Defect zero blocks for finite simple groups

We classify those finite simple groups whose Brauer graph (or decomposition matrix) has a p p -block with defect 0, completing an investigation of many authors. The only finite simple groups whose defect zero p − p- blocks remained unclassified were the alternating groups A n A₍. Here we show that these all have a p p -block with defect 0 for every prime p ≥ 5 p 5. This follows from proving the same result for every symmetric group S n S₍, which in turn follows as a consequence of the t t -core partition conjecture, that every non-negative integer possesses at least one t t -core partition, for any t ≥ 4 t 4. For t ≥ 17 t 17, we reduce this problem to Lagrange’s Theorem that every non-negative integer can be written as the sum of four squares. The only case with t > 17 t>17, that was not covered in previous work, was the case t = 13 t=13. This we prove with a very different argument, by interpreting the generating function for t t -core partitions in terms of modular forms, and then controlling the size of the coefficients using Deligne’s Theorem (née the Weil Conjectures). We also consider congruences for the number of p p -blocks of