Sign or root of unity ambiguities of certain Gauss sums

作     者：Lingli XIA Jing YANG 

作者机构：Basic Courses Department Beijing Union University Beijing 100101 China Department of Mathematical Sciences Tsinghua University Beijing 100084 China 

出 版 物：《Frontiers of Mathematics in China》 (中国高等学校学术文摘·数学（英文）)

年 卷 期：2012年第7卷第4期

页      面：743-764页

核心收录：

学科分类：07[理学] 08[工学] 070104[理学-应用数学] 0835[工学-软件工程] 0701[理学-数学] 081202[工学-计算机软件与理论] 0812[工学-计算机科学与技术（可授工学、理学学位）] 

基　　金：This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 10990011  11001145  61170289) and the Ph. D. Programs Foundation of Ministry of Education of China (No. 20090002120013) 

主　　题：Gauss sum Teichmiiller characters Stickelberger's congruence Stickelberger's relation 

摘      要：Gauss sums play an important role in number theory and arithmetic geometry. The main objects of study in this paper are Gauss sums over the finite field with q elements. Recently, the problem of explicit evaluation of Gauss sums in the small index case has been studied in several papers. In the process of the evaluation, it is realized that a sign （or a root of unity） ambiguity unavoidably occurs. These papers determined the ambiguities by the congruences modulo L, where L is certain divisor of the order of Gauss sum. However, such method is unavailable in some situations. This paper presents a new method to determine the sign （root of unity） ambiguities of Gauss sums in the index 2 case and index 4 case, which is not only suitable for all the situations with q being odd, but also comparatively more efficient and uniform than the previous method.