Reconstructing Abelian Varieties via Model Theory

TL;DR

This work generalizes Zilber’s Jacobian-determination principle from curves to arbitrary (semi)abelian varieties $G$ with a distinguished subvariety $V$, by developing a robust factorization theory within a model-theoretic framework. Central to the approach is the full socle construction, ACP, and a Galois-type correspondence between factorizations of semiabelian pairs $(G,[X])$ and $K$-relics expanding $(G,+,X)$; the Restricted Trichotomy is leveraged to connect definable structures to algebraic data. In the abelian case, the pair $(G,[V])$ is shown to be geometrically determined by $(G(K),+,V(K))$ precisely when $(G,V)$ is simple with $0< ext{dim}(V)< ext{dim}(G)$; more generally, every instance decomposes canonically into simple factors, with uniqueness statements and a precise characterization via the full-socle framework. For semiabelian varieties, a weaker but canonical factorization into subvarieties $W_i$ of subgroups $H_i$ is established, with each $W_i$ geometrically determining $H_i$, yielding a structural analogue of Mordell–Lang phenomena. Overall, the paper provides a model-theoretic reconstruction program that recovers algebraic structure from the group with a distinguished subset and offers a bridge between Hrushovski–Mordell–Lang techniques and Zilber’s Jacobian reconstruction ideas.

Abstract

In 2012, Zilber used model-theoretic techniques to show that a curve of high genus over an algebraically closed field is determined by its Jacobian (viewed only as an abstract group with a distinguished subset for an image of the curve). In this paper, we consider an analogous problem for arbitrary (semi)abelian varieties $A$ over algebraically closed fields $K$ with a distinguished subvariety $V$. Our main result characterizes when the data $(A(K),+,V(K))$ (as a group with distinguished subset) determines the pair $(A,V)$ in the strongest reasonable sense. As it turns out, the situation is best understood by developing a theory of factorizations for such pairs $(A,V)$. In the final sections of the paper, we develop such a theory and prove unique factorization theorems (one for abelian varieties and a weaker one for semi-abelian varieties). In this language, the main theorem mentioned above (in the abelian case) says that the pair $(A,V)$ is determined by the data $(A(K),+,V(K))$ precisely when $(A,V)$ is simple and $0<\dim(V)<\dim(A)$.

Theorems & Definitions (77)

• Definition 1.1
• Theorem 1
• Theorem 2
• Definition 2.1
• Lemma 2.3
• proof
• proof
• Lemma 3.1
• proof
• Definition 3.2