Elliptic Subcovers of Hyperelliptic Curves

Let C be a curve over an arbitrary fieldK. An elliptic subcover is a finiteK-morphism f : C → E to an elliptic curve E/K such that its base-change fK : CK → EK to the algebraic closure does not factor over a non-trivial isogeny of EK . If f ′ : C → E ′ is another elliptic subcover, then f ′ is said to be equivalent to f if there is an isomorphism φ : E ∼ → E ′ such that f ′ = φ ◦ f . Let E(C) denotes the set of all equivalence classes of elliptic subcovers of C. If C(K) 6= ∅, then it is well-known that there is a natural bijection between the set E(C) and the set S(JC) of elliptic subgroups of the Jacobian JC of C. The purpose of this paper is to show that this result is also true when C(K) = ∅, provided that C is a hyperelliptic curve whose genus is even. Thus, we prove in §4: Theorem 1.1. Let C/K be a hyperelliptic curve of genus gC ≡ 0 (2). Then the rule f 7→ f JE induces a bijection ΦC : E(C) ∼ → S(JC).