Results on Selmer Groups Over the Pgl(2) Extension

In this paper we study Iwasawa theory for the Selmer group of an elliptic curve over a certain noncommutative p-adic Lie extension whose Galois group is isomorphic to an open subgroup of PGL 2 (Z p ). Unlike the classes of extension usually considered in noncommutative Iwasawa theory, this extension does not contain the cyclotomic Z p -extension.We give a formula for its Euler characteristic in this case.We also show that the dual Selmer group contains no nonzero pseudonull submodules, a result analogous to a theorem of Ochi and Venjakob for GL 2 -extensions.Finally, we provide an explicit formula computing the change in Iwasawa µ-invariants of dual Selmer groups when two elliptic curves are isogenous; this in the same spirit as that of Howson's result for the GL 2 -extension which extends results in the commutative case by Schneider and Perrin-Riou.