Recently I read a thesis [p-adic Galois representations and elliptic curves](https://www.math.u-bordeaux.fr/%7Eybilu/algant/documents/theses/Abhinandan.pdf). Using Tate's curve, the author proved the p-adic Galois representations atteched to the Tate module of an elliptic curve is de Rham by constructing a basis of $D\_{dR}(V)$ explicitly in Page 53. So the dimension of $D\_{dR}(V)$ implies the representation is de Rham.


Question: Is there a likewise uniformization of an abelian variety over p-adic fields to prove the representation attached to Tate module is de Rham by constructing a basis? And I think I can prove the Hodge-Tate weights are 0 and 1 with the help of a basis.


Thanks in advance!



How to prove the p-adic Galois representations atteched to the Tate module of an abelian variety is de Rham directly?
About the general uniformization you asked for. There is a construction by Raynaud of a general uniformization in the case of an abelian variety with semistable reduction: there exists a semiabelian variety $E$ with good reduction, extension of a good reduction abelian variety $B$ and a good reduction torus $T$: 
$$ 0 \to T \to E \to B\to 0$$ 
(you can suppose $T$ is a split torus by extending a bit you field of definition), and analytic maps 
$$ 0\to M \to E \to A \to 0$$
where $M$ is a lattice inside $E$ (the map from $E$ to $A$ is not algebraic), which you can suppose $M\cong \mathbb{Z}^r$, where $r=\dim(T)$, extending the field (as above). The case of Tate elliptic curves correspond to the case $B=0$ and $M=\mathbb{Z}$. This is what it is used by Coleman and Iovita. They are called Raynaud extension. 


In the non semistable case, you can descent to a "potentially good reduction semiabelian variety" and uniformization the one above by using the semistable reduction theorem by Grothendieck (i.e., that any abelian variety acquires semistable reduction after a finite extension of you local field). 


You can see: 


* M. Raynaud, 1-motifs et monodromie géométrique, Exposé VII, Astérisque 223(1994) 295–319
* Bosch and Lütkebohmert, Degenerating abelian varieties, Topology 30:653–698, 1991
* Werner Lütkebohmert, Rigid Geometry of Curves and Their Jacobians, 2016, Chapter 6.


