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 Aₙ . 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 Sₙ , 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