Title: Rigid local systems and alternating groups,
Abstract: In earlier work, Katz exhibited some very simple one parameter families of
exponential sums which gave rigid local systems on the affine line in
characteristic p whose geometric (and usually, arithmetic) monodromy groups
were SL(2,q), and he exhibited other such very simple families giving SU(3,q).
[Here q is a power of the characteristic p with p odd]. In this paper, we
exhibit equally simple families whose geometric monodromy groups are the
alternating groups Alt(2q). $. We also determine their arithmetic monodromy
groups. By Raynaud's solution of the Abhyankar Conjecture, any finite simple
group whose order is divisible by p will occur as the geometric monodromy group
of some local system on the affine line in characteristic p; the interest here
is that it occurs in our particularly simple local systems.
In the earlier work of Katz, he used a theorem to Kubert to know that the
monodromy groups in question were finite, then work of Gross to determine which
finite groups they were. Here we do not have, at present, any direct way of
showing this finiteness. Rather, the situation is more complicated and more
interesting. Using some basic information about these local systems, a
fundamental dichotomy is proved:
The geometric monodromy group is either Alt(2q) or it is the special
orthogonal group SO(2q-1). An elementary polynomial identity is used to show
that the third moment is 1. This rules out the SO(2q-1) case. This roundabout
method establishes the theorem. It would be interesting to find a "direct"
proof that these local systems have integer (rather than rational) traces; this
integrality is in fact equivalent to their monodromy groups being finite, Even
if one had such a direct proof, it would still require serious group theory to
show that their geometric monodromy groups are the alternating groups.