Moments and Large Geometric Monodromy for Linearly Constrained Kloosterman Families
Abstract
Let q=pa with p prime, λ∈𝔽p×, and ψq=ψ∘Tr𝔽q/𝔽p for a nontrivial additive character ψ of 𝔽p. For n≥2 we study Σn(h)=∑u∈(𝔽q×)n,∑i hi ui=−λψq(∑i ui−1). On 𝔾mn we construct a lisse sheaf 𝒲n, pure of weight n+1 and rank r(n)=2n−binom n⌊ n/2⌋, whose Frobenius traces equal (−1)n−1qΣn(h) minus an explicit polynomial in q, for h∈(𝔽q×)n. For odd p, its geometric monodromy has identity component SLr(n) and cyclic component group of order divisible by p. The full group is {g:det(g)p=1} for n=2, and for n≥3 using boundary unipotence from the companion preprint arXiv:2610.03081. For p=2, the sheaf is self-dual and its geometric monodromy is SOr(n) for odd n and Spr(n) for even n. After normalization to weight zero, the arithmetic and geometric groups coincide, using the same companion input for odd p and n≥3. As q→∞ with p fixed, we obtain Sato-Tate laws for centered, normalized sums with error O(q−1/2) for polynomial tests and formulas for all power moments. For p=2, limiting moments are Gaussian through order r(n)−1, with an excess of one above the orthogonal value at order r(n) for odd n; if n+1 is a power of two, every Frobenius on 𝒲n has eigenvalue q(n+1)/2. The proof uses ∑h|Σn(h)|4=cpq3n−2+O(q3n−2−δ(n)) for n≥3, where cp=2 for odd p, c2=3 and δ(n)=min(1,(n−2)/2), plane sections of the Cayley cubic, exact Frobenius-Schur identities, Laumon's local Fourier transforms and Artin-Schreier theory. For n=2 we identify twisted classical Kloosterman sums and compute the fourth moment exactly.
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BibTeX
@misc{ebadi2026moments,
title = {{Moments and Large Geometric Monodromy for Linearly Constrained Kloosterman Families}},
author = {Hamed Ebadi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04099},
eprint = {2610.04099},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04099},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Moments and Large Geometric Monodromy for Linearly Constrained Kloosterman Families
AU - Hamed Ebadi
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.04099
UR - https://arxiv.org/abs/2610.04099
AB - Let q=p^a with p prime, λ∈𝔽_p^×, and ψ_q=ψ∘Tr_𝔽_q/𝔽_p for a nontrivial additive character ψ of 𝔽_p. For n≥2 we study Σ_n(h)=∑_u∈(𝔽_q^×)^n,∑_i h_i u_i=−λψ_q(∑_i u_i^−1). On 𝔾_m^n we construct a lisse sheaf 𝒲_n, pure of weight n+1 and rank r(n)=2^n−binom n⌊ n/2⌋, whose Frobenius traces equal (−1)^n−1qΣ_n(h) minus an explicit polynomial in q, for h∈(𝔽_q^×)^n. For odd p, its geometric monodromy has identity component SL_r(n) and cyclic component group of order divisible by p. The full group is {g:det(g)^p=1} for n=2, and for n≥3 using boundary unipotence from the companion preprint arXiv:2610.03081. For p=2, the sheaf is self-dual and its geometric monodromy is SO_r(n) for odd n and Sp_r(n) for even n. After normalization to weight zero, the arithmetic and geometric groups coincide, using the same companion input for odd p and n≥3. As q→∞ with p fixed, we obtain Sato-Tate laws for centered, normalized sums with error O(q^−1/2) for polynomial tests and formulas for all power moments. For p=2, limiting moments are Gaussian through order r(n)−1, with an excess of one above the orthogonal value at order r(n) for odd n; if n+1 is a power of two, every Frobenius on 𝒲_n has eigenvalue q^(n+1)/2. The proof uses ∑_h|Σ_n(h)|^4=c_pq^3n−2+O(q^3n−2−δ(n)) for n≥3, where c_p=2 for odd p, c_2=3 and δ(n)=min(1,(n−2)/2), plane sections of the Cayley cubic, exact Frobenius-Schur identities, Laumon's local Fourier transforms and Artin-Schreier theory. For n=2 we identify twisted classical Kloosterman sums and compute the fourth moment exactly.
ER -