2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/129838In this paper we present a new method to show that a principal homogeneous space of the Jacobian of a curve of genus two is nontrivial. The idea is to exhibit a Brauer-Manin obstruction to the existence of rational points on a quotient of this principal homogeneous space. In an explicit example we apply the method to show that a specific curve has infinitely many quadratic twists whose Jacobians have nontrivial Tate-Shafarevich group.37 pagesAlgebraic GeometryNumber Theory14H40; 11G10; 14J27-28text