, Volume 120, Issue 1, pp 361-427

Adjoint selmer groups as Iwasawa modules

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We fix a primep. In this paper, starting from a given Galois representation ϕ having values inp-adic points of a classical groupG, we study the adjoint action of ϕ on thep-adic Lie algebra of the derived group ofG. We call this new Galois representation the adjoint representation Ad(ϕ) of ϕ. Under a suitablep-ordinarity condition (and ramification conditions outsidep), we define, following Greenberg, the Selmer group Sel(Ad(ϕ))/L for each number fieldL. We scrutinize the behavior of Sel(Ad(ϕ))/E as an Iwasawa module for a fixed ℤ p -extensionE /E of a number fieldE and deduce an exact control theorem. A key ingredient of the proof is the isomorphism between the Pontryagin dual of the Selmer group and the module of Kähler differentials of the universal nearly ordinary deformation ring of ϕ. WhenG=GL(2), ϕ is a modular Galois representation and the base fieldE is totally real, from a recent result of Fujiwara identifying the deformation ring with an appropriatep-adic Hecke algebra, we conclude some fine results on the structure of the Selmer groups, including torsion-property and an exact limit formula ats=0 of the characteristic power series, after removing the trivial zero.

To the memory of Professor Kenkichi Iwasawa
Research of the author is partially supported by the NSF grant: DMS-9701017.