  In this paper we prove that if k is a cardinal in L[0^#], then there is an
inner model M such that M |= (V_k,E) has no elementary end extension. In
particular if 0^# 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^#] is Mahlo in every strict inner
model of L[0^#].
