One dimensional commutative groups definable in algebraically closed valued fields and in the pseudo-local fields

TL;DR

The paper provides a complete classification of one-dimensional definable commutative groups in algebraically closed valued fields and pseudo-local fields, up to finite index and finite kernels. Building on the algebraization framework Mop and the notion of opaqueness, it reduces the problem to understanding type-definable subgroups of the one-dimensional algebraic groups (additive, multiplicative, twisted multiplicative, and elliptic) and uses Tate uniformization for elliptic curves to control non-algebraic pieces. The authors develop a detailed description of type-definable subgroups in each basic case and prove that, for $ACVF_{(0,0)}$, all such groups arise from a fixed finite collection of basic types (and their finite quotients) with no extra definable structure beyond algebraic isomorphisms. They also establish that in the mixed characteristic and pseudo-local settings, the finite-kernel phenomenon can be eliminated in many cases, yielding a clean, redundancy-free classification up to definable isomorphism. Overall, the work extends the known $p$-adic classification to broader valued-field contexts and provides tools (opaqueness, algebraization, Tate uniformization) that are broadly applicable to similar model-theoretic classifications.

Abstract

We give a complete list of the one-dimensional groups definable in algebraically closed valued fields and i the pseudo-local fields, up to a finite index subgroup and a quotient by a finite subgroup.

Theorems & Definitions (133)

• Proposition 2.2.1: holly-art
• Proposition 2.2.2
• Proposition 2.2.3: flenner
• Proposition 2.3.2: csqeses
• Lemma 2.4.1
• Proposition 2.4.2
• Proposition 2.5.1
• Proposition 2.5.2
• proof
• Lemma 2.6.1