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

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

2007-06-15

arxiv.org/abs/0706.2340v3

Mathematics

Number Theory

18 pages

Scientific paper

Let P(x) be a one-variable Laurent polynomial of degree (d_1,d_2) over a finite field of characteristic p. For any fixed positive integer s not divisible by p, we prove that the (normalized) p-adic Newton polygon of the L-functions of exponential sums of P(x^s) has a tight lower bound which we call `Hodge-Stickelberger polygon', depending only on d_1,d_2,s, and (p mod s). This Hodge-Stickelberger polygon is a weighted convolution of a `Hodge polygon' for L-function of exponential sum of P(x) and the `Newton polygon' for L-function of exponential sum of x^s (given by the classical Stickelberger theory). We prove an analogous Hodge-Stickelberger lower bound for multivariable Laurent polynomials as well. We prove this Hodge-Stickelberger polygon is the limit of generic Newton polygons of P(x^s) in a sense that was made explicit in the paper.

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