Suppose $T$ is a free finite rank $\mathbb{Z}\_p$-module with a continuous action of $\operatorname{Gal}(\overline{K} / K)$, where $K$ is a number field. There is a definition of local Tamagawa numbers $\operatorname{Tam}(T / K\_v)$ for each prime $v$ of $K$, going back to Fontaine and Perrin-Riou (or to Bloch and Kato for $K = \mathbb{Q}$). For $v \nmid p$ this is the order of the torsion subgroup of $H^1(I\_v, T)^{D\_v}$, where $D\_v$ and $I\_v$ are the decomposition group and intertia group at $v$; for $v \mid p$ it is something more complicated using the Bloch-Kato exponential map. (I'm led to believe that if $T$ is the Tate module of an abelian variety, this recovers the usual description in terms of Neron models.)


If $K\_{v, n} = K\_v(\mu\_{p^n})$, is it true that the factors $\operatorname{Tam}(T / K\_{v, n})$ are eventually constant for large enough $n$? 


(EDIT: In the light of Rob's comment, maybe I should add the assumption that my Galois representation is crystalline at $p$. I'm chiefly interested in the case of the p-adic representation of a modular forms of level prime to $p$ and non-ordinary at $p$.)



Do Tamagawa numbers of Galois representations stabilise in the cyclotomic tower?
As stated, the answer to your question is certainly no. 


For instance, an elliptic curve $E/\mathbb Q$ with split multiplicative ordinary reduction at $p$ will have unbounded Tamagawa number at $p$ in the cyclotomic extension of $\mathbb Q$. To see this, you can check that the Tamagawa number is the order of $H^{2}({\mathbb Q\_{\infty,p}},(T\_{p}E)\_{p}^{+})$. This is what B.Mazur calls the phenomenon of anomalous primes in his Inventiones 18 paper, which is the algebraic counterpart of the existence of exceptional zeroes for $p$-adic $L$-functions.


However, perhaps you meant for the Tamagawa factors outside $p$ to be eventually constant? In that case, I think the answer is yes.


First, $\operatorname {Gal}(K(\mu\_{p^{\infty}})/K)$ has finite prime-to-$p$ part so there exists a large enough $n$ so that $\operatorname {Gal}(K(\mu\_{p^{\infty}})/K(\mu\_{p^n}))$ is unramified outside $p$ (EDIT: Of course this first step is unnecessary). Let $v\nmid p$ be a place of $K(\mu\_{p^{n}})$ and let $w|v$ be a place of $K(\mu\_{p^{n'}})$ with $n'≥n$. Then $H^{1}(I\_{v},T)\simeq H^{1}(I\_{w},T)$ so the Tamagawa number is constant.


