Metaplectic Eisenstein Series on GL(3)

Let F be a global field with ring o of integers, and let μn denote the group of n-th roots of unity in F . Let Γ(f) be the principal congruence subgroup of SL3(o) consisting of elements that are congruent to the identity modulo f. The Kubota symbol κ : Γ(f) −→ μn is a character constructed by Bass, Milnor and Serre [1]. We will give a direct construction, obtaining on the way new formulas for the map. We will handle two cases simultaneously. Case 1: n = 2, F = Q(i), o = Z[i], λ = 1 + i and f = λo. Case 2: n = 3, F = Q(ρ) where ρ = e, o = Z[ρ], λ = 1− ρ, and f = λo = 3o. For these two fields the particular level f may be new. Although we specialize to these particular fields, our formulas should be correct (for some level, with perhaps some other minor modifications) when n > 1 is arbitrary, assuming that F is a totally complex field containing the n-th roots of unity such that −1 is an n-th power in F . We specialize to these particular cases since it is convenient that the class number is 1 and the level f can be chosen so that the map o× −→ (o/f)× is surjective. If c and d are in o and gcd(d, f) = 1 the power residue symbol ( c d ) is defined as follows. First, if c and d are not coprime then ( c d ) = 0. If d = p is prime, then ( c d ) is the unique n-th root of unity such that c(Np−1)/n ≡ ( c