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\((d, d')\)-elliptic curves of genus two. (English) Zbl 1411.14036

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Let \(C\) be a stable curve of genus 2 which fits into a diagram \[ \begin{tikzcd} & C \ar[dl, "f" ', "{d:1}"] \ar[rd, "g", "{d':1}" '] \\ E & & D \end{tikzcd}, \] where \(E\) and \(D\) are elliptic curves and \(f\) of degree \(d\) (resp. \(g\) of degree \(d'\)) is a \(d\)-elliptic (resp. \(d'\)-elliptic) map. If there is no isomorphism \(\psi:E\rightarrow D\) such that \(g=\psi\circ f\), then the above diagram \((C,f,g)\) is called a \((d,d')\)-configuration and \(C\) is a \((d,d')\)-elliptic curve of genus 2.
Building on the description and the construction of \(d\)-elliptic curves (i.e. requiring only one map \(f:C\rightarrow E\) of degree \(d\)) given in [G. Frey and E. Kani, Prog. Math. 89, 153–176 (1991; Zbl 0757.14015)] and [G. Frey and E. Kani, Contemp. Math. 487, 33–81 (2009; Zbl 1198.14024)] and using the compactness of the Jacobian \(J(C)\), the authors provide various explicit constructions for \((d,d')\)-elliptic stable curves of genus 2, and exibit their main arithmetic properties. Such constructions, which also depend on another index, the twisting number \(m:=\deg\{Ker(f_*)\times Ker(g_*) \rightarrow J(C)\}\) (where \(f_*\), resp. \(g_*\), is the natural norm map \(J(C)\rightarrow E\), resp. \(J(C)\rightarrow D\)), yield the existence of \((d,d')\)-elliptic curves in various cases, most notably whenever \(d\) is a prime, and, in the particular case of \(m=1\), the existence of smooth \((d,d')\)-elliptic curves for all \(d,d'>1\).
In the final section, for the case \(d=2\), \(d'=3\), the authors provide a complete description of the 10 isomorphism classes of the configurations \((C,f,g)\) with reducible \(C\), by using elliptic curves \(E\) with an endomorphism \(\varphi\) of degree 2 and building the diagram \[ \begin{tikzcd} & E\cup_O E \ar[dl, "{\mathrm{id}\cup \mathrm{id}}" ', "{2:1}"] \ar[rd, "{\mathrm{id}\cup\varphi}", "{3:1}" '] \\ E & & E \end{tikzcd} \] (where \(C=E\cup_O E\) represents the two copies of \(E\) meeting transversally at the origin \(O\)).

MSC:

14H52 Elliptic curves
14H45 Special algebraic curves and curves of low genus
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