ABSTRACT. Given two elliptic curves E1 and E2 defined over the field of rational numbers, Q, with good reduction at an odd prime p and equivalent mod p Galois representation, we compare the p-Selmer rank, global and local root numbers of E1 and E2 over number fields.
Let p be an odd prime and f be a nearly ordinary Hilbert modular Hecke eigenform defined over a totally real field F . Let I be an irreducible component of the universal nearly ordinary or locally cyclotomic deformation of the representation of Gal F that is associated to f . We study the deformation rings over a p-adic Lie extension F ∞ that contains the cyclotomic Z p -extension of F . More precisely, we prove a control theorem about these rings. We introduce a category M I H (G), where G = Gal(F ∞ /F ) and H = Gal(F ∞ /F cyc ), which is the category of modules which are torsion with respect to a certain Ore set, which generalizes the Ore set introduced by Venjakob. For Selmer groups which are in this category, we formulate a Main conjecture in the spirit of Noncommutative Iwasawa theory. We then set up a strategy to prove the conjecture by generalizing work of Burns, Kato, Kakde, and Ritter and Weiss. This requires appropriate generalizations of results of Oliver and Taylor, and Oliver on Logarithms of certain K-groups, which we have presented here.
The classification of local Galois representations using (ϕ, Γ)-modules by Fontaine has been generalized by Kisin and Ren [8] over Lubin-Tate extensions of local fields using the theory of (ϕq, ΓLT )-modules. We extend the work of Herr [6] by introducing a complex which allows us to compute cohomology over Lubin-Tate extensions and compare it with Galois cohomology groups. That complex is further extended to include certain non abelian extensions. We also generalize the notion of (ϕq, ΓLT )-modules over coefficient rings and build up to show the equivalence with Galois representations over R. This allows us to generalize our results to the case of coefficient rings.
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