Quadratic forms and Genus Theory

William Dallaporta (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

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 = 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.

Mots clés: binary quadratic form; Picard group; Genus Theory; 2-descent on hyperelliptic curves; density on S-integers

Domaines: Mathématiques [math]

William Dallaporta. Quadratic forms and Genus Theory. 2022. ⟨hal-03889227v1⟩