# How to prove the p-adic Galois representations atteched to the Tate module of an abelian variety is de Rham directly?

Recently I read a thesis p-adic Galois representations and elliptic curves. 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.

• There is a famous paper by Colwman and Iovita where they prove that semistable reduction abelian varieties have Tate module semistable: arxiv.org/abs/math/9701229 Jul 2 '19 at 7:02
• @Xarles Thanks for the reference !
– user141691
Jul 2 '19 at 7:06

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.