Jacobians of plane quintic curves of genus one

Abstract

Let K be a number field. By representing genus one curves as plane quintic curves with 5 double points, we construct (up to birational equivalence) the universal elliptic curves defined over the modular curves X₁(5) and X(μ)(5) (X(μ)(5) is the modular curve parameterizing pairs (E, i : (μ)₅ → E) where E is an elliptic curve over Q). We then twist the latter by elements coming from H¹(Gal(K̅/K), (μ)₅) to construct universal families of principal homogeneous spaces for the curves E. Finally we show that every principal homogeneous space arising this way is visible in some abelian variety.