regular Belyi pair and let Gdenote the associated covering
group. Consider a collection
{H1, . . . , Hs}
of proper non-trivial subgroups of G. Let Yidenote the quotient Riemann surface
X/Hiand let gi6= 0 denote its genus, for each i {1, . . . , s}.
Assume the existence of positive integers n, n1, . . . , nsin such a way that
JXnJY n1
1× · · · × J Y ns
s
(we point out that conditions under which an isogeny as above can be obtained
were determined, for example, in [4] and later generalized in [11]).
Consider the following statements:
A. Hiis a normal subgroup of Gand G/Hiis abelian.
B. Yiadmits a large abelian group Kiof automorphisms (namely, its order is
strictly greater than 4(gi1)) with only one exception: Ki
=C6and
YiYi/Ki
ramifies over four values; two marked with 2 and two marked with 3.
The arguments employed in the proof of the theorem are naturally generalized
to provide the following criterion. With the same notations:
Criterion. If for each i {1, . . . , s}either Hisatisfies the statement Aor Yi
satisfies statement B, then JX admits complex multiplication.
It is worth to mention that the statement Bcan be restated in a weaker manner.
Indeed, the same conclusion is obtained if we ask Yito be endowed with a quasi-
large abelian group Kiof automorphisms (namely, its order is strictly greater than
2(gi1)) not belonging to one of the 22 exceptional cases listed in [10, Table 2].
4. Remarks
Remark 1. We should mention that the criterion is, as expected, rather restric-
tive. However, it provides a different approach to find new examples of Jacobian
varieties admitting complex multiplication. In addition, it is worth recalling that a
QUASIPLATONIC RIEMANN SURFACES WITH COMPLEX MULTIPLICATION 5
shorter proof of the theorem can be obtained by noticing that Yand Zin the the-
orem are hyperelliptic. Nevertheless, as our proof is based on significantly simpler
arguments which do not depend on the hyperellipticity of the involved quotients,
its generalization could be used for a possibly wider range of cases.
Remark 2. In [13] Streit provided a representation theoretic sufficient condition
for the Jacobian variety of a quasiplatonic Riemann surface to admit complex
multiplication. Concretely, with the previous notations, if S2(ρa) denotes the sym-
metric square representation of the analytic representation ρaof Gand 1 stands
for the trivial representation of Gthen
hS2(ρa),1iG= 0 =JX admits complex multiplication.
After routine computations, one sees that the previous criterion allows to con-
clude that JX admits complex multiplication provided that n2 mod 4.How-
ever, this criterion does not provide conclusion if n0 mod 4.
Acknowledgements. The author is grateful to the referee for suggesting useful
improvements to the article, and to Jennifer Paulhus and Anita M. Rojas for
valuable conversations and for sharing their computer routines with him.
References
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symmetry group Z2
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[10] R. Pignatelli and C. Raso,Riemann surfaces with a quasi large abelian group of auto-
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[14] J. Wolfart,Triangle groups and Jacobians of CM type. Frankfurt a.M., 2000. URL:
http://www.math.uni-frankfurt.de/wolfart/Artikel/jac.pdf
6 SEBASTI ´
AN REYES-CAROCCA
Departamento de Matem´
atica y Estad
´
ıstica, Universidad de La Frontera, Avenida
Francisco Salazar 01145, Temuco, Chile.
E-mail address:sebastian.reyes@ufrontera.cl
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In the last years, two new tools have been developed for an approach to the question whether a given nonsingular projective algebraic curve over a number field has a Jacobian of CM type. First, such curves can be characterized by the existence of Belyi functions or Grothendieck's dessins d'enfants ([Be], [Gr], [VS], [CIW], [JS3]) — for detailed definitions see Section 1. There is some reasonable hope that dessins also encode deeper properties of curves which has already been proved for places of bad reduction ([Fu], [Bec]), so one should try to read information about the endomorphism algebra of the Jacobian also from combinatorial properties of dessins. We will mainly consider regular dessins or equivalently, Riemann surfaces with many automorphisms. The reduction to this particular case is described in Section 2. Second, there has been considerable progress in transcendence. A classical criterion due to Th. Schneider says that an elliptic curve defined over ¯ Q has complex multiplication if and only if the period quotient is algebraic. From Wüstholz' analytic subgroup theorem follows a generalization to abelian varieties and their period quotients in the Siegel upper half space ([Coh], [SW], see Theorem 3) and has applications in particular to curves with many automorphisms (Sections 4 and 7). The key in joining both tools is the use of the canonical representation of the automorphism group G of the curve X on the space of holomorphic differentials. The Jacobian of X is isogenous to a direct product A k 1 1 × . . . × A km m of simple, pairwise non–isogenous abelian varieties A ν , ν = 1, . . . , m , and among others it will be shown that 1. irreducible subspaces U of this representation belong to isotypic components of Jac X , i.e. consist of pullbacks of differentials on the factors A kν ν (they can even built up from End A ν –invariant subspaces of H 0 (Ω, A ν) , see Section 7), 2. dim U = 1 implies that A ν has complex multiplication (Theorem 4, Section 4), 3. large representation degrees dim U indicate high multiplicity k ν of A ν and/or that End A ν has large degree over its center (Theorems 8 and 9). 1 1 TRIANGLE GROUPS AND DESSINS 2 These canonical representations — which can be effectively constructed as subrepresentations of the regular representation of G ([St1], [St2]) — can also be used to determine the period quotient, hence to apply Theorem 3, but this will be an object of further research ([St3]). In the present paper, the simpler criteria for Jac X to be of CM type (Theorem 4, Remark 3) turn out to give only another look to the well–known fact that Fermat curves and their quotients have Jacobians of CM type ([KR], [Ao]). More interesting is the fact that 'many automorphisms' do not imply 'CM', explained in Section 5 using certain subvarieties of the Siegel upper half space. Surprisingly, even Hurwitz curves (for which the maximal number 84(g − 1) of automorphisms is attained) do not always have Jacobians of CM type, as is shown in Section 6.5 for Macbeath's curve in genus 7. Since in every genus there is only a finite number of curves with many auto-morphisms, Section 6 includes a detailed discussion of all such curves for genera < 5 . It turns out that only one (hyperelliptic) curve in genus 3 and two curves in genus 4 have Jacobians not of CM type. The main new results of the present paper are contained in Sections 4, 5 and 7. The other Sections collect material of which a good part is known to experts but often not presented in the literature in a form we need here. I hope that e.g. Theorem 2 about the regularization of dessins or the examples of Section 6 may be useful in other contexts also.