Explicit construction of decomposable jacobians

In this thesis we give explicit constructions of decomposable hyperelliptic Jacobianvarieties over fields of characteristic 0. These include Jacobians that are isogenous toa product of two absolutely simple varieties, a square of a hyperelliptic Jacobian, anda product of four hyperelliptic Jacobians three of which are of the same dimension.As an application, we produce families of hyperelliptic curves with infinitely manyquadratic twists having at least two rational non-Weierstrass points; and families ofquadruples of hyperelliptic curves together with infinitely many square-free d suchthat the quadratic twists of each of the curves by d possess at least one rationalnon-Weierstrass point.