Hodge-stickelberger polygons for L-functions of exponential sums of P(x^s)

Let F_q be a finite field of cardinality q and characteristic p. Let P̄(x) be any one-variable Laurent polynomial over F_q of degree (d₁, d₂) respectively and p | d₁d ₂. For any fixed s ≥ 1 coprime to p, we prove that the q-adic Newton polygon of the L-functions of exponential sums of P̄(x^s) has a tight lower bound which we call Hodge-Stickelberger polygon, depending only on the d₁, d₂, s and the residue class of (p mod s). This Hodge-Stickelberger polygon is a certain weighted convolution of the Hodge polygon for L-function of exponential sums of P̄(x) and the Newton polygon for the L-function of exponential sums of x^s (which is precisely given by the classical Stickelberger theory). We have an analogous Hodge-Stickelberger lower bound for multivariable Laurent polynomials as well. For any v ∈(ℤ/sℤ)^×, we show that there exists a Zariski dense open subset U_v, defined over ℚ such that for every Laurent polynomial P in U_v(ℚ̄) the q-adic Newton polygon of L(P̄(x^s)/F_q; T) converges to the Hodge-Stickelberger polygon as p approaches infinity and p ≡ u mod s. As a corollary, we obtain a tight lower bound for the q-adic Newton polygon of the numerator of the zeta function of an Artin-Schreier curve given by affine equation y^p - y = P̄(x^s). This estimates the q-adic valuations of reciprocal roots of the numerator of the zeta function of the Artin-Schreier curve.