Norm-trace and Kloosterman sums in finite semi-simple algebras
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914454792306688 |
|---|---|
| author | Wan, Daqing |
| author_facet | Wan, Daqing |
| contents | An asymptotic formula with a square root error term is obtained for the number of elements with given trace and norm in a finite semisimple algebra over a finite field. This extends previous results from finite etale algebras (commutative case) to finite semi-simple algebras (non-commutative case). The main idea is to apply the Eichler formula for Gauss sums over the general linear group and the Hasse-Davenport relation to reduce the problem to the classical geometric case where the result is known to be true. As an application of this reduction, we also obtain a square root estimate for Kloosterman sums over semi-simple algebras. Similar square root estimates are discussed when norm-trace is replaced by product-trace, leading to a new conjecture on product-trace counting over finite semi-simple algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_18511 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Norm-trace and Kloosterman sums in finite semi-simple algebras Wan, Daqing Number Theory 11T24, 11G25, 11L40 An asymptotic formula with a square root error term is obtained for the number of elements with given trace and norm in a finite semisimple algebra over a finite field. This extends previous results from finite etale algebras (commutative case) to finite semi-simple algebras (non-commutative case). The main idea is to apply the Eichler formula for Gauss sums over the general linear group and the Hasse-Davenport relation to reduce the problem to the classical geometric case where the result is known to be true. As an application of this reduction, we also obtain a square root estimate for Kloosterman sums over semi-simple algebras. Similar square root estimates are discussed when norm-trace is replaced by product-trace, leading to a new conjecture on product-trace counting over finite semi-simple algebras. |
| title | Norm-trace and Kloosterman sums in finite semi-simple algebras |
| topic | Number Theory 11T24, 11G25, 11L40 |
| url | https://arxiv.org/abs/2603.18511 |