Adic Lie Extensions Research Articles

On a noncommutative Iwasawa main conjecture for varieties over finite fields

We formulate and prove an analogue of the noncommutative Iwasawa main conjecture for \ell -adic Lie extensions of a separated scheme X of finite type over a finite field of characteristic prime to \ell .

Nekovár duality over p-adic Lie extensions of global fields

Tate duality is a Pontryagin duality between the i th Galois cohomology group of the absolute Galois group of a local field with coefficents in a finite module and the (2-i) th cohomology group of the Tate twist of the Pontryagin dual of the module. Poitou-Tate duality has a similar formulation, but the duality now takes place between Galois cohomology groups of a global field with restricted ramification and compactly-supported cohomology groups. Nekovár proved analogues of these in which the module in question is a finitely generated module T over a complete commutative local Noetherian ring R with a commuting Galois action, or a bounded complex thereof, and the Pontryagin dual is replaced with the Grothendieck dual T^* , which is a bounded complex of the same form. The cochain complexes computing the Galois cohomology groups of T and T^*(1) are then Grothendieck dual to each other in the derived category of finitely generated R -modules. Given a p -adic Lie extension of the ground field, we extend these to dualities between Galois cochain complexes of induced modules of T and T^*(1) in the derived category of finitely generated modules over the possibly noncommutative Iwasawa algebra with R -coefficients.

Fine Selmer group of Hida deformations over non-commutative $p$-adic Lie extensions

We study the Selmer group and the fine Selmer group of $p$-adic Galois representations defined over a non-commutative $p$-adic Lie extension and their Hida deformations. For the fine Selmer group, we generalize the pseudonullity conjecture of J. Coates and R. Sujatha, Fine Selmer group of elliptic curves over $p$-adic Lie extensions, in this context and discuss its invariance in a branch of a Hida family. We relate the structure of the ‘big’ Selmer (resp. fine Selmer) group with the specialized individual Selmer (resp. fine Selmer) groups.

On the structure of Selmer groups of lambda-adic deformations over $p$-adic Lie extensions

In this paper, we consider the \Lambda -adic deformations of Galois representations associated to elliptic curves. We prove that the Pontryagin dual of the Selmer group of a \Lambda -adic deformation over certain p -adic Lie extensions of a number field, that are not necessarily commutative, has no non-zero pseudo-null submodule. We also study the structure of various arithmetic Iwasawa modules associated to such deformations.