Strongly minimal reducts of ACVF

Abstract

Let $\mathbb K=(K,+,\cdot,v,\Gamma)$ be a valued algebraically closed field of characteristic and $(G,\oplus)$ be a $\mathcal K$-interpretable group that is either locally isomorphic to $(K,+)$ or to $(K,\cdot)$. Then if $\mathcal G=(G,\oplus,\ldots)$ is a strongly minimal non locally modular structure intepretable in $\mathbb K$, it interprets a field. We also present an strategy for proving the same without the assumption of having a definable group operation.