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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2504.04307 |
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| _version_ | 1866909567439339520 |
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| author | Castle, Benjamin Hasson, Assaf |
| author_facet | Castle, Benjamin Hasson, Assaf |
| contents | 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 $0<\dim(V)<\dim(A)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_04307 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Reconstructing Abelian Varieties via Model Theory Castle, Benjamin Hasson, Assaf Logic Primary 03C45, Secondary 14K12 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 $0<\dim(V)<\dim(A)$. |
| title | Reconstructing Abelian Varieties via Model Theory |
| topic | Logic Primary 03C45, Secondary 14K12 |
| url | https://arxiv.org/abs/2504.04307 |