Let $K$ be a finite extension of $\mathbb{Q}\_p$ with absolute Galois group $G\_K$. Let $A$ be an abelian variety defined over $K$. The (geometric) Tamagawa number is defined as the order of the quotient $$c(A/K)=A(K)/A\_0(K)$$ where $A\_0(K)$ denotes the $K$-points of $A$ that are sent to the identity component of the special fiber of the Néron model of $A$ via the natural reduction map. 


On the other hand, following Fontaine and Perrin-Riou (in "Cohomologie galoisienne et valeurs de fonctions L", §4), starting with the $p$-adic Tate module $T\_p=T\_p(A)$, one may define the ($p$-adic) Tamagawa number ${\rm Tam}\_\omega(T\_p)$. The latter depends on a choice of a generator $\omega\in\det(t\_{V\_p})^\*$, where $V\_p=T\_p\otimes\mathbb Q\_p$ and $t\_{V\_p}=(B\_{\rm dR}/B\_{\rm dR}^+\otimes V\_p)^{G\_K}$. 


As suggested by the vocabulary, can one expect a direct relation between $c(A/K)$ and ${\rm Tam}\_\omega(T\_p)$ to hold? Is there a proof of this?


The number ${\rm Tam}\_\omega(T\_p)$ is a power of $p$, so one might expect it to be comparable with the $p$-part of $c(A/K)$. However, one would need to have an almost cannonical choice for $\omega$. 



Tamagawa numbers
Denote by $\Phi$ the quotient of $\mathcal A^\vee$, the special fiber of the smooth (but not necessarily proper) model of the dual abelian variety $A^\vee$, by the connected component of $0$ of $\mathcal A^\vee$. Then it is indeed true that the *Tamagawa number* $c\_p$ as defined in *$L$-functions and Tamagawa number of motives* (Bloch, Kato) or in *Autour des conjectures de Bloch-Kato* (Fontaine, Perrin-Riou) is then equal up to a $p$-adic unit to $|\Phi[p^\infty]|$, but this is not quite obvious.


The main step of the proofs are as follows:


* First, relate $\Phi$ to the formal group of $A^\vee$, and then through the $\log$ map to $\operatorname{Lie}\_{\mathbb Z\_p}A^{\vee}$ (which is also the tangent space of the formal group with values in $\mathbb Z\_p$).
* Then notice that this induces isomorphisms between the determinant of $\operatorname{Lie}\_{\mathbb Z\_p}A^{\vee}$ and the determinant of the first Bloch-Kato cohomology group $H^1\_f(\mathbb Q\_p,T\_pA)$ of the $p$-adic Tate module of $A$. This isomorphism involves finite corrective terms which come from the fact that the formal group is a subgroup with finite index of $H^1\_f(\mathbb Q\_p,T\_pA)$ (and similarly for the image of the $\log$ with respect to $\operatorname{Lie}\_{\mathbb Z\_p}A^{\vee}$).
* Now the Tamagawa number $c\_p$ arises as an alternate product of these corrective terms and of the Euler factor at $p$ (because the exponential map of Bloch-Kato coincide in this case with $\log$).
* Finally, relate this alternate product to $\Phi$. This involves the classical description of $\Phi$ in terms of the Neron model as well as a study of the reduction of $A^{\vee}$.


Personally, I don't find any of these steps particularly easy, and the last one certainly isn't. I don't have a good reference to mention that would cover all of them. A good but hard reference for the hardest part is SGA7, Exposé IX.


One thing that I would mention is that, as you say, the computations of the proof above involve in several steps the choice of a $\mathbb Z\_p$-basis, but as this plays a role both in the normalization of maps with target $\operatorname{Lie}A^\vee$ as well as in the definition of $c\_p$, the resulting computation does not depend on the choice of the basis (up to a $p$-adic unit, which plays no role in the Tamagawa number conjectures of Bloch-Kato). 


