### Abstract

For powers q of any odd prime p and any integer n≥2, we exhibit explicit local systems, on the affine line A^{1} in characteristic p>0 if 2|n and on the affine plane A^{2} if 2∤n, whose geometric monodromy groups are the finite symplectic groups Sp_{2n}(q). When n≥3 is odd, we show that the explicit rigid local systems on the affine line in characteristic p>0 constructed in [11] do have the special unitary groups SU_{n}(q) as their geometric monodromy groups as conjectured therein, and also prove another conjecture of [11] that predicted their arithmetic monodromy groups.

Original language | English (US) |
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Article number | 106859 |

Journal | Advances in Mathematics |

Volume | 358 |

DOIs | |

State | Published - Dec 15 2019 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Keywords

- Finite symplectic groups
- Finite unitary groups
- Local systems
- Monodromy groups
- Weil representations

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## Cite this

Katz, N. M., & Tiep, P. H. (2019). Local systems and finite unitary and symplectic groups.

*Advances in Mathematics*,*358*, [106859]. https://doi.org/10.1016/j.aim.2019.106859