Rigid Local Systems and Alternating Groups

In earlier work [Ka-RLSFM], 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 SL2(q), and he exhibited other such very simple families giving SU3(q). [Here q is a power of the characteristic p, and p is 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. See Theorem 3.1. [Of course from the resolution [Ray] 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 A/Fp; 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 [Gross] to determine which finite groups they were. Here we do not