Elements in reductive algebraic groups with abelian connected centralizers

In this paper we prove that, with essentially one exception, an element in a reductive algebraic group has abelian connected centralizer if and only if it is regular. This extends a result of Kurtzke, who proved the statement (without exception) in the case of good characteristic; this assumption allowed results about the group to be deduced from calculations in its Lie algebra. By contrast, the work here relies on explicit calculations in the algebraic groups themselves.