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Consider a finite field $\mathbb{F}_p$ (where $p \equiv 1 \ (\mathrm{mod} \ 3)$, $p \equiv 3 \ (\mathrm{mod} \ 4)$), $\mathbb{F}_{p^2}$-isomorphic elliptic curves (of $j$-invariant $0$) $$ E\!:y_1^2 = x_1^3 + b, \qquad E^{(1)}\!: y_2^2 + x_2^3 + b = 0, \qquad \mathrm{where \qquad } b \in \mathbb{F}_p^* \setminus (\mathbb{F}_p^*)^3. $$

Is there a non-singular irreducible genus $2$ curve $C/\mathbb{F}_p$ and two $\mathbb{F}_p$-coverings $C \to E$, $C \to E^{(1)}$ of some degree $n$?

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  • $\begingroup$ The general question of constructing a curve of genus two mapping to two elliptic curves is solved in Frey, Gerhard; Kani, Ernst Curves of genus 2 covering elliptic curves and an arithmetical application. Arithmetic algebraic geometry (Texel, 1989), 153–176, Progr. Math., 89, Birkhäuser Boston, Boston, MA, 1991. $\endgroup$
    – Felipe Voloch
    Commented Aug 1, 2019 at 8:28
  • $\begingroup$ @FelipeVoloch, I read this article. If I am right, their construction may give another direct product of elliptic curves, not necessary an irreducible curve of genus 2. $\endgroup$
    – Dimitri Koshelev
    Commented Aug 1, 2019 at 15:01

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