### 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) |
---|---|

Article number | 106859 |

Journal | Advances in Mathematics |

Volume | 358 |

DOIs | |

State | Published - Dec 15 2019 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

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

**Local systems and finite unitary and symplectic groups.** / Katz, Nicholas M.; Tiep, Pham Huu.

Research output: Contribution to journal › Article

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

}

TY - JOUR

T1 - Local systems and finite unitary and symplectic groups

AU - Katz, Nicholas M.

AU - Tiep, Pham Huu

PY - 2019/12/15

Y1 - 2019/12/15

N2 - For powers q of any odd prime p and any integer n≥2, we exhibit explicit local systems, on the affine line A1 in characteristic p>0 if 2|n and on the affine plane A2 if 2∤n, whose geometric monodromy groups are the finite symplectic groups Sp2n(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 SUn(q) as their geometric monodromy groups as conjectured therein, and also prove another conjecture of [11] that predicted their arithmetic monodromy groups.

AB - For powers q of any odd prime p and any integer n≥2, we exhibit explicit local systems, on the affine line A1 in characteristic p>0 if 2|n and on the affine plane A2 if 2∤n, whose geometric monodromy groups are the finite symplectic groups Sp2n(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 SUn(q) as their geometric monodromy groups as conjectured therein, and also prove another conjecture of [11] that predicted their arithmetic monodromy groups.

KW - Finite symplectic groups

KW - Finite unitary groups

KW - Local systems

KW - Monodromy groups

KW - Weil representations

UR - http://www.scopus.com/inward/record.url?scp=85073614374&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85073614374&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2019.106859

DO - 10.1016/j.aim.2019.106859

M3 - Article

AN - SCOPUS:85073614374

VL - 358

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

M1 - 106859

ER -