G2 and hypergeometric sheaves

We determine, in every finite characteristic p, those hypergeomettic sheaves of type (7, m) with 7 >= m whose geometric monodromy group G(geom) lies in G(2), cf. Theorems 3.1 and 6.1. For each of these we determine Ggeom exactly, cf. Theorem 9.1. Each of the five primitive irreducible finite subgroups of G,), namely L-2(8), U-3(3), U-3(3).2 = G(2)(2), L-2(7).2 = PGL(2)(7), L-2(13) turns out to occur as G(geom) in a single characteristic p, namely p = 2, 3, 7, 7, 13 for the groups as listed, and for essentially just one hypergeometric sheaf in that characteristic. It would be interesting to find conceptual, rather than classificational/computational, proofs of these results. (c) 2006 Elsevier Inc. All tights reserved.