Realizing Galois representations in abelian varieties by specialization

This thesis comprises two distinct, albeit closely related, parts. In the first part, we improve on a construction of Mestre--Shioda to produce infinite families of hyperelliptic curves having a record number of rational points and Mordell--Weil rank. In the second part, we give a unified treatment of work of Mestre--Shioda and Liu--Lorenzini, and by generalizing a Specialization Theorem of Néron, we construct families of abelian varieties which realize specified Galois representations.