Monodromy of Airy and Kloosterman sheaves

Airy sheaves for reductive groups

“…Also, 𝜒 descends to a morphism [𝐿 − 𝐺 ∩ 𝐽∖𝐽∕𝐽 + ] → 𝔾 𝑎 . Denote the restriction of Hecke 𝜔 1 by GR * 𝜔 1 and let 𝑝 1 ∶= 𝜒 • pr 1 , 𝑝 2 = pr 2 , we get diagram (34) The Hecke eigenvalue is given by 𝜓(𝑓(𝑚 1 ) + 𝑚 1 𝑎).…”

“…Such a realization allowed Katz to generalize this type of equation and define analogous $$\ell$$‐adic local systems [22, 23]. A more general version of $$\ell$$‐adic local systems was studied by Šuch in [34]. These sheaves are defined as the Fourier transform of Artin–Schreier–Witt characters.…”

Finite monodromy of some families of exponential sums

“…In this article we will be concerned with a special class of sheaves, which are a subset of the class of so-called Airy sheaves, smooth sheaves on A 1 k of rank n with a single slope n+1 n at infinity, which can also be characterized as the Fourier transform of smooth sheaves of rank 1 with slope > 1 at infinity. The monodromy of these sheaves was extensively studied by O. Šuch in [ Šuc00], who gave a full classification of their possible non-finite monodromy groups [ Šuc00,Propositions 11.6,11.7].…”

“…Suppose that F d were Artin-Schreier induced. Then the proof of[ Šuc00, Proposition 11.1] shows that F d ⊗ F d contains an Artin-Schreier subsheaf L ψ(at) for some a ∈ k * . That is,Hom(F d ⊗ F d , L ψ(at) ) = 0for some a ∈ k * or, equivalently,H 0 (A 1 k, F d ⊗ F d ⊗ L ψ(at) ) = 0 for some a ∈ k * .…”

Estimates for exponential sums with a large automorphism group

“…(2) Suppose that p > 2, and let G ⊆ GL(V ) be the geometric monodromy group of F g , where V is its stalk at a geometric generic point. Then by [16,Propositions 11 for some other criterions that rule out the finite monodromy case in the p ≤ 2d − 1 case. The determinant of F g is computed over k in [7,Theorem 17].…”