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Katz, Nicholas M.; Rojas‐León, Antonio; Tiep, Pham Huu

doi:10.1016/j.jnt.2019.06.016

Cited by 8 publications

(3 citation statements)

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“…Because

scriptF

is pure of weight zero, the fact that its traces are algebraic integers forces its

G_{arith}

to be finite, cf. [11, Proposition 2.1 and Remark 2.2]. We now explain why

scriptF

is arithmetically semisimple.…”

Section: A Miscellany On Moments Irreducibility and Van Der Geer–van Der Vlugtmentioning

confidence: 94%

Exponential sums and total Weil representations of finite symplectic and unitary groups

Katz

Tiep

2020

Proc. London Math. Soc.

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We construct explicit local systems on the affine line in characteristic p>2, whose geometric monodromy groups are the finite symplectic groups normalSp2nfalse(qfalse) for all n⩾2, and others whose geometric monodromy groups are the special unitary groups normalSUnfalse(qfalse) for all odd n⩾3, and q any power of p, in their total Weil representations. One principal merit of these local systems is that their associated trace functions are one‐parameter families of exponential sums of a very simple, that is, easy to remember, form. We also exhibit hypergeometric sheaves on Gm, whose geometric monodromy groups are the finite symplectic groups normalSp2nfalse(qfalse) for any n⩾2, and others whose geometric monodromy groups are the finite general unitary groups normalGUnfalse(qfalse) for any odd n⩾3.

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“…Because

scriptF

is pure of weight zero, the fact that its traces are algebraic integers forces its

G_{arith}

to be finite, cf. [11, Proposition 2.1 and Remark 2.2]. We now explain why

scriptF

is arithmetically semisimple.…”

Section: A Miscellany On Moments Irreducibility and Van Der Geer–van Der Vlugtmentioning

confidence: 94%

Exponential sums and total Weil representations of finite symplectic and unitary groups

Katz

Tiep

2020

Proc. London Math. Soc.

Self Cite

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show abstract

“…Namely, the respective hypergeometric sheaves H (in characteristic p in (b)-(e)) are explicitly constructed in Theorem 9.3 for case (a), in [KT5] and Theorem 8.6 for case (b), in [KT6] for type (α) in (c) and for (d) with 2 ∤ q, in [KT7] for type (β) in (c) and for (e), and in [KT8] for case (d) with 2|q. The extraspecial normalizers, and the sporadic and non-generic cases in (f), are handled in [KT8] and [KRL], [KRLT1]- [KRLT4].…”

Section: Converse Theoremsmentioning

confidence: 99%

“…Further constraints for a finite group G to occur as G geom of a hypergeometric sheaf are established in § §4, 5, 9. With an explicit, finite, list of exceptions, all the almost quasisimple triples (G, V, g), that satisfy the Abhyankar condition at p and in addition these extra constraints, are then shown (modulo a central subgroup) to occur hypergeometrically; the respective hypergeometric sheaves H are explicitly constructed in a series of companion papers [KRL], [KRLT1]- [KRLT4], [KT1]- [KT3], [KT5]- [KT8].…”

Section: Introductionmentioning

confidence: 99%