A rigid local system with monodromy group the Big Conway group 2.CO₁ and two others with monodromy group the Suzuki Group 6.Suz

We first develop some basic facts about hypergeometric sheaves on the multiplicative group G_m in characteristic p > 0. Specializing to some special classes of hypergeometric sheaves, we give relatively “simple” formulas for their trace functions, and a criterion for them to have finite monodromy. We then show that one of our local systems, of rank 24 in characteristic p = 2, has the big Conway group 2.Co₁, in its irreducible orthogonal representation of degree 24 as the automorphism group of the Leech lattice, as its arithmetic and geometric monodromy groups. Each of the other two, of rank 12 in characteristic p = 3, has the Suzuki group 6.Suz, in one of its irreducible representations of degree 12 as the Q(ζ3)-automorphisms of the Leech lattice, as its arithmetic and geometric monodromy groups. We also show that the pullback of these local systems by x → x^N mappings to the affine line A¹ yields the same arithmetic and geometric monodromy groups.