Defect zero blocks for finite simple groups, by Andrew Granville and
Ken Ono

Trans. Amer. Math. Soc. 348 (1996), no. 1, 331-347.

We classify those finite simple groups whose Brauer graph (or decomposition 
matrix) has a $p$-block with defect 0, completing an investigation of many
authors. The only finite simple groups previously left unclassified were the 
alternating groups $A_{n}$. Here we show that these all have a $p$-block with 
defect 0 for every prime $p\geq 5$. This follows from proving the same result 
for every symmetric group $S_{n}$, which in turn follows as a consequence of 
the {\sl $t$-core partition conjecture}, that every non-negative integer 
possesses at least one $t$-core partition, for any $t\geq 4$. For $t\geq 17$, 
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 $t<17$, that 
was not covered in previous work, was the case $t=13$. This we prove with a 
very different argument, by interpreting the generating function for $t$-core 
partitions in terms of modular forms, and then controlling the size of the 
coefficients using Deligne's Theorem (n\'ee the {\sl Weil Conjectures}). We 
also consider congruences for the number of $p$-blocks of $S_{n}$, proving a 
conjecture of Garvan, that establishes certain multiplicative congruences when 
$5\leq p \leq 23$. By using a result of Serre concerning the divisibility of 
coefficients of modular forms, we show that for any given prime $p$ and 
positive integer $m$, the number of $p-$blocks with defect 0 in $S_n$ is a 
multiple of $m$ for almost all $n$. We also establish that any given prime 
$p$ divides the number of $p-$modularly irreducible representations of 
$S_{n}$, for almost all $n$.