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

TL;DR

This paper establishes a connection between the Picard groups of certain hyperelliptic curves over rationals and the class groups of imaginary quadratic fields, showing that non-torsion line bundles can be specialized to ideal classes of arbitrarily large order.

Contribution

It generalizes a result from elliptic to hyperelliptic curves, linking their Picard groups to class groups of quadratic fields and answering a specific open question.

Findings

Non-torsion degree 0 line bundles can be specialized to ideal classes of quadratic fields.

The order of these ideal classes can be made arbitrarily large.

The result applies to hyperelliptic curves with specific properties of Weierstrass points.

Abstract

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.