Rationality of blocks of quasi-simple finite groups

Let l be a prime number. We show that the Morita Frobenius number of an l-block of a quasi-simple finite group is at most 4 and that the strong Frobenius number is at most 4|D |²!
D, where denotes a defect group of the block. We deduce that a basic algebra of any block of the group algebra of a quasi-simple finite group over an algebraically closed field of characteristic l  is defined over a field with l ^a elements for some a≤4. We derive consequences for Donovan’s conjecture. In particular, we show that Donovan’s conjecture holds for l-blocks of special linear groups.