Twisted T -adic Exponential Sums

The L-function and C-function of twisted T -adic exponential sums are defined. The Hodge bound for the T -adic Newton polygon of the C-function is established. As an application, the T -adic Newton polygons of the L-functions of twisted p-power order exponential sums associated to diagonal forms are explicitly given. 1. Preliminaries Let Fq be the field of characteristic p with q elements, and Zq = W (Fq). Let T and s be two independent variables. In this section we are concerned with the ring Zq[[T ]][[s]], elements of which are regarded as power series in s with coefficients in Zq[[T ]]. Let Qp = Zp[ 1 p ], Qp the algebraic closure of Qp, and Q̂p the p-adic completion of Qp. Definition 1.1. A (vertical) specialization is a morphism T 7→ t from Zq[[T ]] into Q̂p with 0 6= |t|p < 1. We shall prove the vertical specialization theorem. Theorem 1.2 (Vertical specialization). Let A(s, T ) ∈ 1 + sZq[[T ]][[s]] be a T -adic entrie series in s. If 0 6= |t|p < 1, then t− adic NP of A(s, t) ≥ T − adic NP of A(s, T ), where NP is the short for Newton polygon. Moreover, the equality holds for one t iff it holds for all t. By the vertical specialization, the Newton polygon of a T -adic entire series in 1+sZq[[T ]][[s]] goes up under vertical specialization, and is stable under all specializations if it is stable under one specialization. Definition 1.3. A T -adic entire series in 1 + sZq[[T ]][[s]] is said to be stable if its Newton polygon is stable under specialization. Definition 1.4 (Tensor product). Let A(s, T ), B(s, T ) ∈ 1+sZq[[T ]][[s]] be two T -adic entire power series in s. If A(s, T ) = ∏ α(1− αs), and B(s, T ) = ∏ β(1− βs), we define A⊗B(s, T ) = ∏