This is a followup to [this question](https://mathoverflow.net/questions/65451/do-tamagawa-numbers-of-galois-representations-stabilise-in-the-cyclotomic-tower).


Let $p \ge 3$ be prime, and let $V$ be a crystalline 2-dimensional representation of $G\_{\mathbb{Q}\_p}$ and $T$ a lattice in $V$. I'm going to assume just about every niceness condition on $V$ that I can think of:


* $V$ is irreducible;
* $\operatorname{Fil}^0 \mathbb{D}\_{\mathrm{cris}}(V)$ is 1-dimensional (so one Hodge-Tate weight of $V$ is $\le 0$ and the other is $> 0$)
* none of the eigenvalues of Frobenius on $\mathbb{D}\_\mathrm{cris}(V)$ are integral powers of $p$;
* the Hodge filtration of $V$ has length $< (p-1)$, so $T$ corresponds to a strongly divisible $\mathbb{Z}\_p$-lattice $\mathbb{D}(T)$ in $\mathbb{D}\_{\mathrm{cris}}(V)$ via Fontaine-Laffaille theory.


Let $T$ be a lattice in $V$, and let $\omega$ be a $\mathbb{Z}\_p$-basis of the "tangent space" $t\_T = \mathbb{D}(T) / \operatorname{Fil}^0 \mathbb{D}(T)$. The Tamagawa number of $T$ over $K\_n = \mathbb{Q}\_p(\mu\_{p^n})$ is given by
$$ \operatorname{Tam}^0\_{K\_n, \omega}(T) = \frac{[H^1\_f(K\_n, T) : \exp(\mathcal{O}\_{K\_n} \omega)]}{[\mathbb{D}(T) : (1- \varphi) \mathbb{D}(T)]}$$


where $[ A : B ]$ is a generalised index (so $[ \mathbb{Z}\_p : \tfrac{1}{p} \mathbb{Z}\_p] = \tfrac{1}{p}$ etc).



> 
> Question: Is it true that under the above hypotheses this Tamagawa number is always 1?
> 
> 
> 


I know this is true for all $n$ if $V$ corresponds to an elliptic curve (because the Tamagawa number has an alternative definition in terms of the index of the nonsingular points in the special fibre of the Neron model) and, if I've understood correctly, for $n = 0$ it is true for any $V$ satisfying the hypotheses above (by a theorem of Bloch and Kato).



Tamagawa numbers of crystalline Galois representations 
I think that for $K\_n=\mathbf{Q}\_p$ what you're looking for is in my (unpublished) paper


<http://perso.ens-lyon.fr/laurent.berger/autrestextes/tamag0919.pdf>


see proposition II.2 for instance.


This paper is unpublished because it was rewritten and massively expanded with/by Denis Benois. There is some stuff in the paper with Benois about going up the cyclotomic tower which may help. I'm sorry I don't have time to look in Benois-Berger to see if there's an answer to your full question.


