Dear Rob, If a_p is congruent to 1, then one of the eigenvalues is actually equal to 1, isn't it? So E is Steinberg at p and there is Tamagawa number explosion.

Sure. For obscure reasons, I wrote the answer for a general Zp[G]-extension, where G is a finite group. Sorry about that.

After your edit, I think you are on better ground, but I am still unsure. By Sen and the crystalline property, you have eliminated one of the potential source of Tamagawa number explosion (i.e Frobenius of weight 0), but I worry about the possibility that your representation might have weights -2 (in which case there might be Tamagawa explosion in the dual, and this should cause Tamagawa explosion in the original representation). Anyways, I would tend to think it much easier to prove the result in the specific context you have in mind than trying to formulate a general theorem.

"What is known?" A lot; "and not known?" Also a lot.

@Mephisto I have explained myself clumsily. What I meant more precisely was something along these lines. Say you are in the late 80s/early 90s and you are listing the tools you need for the conjectures you have in mind: duality for cohomology over l.c.n rings, some results on perfect complexes, comparison theorems, base-change for some cohomology complexes, excision... None of these results had clear, complete references at the time but there were no point restricting yourself to Galois cohomology (SGA XVII, Mazur 1973 and being Bloch-Kato would in fact rather suggest otherwise).

Or think of the excision map in Galois cohomology with restricted ramification.

@Mephisto Central to the approach of Kato is the fact that the étale cohomology of a perfect complex of étale sheaves is a perfect complex. In the early 90s, I wouldn't know what source to quote for the corresponding statement in Galois cohomology. But even something as "basic" as Poitou-Tate had no clear reference before the first edition of Milne's ADT, and that's only from 1986.

For a layman (nay, this is already too generous, an ignoramus) as myself, this is a quite impressive answer.

If you read the proofs of the construction of the $p$-adic Langlands correspondence, you will see that the restriction $F=\mathbb Q_{p}$ is ubiquitous, starting from (but not restricted to) the fact that you want the residual field to be cyclic. @Chandan Pierre Berger is an entrepreneur and benefactor unlikely ever to contribute to Bourbaki, it is of course Laurent you want.

Not three weeks ago, I heard Fontaine said that he didn't know how to formulate the local Fontaine-Mazur conjecture (that is , he didn't know how to characterize local representations coming from varieties over local fields).