Content uploaded by Sebastián Reyes-Carocca
Author content
All content in this area was uploaded by Sebastián Reyes-Carocca on Nov 09, 2020
Content may be subject to copyright.
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
A∼An1
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:X→P1
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, M¨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:X→P1∼
=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, tat−1=a, tbt−1=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 m≡2 mod 4.Following [3, Theorem 1(1)], the regular cov-
ering map
X→X/Gm∼
=P1
ramifies over three values marked with 2, m and 2m, and the genus of Xis
gX=m−2.
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=xm−1.
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:Y→Y/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 m≡0 mod 4. Following [3, Theorem 1(2b)], the regular
covering map
X→X/Gm∼
=P1
ramifies over three values marked with 2, m and mand the genus of Xis
gX=m−3.
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=xm−1 and y2=xm
2−1
respectively; their genera are gY=m
2−1 and gZ=m
4−1.
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(gZ−1) 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:Z→Z/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
JXn∼JY 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(gi−1)) with only one exception: Ki∼
=C6and
Yi→Yi/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(gi−1)) 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 n≡2 mod 4.How-
ever, this criterion does not provide conclusion if n≡0 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
[1] G. V. Bely
˘
ı,On Galois extensions of a maximal cyclotomic field Math. USSR Izv. 14, 247–
256 (1980).
[2] A. Carocca, H. Lange and R. E. Rodr
´
ıguez, Jacobians with complex multiplication,
Trans. Amer. Math. Soc. 363 (2011), no. 12, 6159–6175.
[3] R. A. Hidalgo, L. Jim´
enez, S. Quispe and S. Reyes-Carocca, Quasiplatonic curves with
symmetry group Z2
2o Zmare definable over Q,Bull. London Math. Soc. 49 (2017) 165–183.
[4] E. Kani and M. Rosen, Idempotent relations and factors of Jacobians, Math. Ann. 284
(1989), 307–327.
[5] C. Lomuto, Riemann surfaces with a large abelian group of automorphisms, Collect. Math.
57 (2006), no. 3, 309–318.
[6] N. M¨
uller and R. Pink, Hyperelliptic curves with many automorphisms, Preprint arXiv:
1711.06599
[7] A. Obus and T. Shaska, Superelliptic curves with many automorphisms and CM Jacobians,
Preprint arXiv:2006.12685
[8] F. Oort, CM Jacobians. In: Conference Galois covers and deformations Bordeaux, June
25–29, 2012, http://www.staff.science.uu.nl/∼oort0109/Bord2-VI-12.pdf
[9] F. Oort Moduli of abelian varieties in mixed and in positive characteristic. In: Handbook of
moduli. Vol. III, Adv. Lect. Math. (ALM) 26 (2013) 75–134.
[10] R. Pignatelli and C. Raso,Riemann surfaces with a quasi large abelian group of auto-
morphisms, Matematiche (Catania) 66 (2011), no. 2, 77–90.
[11] S. Reyes-Carocca and R. E. Rodr
´
ıguez, A generalisation of Kani-Rosen decomposition
theorem for Jacobian varieties, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 19 (2019), no. 2,
705–722.
[12] G. Shimura and Y. Taniyama,Complex multiplication of abelian varieties and its applica-
tions to number theory, Publ. Math. Soc. Japan 6, 1961.
[13] M. Streit, Period matrices and representation theory, Abh. Math. Sem. Univ. Hamburg
71 (2001), 279–290.
[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
A preview of this full-text is provided by Springer Nature.
Content available from Geometriae Dedicata
This content is subject to copyright. Terms and conditions apply.