### Abstract

We determine, in every finite characteristic p, those hypergeometric 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 G_{geom} exactly, cf. Theorem 9.1. Each of the five primitive irreducible finite subgroups of G_{2}, 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.

Original language | English (US) |
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Pages (from-to) | 175-223 |

Number of pages | 49 |

Journal | Finite Fields and their Applications |

Volume | 13 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2007 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Algebra and Number Theory
- Engineering(all)
- Applied Mathematics

### Keywords

- Exceptional groups
- Finite fields
- Monodromy