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

TL;DR

This paper extends classical genus theory to twisted quadratic forms over PIDs, linking it with 2-descent on hyperelliptic curves and applying it to divisor class specialization problems.

Contribution

It generalizes genus theory to twisted forms over PIDs and connects it with 2-descent on hyperelliptic curves, providing new insights into divisor class specializations.

Findings

Genus theory map corresponds to 2-descent map on hyperelliptic curves.

Proves the density of non-trivial specializations is 1 under certain conditions.

Extends classical quadratic form and ideal class correspondence to broader algebraic settings.

Abstract

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 $1$.