Visual aspects of Gaussian periods and analogues (Q6628927)

The author gave a visual perspective of Gaussian periods(which are certain exponential sums). She studied various conjectures, theorems and their properties that emerge from plotting Gaussian periods in complex plane and generalized the description of Gaussian periods in complex plane, thereby relating to the supercharacter theory and classical field theory.\N\NThe work on Gaussian periods started mainly with works from \textit{J. L. Brumbaugh} et al. [J. Number Theory 144, 151--175 (2014; Zbl 1296.20004); Exp. Math 22, No. 4, 421--442 (2013; Zbl 1294.20013)], \textit{W. Duke} et al. [Proc. Am. Math. Soc. 143, No. 5, 1849--1863 (2015; Zbl 1329.11083)] and \textit{S. R. Garcia} et al. [Notices Am. Math. Soc. 62, No. 8, 878--888 (2015; Zbl 1338.11098)], where they studied Gaussian periods through framework of supercharacter theory. The author generalized results of \textit{W. Duke} et al. [Proc. Am. Math. Soc. 143, No. 5, 1849--1863 (2015; Zbl 1329.11083)] and \textit{S. R. Garcia} et al. [Notices Am. Math. Soc. 62, No. 8, 878--888 (2015; Zbl 1338.11098)] in the following Theorem.\N\NTheorem 1. Let \(n\in\mathbb Z_{\geq2}\) and \(m\in\mathbb Z_{\geq1}\). Suppose \(d\mid(\#GL_m(\mathbb Z/n\mathbb Z)),\) and choose a matrix \(A\in GL_m(\mathbb Z/n\mathbb Z)\) of order \(d\) such that \(\Phi_d(A)=0\) in Mat\(_m(\mathbb Z/n\mathbb Z)\), where \(\Phi_d\) is the \(d\)th cyclotomic polynomial. Let \(\theta_{n,m,A}:(\mathbb Z/n\mathbb Z)^m\rightarrow\mathbb C\) be the cyclic supercharacter corresponding to \(n,m,\) and A. Then img\((\theta_{n,m,A})\) is contained in the image of the Laurent polynomial function \(g_d:\mathbb T^{\phi(d)}\rightarrow\mathbb C\) defined by the following:\N\[\Ng_d(z_1,z_2,\ldots,z_{\phi(d)})=\sum_{k=0}^{d-1}\prod_{j=0}^{\phi(d)-1}{z{_{j+1}}^{c_{jk}}},\N\]\Nwhere the \(c_{jk}\) are given by the relations\N\[\Nx^k\equiv\sum_{j=0}^{\phi(d)-1}c_{jk}x^j\bmod\Phi(d)(x).\N\]\NAdditionally, for a fixed order \(d\), and as \(n\) tends to infinity \(-\) assuming there exists a matrix \(A\in G_m(\mathbb Z/n\mathbb Z)\) such that \(\Phi_d(A)=0\) mod \(n\) \(-\) every nonempty open disk in the image of \(g_d\) eventually contains points in img(\(\theta_{n,m,A}\)). In other words, the image of \(g_d\) is ``filled out'' by cyclic supercharacter plots as the modulus grows without bound.\N\NThe author also introduced a more dynamic perspective to study Gaussian periods. She proved ceratin result which describes and explains certain dynamic behaviours of Gaussian period plots. Also, the number theory and cassical field theory perspective of Gaussian period plots is discussed in this paper. The author gave the another result which computes the Galois group of ray class fields of quadratic imaginary fields over their Hilbert class fields. \N\NFinally, the end section deals with certain open questions and strategies to solve them.