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The number of curves of genus two with elliptic differentials.

Let C be a curve of genus 2 defined over an algebraically closed field K, and suppose that C admits a non-constant morphism f : C → E to an elliptic curve E. If f does not factor over an isogeny of E, then we say that f is an elliptic subcover of C. Note that this last condition imposes no essential restriction since every nonconstant f : C → E factors over a unique elliptic subcover fmin : C → Emin. A classical theorem due to Picard [Pi] and Bolza [Bo] of 1882/86 states that a curve C of genus 2 has either none, two or infinitely many elliptic subcovers. This is in part due to the fact that the elliptic subcovers occur in pairs. More precisely, given an elliptic subcover f : C → E, there is a canonical “complementary” elliptic subcover f ′ : C → E ′ of the same degree N := deg(f) = deg(f ′) which is characterized by the requirement that the induced maps on the associated Jacobian varieties fit into an exact sequence