HODGE-STICKELBERGER POLYGONS FOR L-FUNCTIONS OF EXPONENTIAL SUMS OF P (x)

Let Fq be a finite field of cardinality q and characteristic p. Let P (x) be any one-variable Laurent polynomial over Fq of degree (d1, d2) respectively and p d1d2. For any fixed s ≥ 1 coprime to p, we prove that the q-adic Newton polygon of the L-functions of exponential sums of P (xs) has a tight lower bound which we call HodgeStickelberger 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 P (x) and the Newton polygon for the L-function of exponential sums of xs (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 ν ∈ (Z/sZ)×, we show that there exists a Zariski dense open subset Uν defined over Q such that for every Laurent polynomial P in Uν(Q) the q-adic Newton polygon of L(P (x)/Fq ;T ) converges to the Hodge-Stickelberger polygon as p approaches infinity and p ≡ ν 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 yp−y = P (xs). This estimates the q-adic valuations of reciprocal roots of the numerator of the zeta function of the Artin-Schreier curve.