Jacobi sums and the Hilbert symbol for a power of two
Let \(J_m^{(a)}(\mathfrak p)\) be the Jacobi sum defined for a tuple of integers \((a)\) and a prime ideal \(\mathfrak p\), not dividing \(m\), of the \(m\)th cyclotomic field. Extending multiplicatively yields a map \(\mathfrak a \mapsto J_m^{(a)}(\mathfrak a)\), which \textit{A. Weil} [Trans. Am. Math. Soc. 73, 487-495 (1952; Zbl 0048.27001)] showed to be a Hecke character of conductor dividing \(m^2\). \textit{R. Coleman} and \textit{W. McCallum} [J. Reine Angew. Math. 385, 41-101 (1988; Zbl 0654.12003)] determined the exact conductor when \(m\) is a power of an odd prime, using the arithmetic of Fermat curves. Recently, the author [\textit{H. Miki}, J. Number Theory 50, 87-105 (1995; Zbl 0829.11053)] gave a purely arithmetic proof of this result using explicit formulas for the norm residue symbol. In the present paper, he extends his methods to include the case where \(m\) is a power of 2. A major step is the determination of the conductor of the character \(\alpha \mapsto (\alpha, 2^g(1+4)^h(-1)^s)_n\), where \(g,s\) are integers, \(h\) is a 2-adic integer, and \(( , )_n\) denotes the norm residue symbol in \(\mathbb Q_2(\zeta_{2^n})\) for the power \(2^n\).
scientific article; zbMATH DE number 952710