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.

Thanks in advance!

  • 3
    $\begingroup$ 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 $\endgroup$
    – Xarles
    Jul 2 '19 at 7:02
  • $\begingroup$ @Xarles Thanks for the reference ! $\endgroup$
    – 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.

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.

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