A STICKLEBERGER THEOREM FOR p-ADIC GAUSS SUMS

Character sums over finite fields have been widely studied since Gauss. During the last decade, coding theorists initiated the study of a new type of sums, over points with coordinates in a p-adic field. Let O (u) m be the ring of integers of Q p(ζpm−1), the unramified extension of degree m of the field of p-adic numbers Q p obtained by adjoining the (p−1)-th roots of unity, Tm = T ∗ m∪{0}, where T ∗ m is the multiplicative subgroup of elements of finite order in O (u)∗ m . Many character sums over finite fields can be extended to this situation ; the first studied was ∑ x∈Tm ψ̃l,m(f(x)), ψ̃l,m an additive character of order p l over O (u) m , f a polynomial in O (u) m [X ], in order to give a generalization of the Weil-Carlitz-Uchiyama bound.