Jacobi Sums and the Hilbert Symbol for a Power of Two

摘要

Many number theorists have taken up the problem of determining the exactconductor C(sub m, sup (a)) of the Jacobi sum Hecke character alpha since Weil raised its interesting problem in 1952. Recently, Coleman-McCallum determined the exact conductor C(sub m, sup (a)) when m is a power of any odd prime number l, using the arithmetic geometry of Fermant curves, and Miki gave a purely number theoretic proof to their results. But the case l = 2 is still an unsolved more difficult open problem, and it seems that Coleman-McCallum's method is not applicable to the case l = 2, though Coleman (with G. Anderson) gave a partial result by using Ihara-Anderson's theory. The purpose of the present paper is to give the complete determination of the conductor f(sub n)(g, h, s) of the character alpha.