Jacobians of curves in abelian surfaces

In this thesis we decompose the Jacobians of certain smooth curves into smaller abelian varieties up to isogeny. The curves lie in the linear systems of (1,d)-polarizations of simple abelian surfaces. This goal is motivated by Poincaré's reducibility theorem which states that any abelian variety is isogenous to a product of simple abelian varieties such that the simple factors are unique up to isogeny. We construct curves inside simple abelian surfaces and determine the isogenous decomposition of their Jacobians into simple factors when the degrees of polarizations are (1,2), (1,3) and (1,4). We establish isogeny relations for Jacobians of similar curves lying in the linear systems of polarizations of higher degrees. Sufficient conditions for these curves to cover elliptic curves and thus have elliptic factors in the decomposition of their Jacobians have also been established.