A note on Jacobians of quasiplatonic Riemann surfaces with complex multiplication
aa r X i v : . [ m a t h . AG ] O c t A NOTE ON JACOBIANS OF QUASIPLATONIC RIEMANNSURFACES WITH COMPLEX MULTIPLICATION
SEBASTI ´AN REYES-CAROCCA
Abstract.
Let m > C ⋊ C m admits complex multiplication. Wethen extend this result to provide a criterion under which the Jacobian varietyof a quasiplatonic Riemann surface admits complex multiplication. Introduction
A simple complex polarized abelian variety A of dimension g is said to admitcomplex multiplication if its rational endomorphism algebra E = End( A ) ⊗ Z Q is a number field of degree 2 g . In this case E is a CM field; namely, a totallyimaginary quadratic extension of a totally real field of degree g . If A is not simplethen, by Poincar´e Reducibility theorem, there exist pairwise non isogenous simpleabelian varieties A , . . . , A s and positive integers n , . . . , n s in such a way that A ∼ A n × · · · × A n s s where ∼ stands for isogeny. By definition, A admits complex multiplication if eachsimple factor A j does.Let X be a compact Riemann surface (or, equivalently, a complex algebraiccurve) and let J X denote its Jacobian variety. Classical examples of compact Rie-mann surfaces with Jacobian variety admitting complex multiplication are Fermatcurves and their quotients. However, in general, it is a difficult task to decidewhether or not the Jacobian variety of a given compact Riemann surface admitscomplex multiplication, and much less is known about their distribution in themoduli space A g of principally polarized abelian varieties; see, for example, [8].As a matter of fact, a well-known conjecture due to Coleman predicts that if g > g with Jacobian variety admitting complex multiplication is finite. In spite of thefact that this conjecture has been proved to be false for g , currently it stillremains as an open problem for g > . This conjecture is closely related to impor-tant open problems of Shimura varieties, special points in the Torelli locus of A g and the theory of unlikely intersections. Mathematics Subject Classification.
Key words and phrases.
Riemann surfaces, Jacobian varieties, complex multiplication.Partially supported by Fondecyt Grants 11180024, 1190991 and Redes Grant 170071.
If the Jacobian variety of a compact Riemann surface X admits complex multi-plication then X can 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 thesecompact Riemann surfaces, particularly in their applications to number theory andarithmetic geometry.By the classical Belyi’s theorem [1], a compact Riemann surface X can be definedover a number field if and only if there exists a holomorphic map β : X → P with at most three critical values; the pair ( X, β ) is called a Belyi pair. Possiblythe more interesting examples of Belyi pairs are the regular ones: namely, those forwhich β is given by the action of a group of automorphisms of X . In this case, X isknown to be quasiplatonic (or to have many automorphisms); that is, it cannot bedeformed non-trivially in the moduli space together with its automorphism group.Oort in [9, p.18] considered quasiplatonic Riemann surfaces and discussed theproblem of determining which among them have Jacobian variety admitting com-plex multiplication. For genus at most four, this problem was completely solvedby 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 indetermining which among them have Jacobian variety admitting complex multi-plication.A different approach can be done by considering regular Belyi pairs β : X → P ∼ = X/H whose covering groups H share a common property or have the same algebraicstructure. In this direction, it was proved in [14] that if H is abelian then J X admits complex multiplication, and in [2] the same conclusion was obtained for twoinfinite series of compact Riemann surfaces arising as quotients of regular Belyicurves with a metacyclic group of automorphisms.In this short note we consider a infinite series of quasiplatonic Riemann surfaceswith associated covering group isomorphic to the semidirect product G m := h a, b, t : a = b = ( ab ) = t m = 1 , tat − = a, tbt − = ab i ∼ = C ⋊ C m where m > C ⋊ C m , we prove the following result. Theorem.
Let m > X, β ) is a regular Belyi pair withassociated covering group isomorphic to G m then the Jacobian variety J X admitscomplex multiplication.The proof of the theorem –which is rather simple and based on the classicaltheory of covering of Riemann surfaces– is done in Section §
2. Then, in Section § UASIPLATONIC RIEMANN SURFACES WITH COMPLEX MULTIPLICATION 3 multiplication. Finally, we end this short note by recalling a couple of observationsin Section §
4. 2.
Proof of the theorem
Let (
X, β ) be a regular Belyi pair with associated covering group isomorphic to G m where m > Case A.
Assume m ≡ . Following [3, Theorem 1(1)], the regular cov-ering map X → X/G m ∼ = P ramifies over three values marked with 2 , m and 2 m, and the genus of X is g X = m − . Moreover, its Jacobian variety decomposes isogenously as the product
J X ∼ J Y (2.1)where Y = X/ h a i is the quotient Riemann surface represented by the affine alge-braic curve y = x m − . Note that h a i is a normal subgroup of G m and the quotient H = G m / h a i is anabelian group of order 2 m acting as a group of automorphisms of Y. Clearly, thecorresponding orbit space
Y /H has genus zero and the associated regular coveringmap β H : Y → Y /H ramifies over at most three values. It follows that (
Y, β H ) is a regular Belyi pairwith abelian covering group; thus, by [14, Theorem 4], we obtain that J Y admitscomplex multiplication. The result follows from the isogeny (2.1).
Case B.
Assume m ≡ X → X/G m ∼ = P ramifies over three values marked with 2 , m and m and the genus of X is g X = m − . Moreover, as observed in [3, Subsection § J X ∼ J Y × J Z (2.2)where Y = X/ h a i and Z = X/ h b i are the quotient Riemann surfaces representedby the affine algebraic curves y = x m − y = x m − g Y = m − g Z = m − . We argue analogously as done in the case A to ensure that J Y admits complexmultiplication. Besides, in order to prove that
J Z also does, define ι ( x, y ) = ( x, − y ) and τ ( x, y ) = (exp( πim ) x, y ) SEBASTI ´AN REYES-CAROCCA and let K = h ι, τ i ∼ = C × C m/ . We observe that the abelian group K satisfies | K | > g Z −
1) for all m > . Then, by the classification of large abelian groups of automorphisms of compactRiemann surfaces given in [5, Theorem 3.1], we see that the branched regularcovering map β K : Z → Z/K ∼ = P ramifies over three values, marked with 2, m and m . Thus, again by [14, Theorem4], we conclude that J Z admits complex multiplication and the result follows fromthe isogeny (2.2). 3.
A generalization
Let (
X, β ) be a regular Belyi pair and let G denote the associated coveringgroup. Consider a collection { H , . . . , H s } of proper non-trivial subgroups of G. Let Y i denote the quotient Riemann surface X/H i and let g i = 0 denote its genus, for each i ∈ { , . . . , s } . Assume the existence of positive integers n, n , . . . , n s in such a way that J X n ∼ J Y n × · · · × J Y n s s (we point out that conditions under which an isogeny as above can be obtainedwere determined, for example, in [4] and later generalized in [11]).Consider the following statements: A. H i is a normal subgroup of G and G/H i is abelian. B. Y i admits a large abelian group K i of automorphisms (namely, its order isstrictly greater than 4( g i − K i ∼ = C and Y i → Y i /K i ramifies over four values; two marked with 2 and two marked with 3.The arguments employed in the proof of the theorem are naturally generalizedto provide the following criterion. With the same notations: Criterion.
If for each i ∈ { , . . . , s } either H i satisfies the statement A or Y i satisfies statement B , then J X admits complex multiplication.It is worth to mention that the statement B can be restated in a weaker manner.Indeed, the same conclusion is obtained if we ask Y i to be endowed with a quasi-large abelian group K i of automorphisms (namely, its order is strictly greater than2( g i − 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 Jacobianvarieties admitting complex multiplication. In addition, it is worth recalling that a
UASIPLATONIC RIEMANN SURFACES WITH COMPLEX MULTIPLICATION 5 shorter proof of the theorem can be obtained by noticing that Y and Z in the the-orem are hyperelliptic. Nevertheless, as our proof is based on significantly simplerarguments 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 conditionfor the Jacobian variety of a quasiplatonic Riemann surface to admit complexmultiplication. Concretely, with the previous notations, if S ( ρ a ) denotes the sym-metric square representation of the analytic representation ρ a of G and 1 standsfor the trivial representation of G then h S ( ρ a ) , i G = 0 = ⇒ J X admits complex multiplication.After routine computations, one sees that the previous criterion allows to con-clude that
J X admits complex multiplication provided that n ≡ . How-ever, this criterion does not provide conclusion if n ≡ . Acknowledgements.
The author is grateful to the referee for suggesting usefulimprovements to the article, and to Jennifer Paulhus and Anita M. Rojas forvaluable 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. , 247–256 (1980).[2] A. Carocca, H. Lange and R. E. Rodr´ıguez,
Jacobians with complex multiplication,
Trans. Amer. Math. Soc. (2011), no. 12, 6159–6175.[3]
R. A. Hidalgo, L. Jim´enez, S. Quispe and S. Reyes-Carocca,
Quasiplatonic curves withsymmetry group Z ⋊ Z m are definable over Q , Bull. London Math. Soc. (2017) 165–183.[4] E. Kani and M. Rosen,
Idempotent relations and factors of Jacobians , Math. Ann. (1989), 307–327.[5]
C. Lomuto,
Riemann surfaces with a large abelian group of automorphisms,
Collect. Math. (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. ∼ oort0109/Bord2-VI-12.pdf[9] F. Oort
Moduli of abelian varieties in mixed and in positive characteristic.
In: Handbook ofmoduli. Vol. III, Adv. Lect. Math. (ALM) (2013) 75–134.[10] R. Pignatelli and C. Raso , Riemann surfaces with a quasi large abelian group of auto-morphisms,
Matematiche (Catania) (2011), no. 2, 77–90.[11] S. Reyes-Carocca and R. E. Rodr´ıguez,
A generalisation of Kani-Rosen decompositiontheorem for Jacobian varieties , Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) (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 , 1961.[13] M. Streit,
Period matrices and representation theory , Abh. Math. Sem. Univ. Hamburg (2001), 279–290.[14] J. Wolfart , Triangle groups and Jacobians of CM type ∼ wolfart/Artikel/jac.pdf SEBASTI ´AN REYES-CAROCCA
Departamento de Matem´atica y Estad´ıstica, Universidad de La Frontera, AvenidaFrancisco Salazar 01145, Temuco, Chile.
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