Cohomology of Linearly Constrained Kloosterman Families: Newton Polytopes, Weights, and Boundary Monodromy
Abstract
Let p be an odd prime, ψ a nontrivial additive character of 𝔽p, ψq=ψ∘Tr𝔽q/𝔽p and λ∈𝔽p×. For h∈(𝔽q×)n let Σn(h) be the sum of ψq(u1−1+⋯+un−1) over u∈(𝔽q×)n with h1u1+⋯+hnun=−λ, an additively twisted average of products of Kloosterman sums. For every n≥2, the cohomology computing Σn(h) is concentrated in one degree and has dimension 2n−1. It has r(n)=2n−binom n⌊ n/2⌋ eigenvalues of weight n−1, and its other eigenvalues are the integers qk−1, 1≤ k≤⌊ n/2⌋, with ballot numbers as multiplicities. The part of top weight forms a lisse sheaf 𝒲n of rank r(n) on the torus 𝔾mn. We determine its local monodromy at the coordinate boundary and its determinant: det𝒲n is the Artin--Schreier sheaf of the sum −2n−1λ−1(h1+⋯+hn) of the critical values of the phase, twisted by p(n+1)r(n)/2. The geometric monodromy group of 𝒲n, and the arithmetic one after normalization to weight $0$, are equal to {g∈GLr(n):det(g)p=1}. Consequently the normalized sums become equidistributed like traces of Haar-random elements of {g∈U(r(n)):det(g)p=1}. The proofs use Laumon's Fourier transform, middle convolution, the Cayley cubic surface and Larsen's alternative, but no classification of finite groups.
Related
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BibTeX
@misc{ebadi2026cohomology,
title = {{Cohomology of Linearly Constrained Kloosterman Families: Newton Polytopes, Weights, and Boundary Monodromy}},
author = {Hamed Ebadi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03081},
eprint = {2610.03081},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03081},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Cohomology of Linearly Constrained Kloosterman Families: Newton Polytopes, Weights, and Boundary Monodromy
AU - Hamed Ebadi
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.03081
UR - https://arxiv.org/abs/2610.03081
AB - Let p be an odd prime, ψ a nontrivial additive character of 𝔽_p, ψ_q=ψ∘Tr_𝔽_q/𝔽_p and λ∈𝔽_p^×. For h∈(𝔽_q^×)^n let Σ_n(h) be the sum of ψ_q(u_1^−1+⋯+u_n^−1) over u∈(𝔽_q^×)^n with h_1u_1+⋯+h_nu_n=−λ, an additively twisted average of products of Kloosterman sums. For every n≥2, the cohomology computing Σ_n(h) is concentrated in one degree and has dimension 2^n−1. It has r(n)=2^n−binom n⌊ n/2⌋ eigenvalues of weight n−1, and its other eigenvalues are the integers q^k−1, 1≤ k≤⌊ n/2⌋, with ballot numbers as multiplicities. The part of top weight forms a lisse sheaf 𝒲_n of rank r(n) on the torus 𝔾_m^n. We determine its local monodromy at the coordinate boundary and its determinant: det𝒲_n is the Artin--Schreier sheaf of the sum −2^n−1λ^−1(h_1+⋯+h_n) of the critical values of the phase, twisted by p^(n+1)r(n)/2. The geometric monodromy group of 𝒲_n, and the arithmetic one after normalization to weight $0$, are equal to {g∈GL_r(n):det(g)^p=1}. Consequently the normalized sums become equidistributed like traces of Haar-random elements of {g∈U(r(n)):det(g)^p=1}. The proofs use Laumon's Fourier transform, middle convolution, the Cayley cubic surface and Larsen's alternative, but no classification of finite groups.
ER -