Hypergeometric sheaves and finite symplectic and unitary groups

Nicholas M. Katz, Pham Huu Tiep

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

We construct hypergeometric sheaves whose geometric monodromy groups are the finite symplectic groups Sp2n(q) for any odd n ≥ 3, for q any power of an odd prime p. We construct other hypergeometric sheaves whose geometric monodromy groups are the finite unitary groups GUn(q), for any even n ≥ 2, for q any power of any prime p. Suitable Kummer pullbacks of these sheaves yield local systems on A1, whose geometric monodromy groups are Sp2n(q), respectively SUn(q), in their total Weil representation of degree qn, and whose trace functions are simple-to-remember one-parameter families of two-variable exponential sums. The main novelty of this paper is three-fold. First, it treats unitary groups GUn(q) withn even via hypergeometric sheaves for the first time. Second, in both the symplectic and the unitary cases, it uses a maximal torus which is a product of two sub-tori to furnish a generator of local monodromy at 0. Third, this is the first natural occurrence of families of two-variable exponential sums in the context of finite classical groups.

Original languageEnglish (US)
Pages (from-to)577-691
Number of pages115
JournalCambridge Journal of Mathematics
Volume9
Issue number3
DOIs
StatePublished - 2021

All Science Journal Classification (ASJC) codes

  • General Mathematics

Keywords

  • Finite simple groups
  • Hypergeometric sheaves
  • Local systems
  • Monodromy groups
  • Weil representations

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