The ABCD hypothesis

If [Formula: see text] and [Formula: see text] are cardinals, then the cardinal exponentiation [Formula: see text] is the set of all functions from [Formula: see text] into [Formula: see text]. The ABCD hypothesis is the statement that for all infinite cardinals [Formula: see text] and [Formula: see text], [Formula: see text] if and only if [Formula: see text] and [Formula: see text]. The Axiom of Determinacy, AD, states that every two-player infinite game on [Formula: see text] has a winning strategy for one of the two players. Let [Formula: see text] be the supremum of the ordinals onto which [Formula: see text] surjects. This paper will show that if AD holds and [Formula: see text], then the ABCD hypothesis holds below [Formula: see text].