Tate models for semistable algebraic varieties with mixed reduction over a local field

2011-01-08T05:28:45Z

It's known that if $A$ is an abelian variety of totally multiplicative reduction over a p-adic field K, then, after taking a finite field extension, it becomes isomorphic, as a rigid analytic group, to $G_m^g/\prod_{i=1}^g q_i^{\mathbb{Z}}$ where $q_i$ are points (after a finite field extension) of $G_m^g$ which generate a discrete subgroup.

My question is, what can be said when $A$ is not totally multiplicative or good reduction, but is some semistable abelian variety of mixed multiplicative-good reduction. I would guess that instead of being a quotient of $G_m$ by a discrete subgroup, $A$ will be (as a rigid analytic variety) a quotient of a good-reduction semi-abelian variety (i.e. a variety which is an extension by $G_m^m$ of an abelian variety of good reduction) by a free discrete subgroup. Does anyone know whether this is true, or whether there's something else replacing the Tate uniformization in this case?

Answer by Emerton for Tate models for semistable algebraic varieties with mixed reduction over a local field

2011-01-08T05:42:41Z

If $A$ has semi-abelian reduction, then $A$ is uniformized by a semi-abelian variety $G_A$, namely there is an exact sequence $$0 \to \Gamma_A \to G_A \to A \to 0,$$ where $\Gamma_A$ is free of finite rank, $G_A$ is semi-abelian, and the maximal abelian variety quotient of $G_A$ has good reduction.

This is due to Raynaud, I believe, and is also discussed in SGA 7. For precise references, allow me (out of laziness) to cite one of my papers --- see the discussion at the beginning of Section 3.

Tate models for semistable algebraic varieties with mixed reduction over a local field - MathOverflow

Tate models for semistable algebraic varieties with mixed reduction over a local field

Answer by Emerton for Tate models for semistable algebraic varieties with mixed reduction over a local field