2022/10/24 by William Dallaporta, Dallaporta, William
Mathematics · #11E16 #11R45 (Secondary) #14H25 #14H40 (Primary) #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2210.13045
openalex publication_date 2022/10/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 = \mathbbK[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.