Selmer groups and the Fontaine -Mazur conjecture

A classical theorem of Mazur proves a control theorem for the p-primary Selmer group of an abelian variety with respect to a Zp -extension of a number field. This theorem has been generalized in various ways by Greenberg, where he considers certain p-adic Lie extensions of a number field. In this dissertation, a control theorem is proven for the p-primary Selmer group of an abelian variety with respect to extensions of the form: maximal pro-p extension of a number field unramified outside a finite set of primes R which does not include any primes dividing p in which another finite set of primes S split completely. In a case related to the Fontaine-Mazur conjecture, the control theorem gives information about p-ranks of Selmer and Tate-Shafarevich groups of the abelian variety.;This dissertation also discusses what can be said in regards to a control theorem when the set R contains all the primes of the number field dividing p. In this case, it is shown that a control theorem can fail in the sense that the maps between the Selmer groups can have infinite kernels and can have infinite cokernels of unbounded Zp -corank. The latter case gives information about the structure of the Selmer group at the top of the extension.