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Genus Theory is a classical feature of integral binary quadratic forms. Usingthe author's generalization of the well-known correspondence between quadraticform classes and ideal classes of quadratic algebras, we extend it to the casewhen 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 formversion of the $2$-descent map on a certain hyperelliptic curve. As anapplication, we make a contribution to a question of Agboola and Pappasregarding a specialization problem of divisor classes on hyperelliptic curves.Under suitable assumptions, we prove that the set of non-trivialspecializations has density $1$.

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