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Let $C$ be a curve of genus $g\geq 2$ over complex number. Assume that $C$ has complex multiplication (CM).

Does there exist such a curve $C$ such that $C'$ is also of CM type for any unbranched cover $C' \to C$?

PS: when $C$ is of genus one, the answer is yes. Since any unbranched cover of an elliptic curve is still an elliptic curve and is isogenous to the original one. The question now ask for the existence of such a curve of genus $g\geq 2$.

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    $\begingroup$ Could you explain what you mean by "$C$ has complex multiplication"? $\endgroup$
    – abx
    Commented Feb 18, 2014 at 14:03
  • $\begingroup$ @abx: $C$ has complex multiplication means the Jacobian $Jac(C)$ of $C$ has complex multiplication. The Jacobian $Jac(C)$ is an abelian variety. an abelian variety $A$ of dimension $g$ is said to have complex multiplication if the Endmorphism algebra $End_{\mathbb Q}(A)$ contains a field $K$ such that $[K:Q]=2g$ (See Mumford's <abelian variety>) $\endgroup$
    – Pyramid
    Commented Feb 18, 2014 at 14:28
  • $\begingroup$ Presumably you know concretely at least one CM-type complex curve. Could you please explain how you know this one particular curve is not an example? $\endgroup$
    – JHM
    Commented Feb 18, 2014 at 15:05
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    $\begingroup$ Well with your definition of CM in the comment there is not going to exists such a $C$ because $Jac(C')$ is not going to be simple and hence $End(Jac(C')) \otimes \mathbb Q$ is not going to contain a field $K$ of degree $2g(C')$ over $\mathbb Q$. You should take as a definition of CM that $End(Jac(C')) \otimes \mathbb Q$ contains a commutative algebra of rank $2g(C')$ to make this question interesting. $\endgroup$
    – Maarten Derickx
    Commented Feb 18, 2014 at 16:39
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    $\begingroup$ P.s. do you mean: "$C'$ is also of CM type for an unbranched cover $C' \to C$." Or do you mean: "$C'$ is of CM type for all unbranched covers $C'\to C$." $\endgroup$
    – Maarten Derickx
    Commented Feb 18, 2014 at 16:44

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