$0^{\sharp }$ and elementary end extensions of $V_{\kappa }$

Abstract:

In this paper we prove that if $\kappa$ is a cardinal in $L[0^{\sharp }]$, then there is an inner model $M$ such that $M \models (V_{\kappa },\in )$ has no elementary end extension. In particular if $0^{\sharp }$ exists, then weak compactness is never downwards absolute. We complement the result with a lemma stating that any cardinal greater than $\aleph _1$ of uncountable cofinality in $L[0^{\sharp }]$ is Mahlo in every strict inner model of $L[0^{\sharp }]$.