Generic Newton polygon for exponential sums in two variables with triangular base

2017/01/01 by Ren, Rufei
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1701.00254

Abstract

Let p be a prime number. Every two-variable polynomial f(x₁, x₂) over a finite field of characteristic p defines an Artin--Schreier--Witt tower of surfaces whose Galois group is isomorphic to \mathbb Z_p. Our goal of this paper is to study the Newton polygon of the L-functions associated to a finite character of ℤ_p and a generic polynomial whose convex hull is a fixed triangle Δ. We denote this polygon by \textrmGNP(Δ). We prove a lower bound of \textrmGNP(Δ), which we call the improved Hodge polygon \textrmIHP(Δ), and we conjecture that \textrmGNP(Δ) and \textrmIHP(Δ) are the same. We show that if \textrmGNP(Δ) and \textrmIHP(Δ) coincide at a certain point, then they coincide at infinitely many points. When Δ is an isosceles right triangle with vertices (0,0), (0, d) and (d, 0) such that d is not divisible by p and that the residue of p modulo d is small relative to d, we prove that \textrmGNP(Δ) and \textrmIHP(Δ) coincide at infinitely many points. As a corollary, we deduce that the slopes of \textrmGNP(Δ) roughly form an arithmetic progression with increasing multiplicities.

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