A note on Jacobians of quasiplatonic Riemann surfaces with complex multiplication
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A note on Jacobians of quasiplatonic Riemann surfaces with complex multiplication Sebastián Reyes-Carocca1 Received: 14 July 2020 / Accepted: 17 October 2020 © Springer Nature B.V. 2020
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 C22 2 Cm 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. Keywords Riemann surfaces · Jacobian varieties · Complex multiplication Mathematics Subject Classification (2010) 11G15 · 11G10 · 14K22 · 14H37
1 Introduction A simple complex polarized abelian variety A of dimension g is said to admit complex multiplication if its rational endomorphism algebra E = End(A) ⊗Z Q is a number field of degree 2g. In this case E is a CM field; namely, a totally imaginary quadratic extension of a totally real field of degree g. If A is not simple then, by Poincaré Reducibility theorem, there exist pairwise non isogenous simple abelian varieties A1 , . . . , As and positive integers n 1 , . . . , n s in such a way that A ∼ An1 1 × · · · × Ans s where ∼ stands for isogeny. By definition, A admits complex multiplication if each simple factor A j does. Let X be a compact Riemann surface (or, equivalently, a complex algebraic curve) and let J X denote its Jacobian variety. Classical examples of compact Riemann surfaces with Jacobian variety admitting complex multiplication are Fermat curves and their quotients.
Partially supported by Fondecyt Grants 11180024, 1190991 and Redes Grant 170071.
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Sebastián Reyes-Carocca [email protected] Departamento de Matemática y Estadística, Universidad de La Frontera, Avenida Francisco Salazar, 01145 Temuco, Chile
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Geometriae Dedicata
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 Ag 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 ≥ 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 g ≤ 7, currently it still remains as an open problem for g ≥ 8. This conjecture is closely related to important open problems of Shimura varieties, special points in the Torelli locus of Ag and the theory of unlikely intersections. If the Jacobian variety of a compact Riemann surface X admits complex multiplication 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 these compact Riemann surfaces, particularly in their applications to number theory and arithmetic geometry. B
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