From Picard groups of hyperelliptic curves to class groups of quadratic fields

Let $C$ be a hyperelliptic curve defined over $\mathbb{Q}$, whose Weierstrass points are defined over extensions of $\mathbb{Q}$ of degree at most three, and at least one of them is rational. Generalizing a result of R. Soleng (in the case of elliptic curves), we prove that any line bundle of degree $0$ on $C$ which is not torsion can be specialised into ideal classes of imaginary quadratic fields whose order can be made arbitrarily large. This gives a positive answer, for such curves, to a question by Agboola and Pappas.
Comment: 28 pages, LaTeX. Minor improvements following the referee's suggestions. New proof of Lemma 3.10. To appear in Trans. Amer. Math. Soc