Integral Monodromy Groups of Kloosterman Sheaves

Abstract

We show that integral monodromy groups of Kloosterman ℓ-adic sheaves of rank n ě 2 on Gm{Fq are as large as possible when the characteristic ℓ is large enough, depending only on the rank. This variant of Katz's results over C was known by works of Gabber, Larsen, Nori and Hall under restrictions such as ℓ large enough depending on charpFqq with an ineffective constant, which is unsuitable for applications. We use the theory of finite groups of Lie type to extend Katz's ideas, in particular the classification of maximal subgroups of Aschbacher and Kleidman-Liebeck. These results will apply to study hyper-Kloosterman sums and their reductions in forthcoming work.

Roots of $L$-functions of characters over function fields, generic linear independence and biases

“…Remark 5.9. By work of Hall [Hal08] or J-K. Yu (unpublished) when r " 2, and the author [PG18b] for any r ě 2, one may actually take Λ r,p " tλ P Spec 1,p pOq above ℓ : ℓ " r 1u.…”

Distribution questions for trace functions with values in cyclotomic integers and their reductions

“…Monodromy groups. For Kloosterman sums, an important input is the determination of the O λ -integral monodromy groups of Kloosterman sheaves [PG16a], analogous to the determination of the monodromy groups over Q ℓ by Katz [Kat88], when ℓ is large enough depending only on the rank.…”

“…Monodromy groups. For Kloosterman sums, an important input is the determination of the O λ -integral monodromy groups of Kloosterman sheaves [PG16a], analogous to the determination of the monodromy groups over Q ℓ by Katz [Kat88], when ℓ is large enough depending only on the rank. This was already known by results of Gabber, Larsen and Nori, but for ℓ large enough depending on q and with an ineffective constant, which would have been unusable for our applications.…”

Exponential sums over finite fields and the large sieve

“…(3) Under the general Riemann hypothesis (GRH) for the Dedekind zeta function of Qpζ 4p q, one may take ε " 0 and e ě 4B n `1. (4) By relying on the determination of the monodromy groups over Q ℓ by Katz and the results of Larsen-Pink, instead of [PG16c], these results would hold when p is fixed, e Ñ `8, with an implicit constant depending on p.…”

“…We state some of the results for hyper-Kloosterman sums of rank n ě 2. The bounds are uniform in p, thanks to the determination of the monodromy groups over F λ in [PG16c] when ℓ " n 1 (with no dependency with respect to p). Proposition 1.1.…”