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arXiv2026

Distinguishing Gauss sums

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 $\F_q^\times$, 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 $\F_q^\times$ are explained by Frobenius-conjugacy. For example, if $χ_1$ has order $q-1$ or $\frac{1}{2}(q-1)$ 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.

math.NT↗

arXiv2016

P-adic Line Integrals and Cauchy's Theorems

Working in the p-adic analog of the complex numbers, we'll define a line integral on a small arc of a circle. This allows new versions of the Residue Theorem, the Cauchy-Goursat Theorem on discs with and without holes, Cauchy's Integral Formula and the Z-P Theorem. In contrast to results in complex analysis, these integrals allow the points on a boundary circle, the bulk of a p-adic disc, to be treated the same as points interior to the boundary circle. The theory of the integral is developed, especially for functions holomorphic on an open disc, and integrals will be calculated for rational functions, Krasner analytic functions and some well-known functions that are not Krasner analytic. Some computations will produce values of Kubota-Leopoldt L-functions at ordinary integers.

math.NT↗