Distinguishing Gauss sums
Abstract
Let Fq be the field of order q = pf for prime p. If G(χ) is the Gauss sum attached to a multiplicative character χ on Fq×, then G(χ) = G(χp). We investigate the converse question: when does the equality of Gauss sums G(χ2) = G(χ1) imply that χ2 = χ1pj for some integer j. If χ2 = χ1pj, we say that χ1 and χ2 are Frobenius-conjugate. We use the Stickelberger factorization of ideals in cyclotomic fields to give an easily testable criterion for equality of Gauss sums, based on p-adic digit expansion. As an application, we develop several conditions under which Gauss sum equalities between characters on Fq× are explained by Frobenius-conjugacy. For example, if χ1 has order q−1 or 1/2(q−1) and G(χ2) = G(χ1), then χ1 and χ2 are Frobenius-conjugate. We also include several examples, based on the explicit evaluation of certain {\em pure} Gauss sums by R.J. Evans.
Related
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BibTeX
@misc{adrian2026distinguishing,
title = {{Distinguishing Gauss sums}},
author = {Moshe Adrian and Jack Diamond and Kenneth Kramer and Geo Kam-Fai Tam},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28211},
eprint = {2609.28211},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28211},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Distinguishing Gauss sums
AU - Moshe Adrian
AU - Jack Diamond
AU - Kenneth Kramer
AU - Geo Kam-Fai Tam
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28211
UR - https://arxiv.org/abs/2609.28211
AB - Let F_q be the field of order q = p^f for prime p. If G(χ) is the Gauss sum attached to a multiplicative character χ on F_q^×, then G(χ) = G(χ^p). We investigate the converse question: when does the equality of Gauss sums G(χ_2) = G(χ_1) imply that χ_2 = χ_1^p^j for some integer j. If χ_2 = χ_1^p^j, we say that χ_1 and χ_2 are Frobenius-conjugate. We use the Stickelberger factorization of ideals in cyclotomic fields to give an easily testable criterion for equality of Gauss sums, based on p-adic digit expansion. As an application, we develop several conditions under which Gauss sum equalities between characters on F_q^× are explained by Frobenius-conjugacy. For example, if χ_1 has order q−1 or 1/2(q−1) and G(χ_2) = G(χ_1), then χ_1 and χ_2 are Frobenius-conjugate. We also include several examples, based on the explicit evaluation of certain {\em pure} Gauss sums by R.J. Evans.
ER -