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

Abstract

Let C C be a hyperelliptic curve defined over Q \mathbb {Q} , whose Weierstrass points are defined over extensions of Q \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 0 on C 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.