Rationality of blocks of quasi-simple finite groups

Let $\mathrm{\ell }$ be a prime number. We show that the Morita Frobenius number of an $\mathrm{\ell }$-block of a quasi-simple finite group is at most 4 and that the strong Frobenius number is at most $4\mathrm{\mid }D{\mathrm{\mid }}^2!$, where D 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 $\mathrm{\ell }$ is defined over a field with ${\mathrm{\ell }}^a$ elements for some $a\le 4$. We derive consequences for Donovan’s conjecture. In particular, we show that Donovan’s conjecture holds for $\mathrm{\ell }$-blocks of special linear groups.