Quadratic forms and Genus Theory : a link with 2-descent and an application to non-trivial specializations of ideal classes

Genus Theory is a classical feature of integral binary quadratic forms. Using the author’s generalization of the well-known correspondence between quadratic form classes and ideal classes of quadratic algebras, we extend it to the case when quadratic forms are twisted and have coefficients in any PID $R$. When $R=\mathbb{K}[X]$, we show that the Genus Theory map is the quadratic form version of the $2$-descent map on a certain hyperelliptic curve. As an application, we make a contribution to a question of Agboola and Pappas regarding a specialization problem of divisor classes on hyperelliptic curves. Under suitable assumptions, we prove that the set of non-trivial specializations has density $1$.