On Shioda's problem about Jacobi sums II

Abstract

In the present paper, we will give a complete affirmative answer to the l-part of Shioda’s problem ([5, Question 3.4]) on Jacobi sums J (a) l (p), and to the conjecture (F. Gouvea and N. Yui [1, Conjecture (1.9)]) which comes from Shioda’s problem and my congruences for Jacobi sums (see [3, Theorem 2]) (see Theorem 1 and its Corollary of the present paper). We retain the notation of [4], but l is any odd prime number here. Furthermore, let n be any positive integer and let ζm be a primitive mth root of unity in C for any positive integer m. Let Q be the algebraic closure of Q in C. We fix an algebraic closure Ql of Ql, and by a fixed imbedding Q ↪→ Ql we consider Q as a subfield of Ql. Let M be any finite unramified extension of Ql in Ql, and put Mn = M(ζln) and πn = ζln − 1. Then πn is a prime element of Mn. Let σ−1 ∈ G = Gal(Mn/M) (the Galois group of Mn over M) be such that ζ σ−1 ln = ζ −1 ln . Let ordMn denote the normalized additive valuation of Mn, and let Un = U(Mn) be the group of principal units in Mn: Un = U(Mn) = {x ∈Mn | ordMn(x− 1) ≥ 1}.