G2 and hypergeometric sheaves

Nicholas M. Katz

doi:10.1016/j.ffa.2006.02.004

G₂ and hypergeometric sheaves

Nicholas M. Katz

Mathematics

Research output: Contribution to journal › Article › peer-review

12 Scopus citations

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 G₂, 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 G₂, namely L₂ (8), U₃ (3), U₃ (3) . 2 = G₂ (2), L₂ (7) . 2 = PGL₂ (7), L₂ (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)
Pages (from-to)	175-223
Number of pages	49
Journal	Finite Fields and their Applications
Volume	13
Issue number	2
DOIs	https://doi.org/10.1016/j.ffa.2006.02.004
State	Published - Apr 2007

All Science Journal Classification (ASJC) codes

Theoretical Computer Science
Algebra and Number Theory
General Engineering
Applied Mathematics

Keywords

Exceptional groups
Finite fields
Monodromy

Access to Document

10.1016/j.ffa.2006.02.004

Cite this

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AU - Katz, Nicholas M.

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N2 - 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.

AB - 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.

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KW - Monodromy

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