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

This paper has appeared in the Trans. Amer. Math. Soc. 348 (1996), no.
1, 331-347, and so the dvi version has been removed.
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 $n$.

Andrew Granville and Ken Ono <andrew@sophie.math.uga.edu, ono@symcom.math.uiuc.edu>