Zilber’s restricted trichotomy in characteristic zero

Castle, Benjamin

doi:10.1090/jams/1037

Zilber’s restricted trichotomy in characteristic zero
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by Benjamin Castle

J. Amer. Math. Soc.

DOI: https://doi.org/10.1090/jams/1037

Published electronically: October 3, 2023

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Abstract:

We prove the characteristic zero case of Zilber’s Restricted Trichotomy Conjecture. That is, we show that if $\mathcal M$ is any non-locally modular strongly minimal structure interpreted in an algebraically closed field $K$ of characteristic zero, then $\mathcal M$ itself interprets $K$; in particular, any non-1-based structure interpreted in $K$ is mutually interpretable with $K$. Notably, we treat both the ‘one-dimensional’ and ‘higher-dimensional’ cases of the conjecture, introducing new tools to resolve the higher-dimensional case and then using the same tools to recover the previously known one-dimensional case.

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