G2 and hypergeometric sheaves

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Abstract

We determine, in every finite characteristic p, those hypergeometric sheaves of type (7, m) with 7 ≥ m whose geometric monodromy group Ggeom lies in G2, 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 G2, namely L2 (8), U3 (3), U3 (3) . 2 = G2 (2), L2 (7) . 2 = PGL2 (7), L2 (13) turns out to occur as Ggeom 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 languageEnglish (US)
Pages (from-to)175-223
Number of pages49
JournalFinite Fields and their Applications
Volume13
Issue number2
DOIs
StatePublished - 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

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