The local obstruction to semi-stable reduction for abelian varieties

TL;DR

The paper analyzes Grothendieck's local obstruction groups $\Phi_{A,v}$ governing semi-stable reduction of abelian varieties, and develops a comprehensive framework to classify these obstructions via $(p,t,a)$-inertial groups. It first establishes structural and finiteness properties of $\Phi_{A,v}$, including their behavior as ramification groups and their Minkowski-based order bounds, and then defines $(p,t,a)$-inertial groups to organize possible finite monodromy groups. A geometric p-adic characterization uses descent data on polarized semi-abelian varieties over finite fields together with $p$-adic Hodge theory, while equal-characteristic cases are treated via Galois twisting and degeneration techniques. The core contribution is a complete (and in many cases explicit) description of finite monodromy groups as precisely the $(p,t,a)$-inertial groups, along with constructions realizing these groups via deformation, twisting, and Honda–Tate theory, thereby linking local obstructions to endomorphism algebras of abelian varieties over finite fields and their deformations.

Abstract

Grothendieck defined a group that represents the local obstruction for an abelian variety to have semi-stable reduction. These groups were studied by Silverberg and Zarhin and more recently by the author in order to give a group theoretic characterization of them depending only on the dimension. We give an overview of the developments since Grothendieck's definition with the added novelty of the case of equal characteristic local fields.

Figures (1)

• Figure 1: Navigating through the categories in play

Theorems & Definitions (30)

• Definition 1.1
• Theorem 1.2: sga exposé IX
• Theorem 1.4
• Proposition 2.1
• Definition 2.2
• Theorem 2.3: SZ95, Theorem 5.2
• Remark 2.4
• Definition 2.5
• Proposition 2.6
• Proposition 2.7