Matrix Kloosterman sums and product-trace estimates for semisimple algebras

Abstract

Let k=𝔽_q, E=𝔽_qⁿ and Tr=Tr_E/k. For r≥ 2, a∈ k^× and x∈ E^×, let N(E,r,x,a) be the number of r-tuples (x₁,⋯,x_r) in (E^×)^r satisfying x₁⋯ x_r=x and Tr(x₁+⋯+x_r)=a. We prove |N(E,r,x,a)−((qⁿ−1)^r−1+(−1)^r)/q|≤ (rⁿ−1) q^{((r−1)n−1)/2}. This proves the square-root estimate predicted in Wan's conjecture and generalizes a previous result of Moisio and Wan. For a finite semisimple algebra B=∏_i=1^s M_{d_i}(𝔽_{q^n_i}) over k and a regular element x∈ B^×, the same method combined with Zelingher's formula leads to analogous square-root estimates.

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BibTeX

@misc{guo2026matrix,
  title = {{Matrix Kloosterman sums and product-trace estimates for semisimple algebras}},
  author = {Xuejun Guo and Chen Lin and Chenhao Tang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.23275},
  eprint = {2607.23275},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.23275},
}

RIS

TY  - UNPB
TI  - Matrix Kloosterman sums and product-trace estimates for semisimple algebras
AU  - Xuejun Guo
AU  - Chen Lin
AU  - Chenhao Tang
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2607.23275
UR  - https://arxiv.org/abs/2607.23275
AB  - Let k=𝔽_q, E=𝔽_q^n and Tr=Tr_E/k. For r≥ 2, a∈ k^× and x∈ E^×, let N(E,r,x,a) be the number of r-tuples (x_1,⋯,x_r) in (E^×)^r satisfying x_1⋯ x_r=x and Tr(x_1+⋯+x_r)=a. We prove |N(E,r,x,a)−((q^n−1)^r−1+(−1)^r)/q|≤ (r^n−1) q^((r−1)n−1)/2. This proves the square-root estimate predicted in Wan's conjecture and generalizes a previous result of Moisio and Wan. For a finite semisimple algebra B=∏_i=1^s M_d_i(𝔽_q^n_i) over k and a regular element x∈ B^×, the same method combined with Zelingher's formula leads to analogous square-root estimates.
ER  -

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