Generic A-family of exponential sums

Abstract

Let s→:=(s1,s2,…,sm) with s1<⋯<sm being positive integers. Let A(s→) be the space of all 1-variable polynomials f(x)=∑ℓ=1maℓxsℓ parameterized by coefficients a→=(a1,…,am) with am≠0. We study the p-adic valuation of the roots of the L-function of exponential sum of f¯ for modulo p reduction of any generic point f∈A(s→)(Q¯). Let NP(f¯) be the normalized p-adic Newton polygon of the L function of exponential sums of f¯. Let GNP(A(s→),F¯p) be the generic Newton polygon for A(s→) over F¯p, and let HP(A(s→)):=NPp(∏i=1d−1(1−pidT)) be the absolute lower bound of NP(A(s→)). One knows that NP(f¯)≺GNP(A(s→);F¯p)≺HP(f¯) for all prime p and for all f¯∈A(s→)(Q¯), and these equalities hold only when p≡1modd. In the case s→=(s,d) with s<d coprime we provide a computational method to determine GNP(A(s,d),F¯p) explicitly by constructing its generating polynomial Hr∈Q[Xr,1,Xr,2,…,Xr,d−1] for each residue class p≡rmodd. For p≡rmodd (with 2≤r≤d−1 coprime to d) large enough Hr=∑n=1d−1hr,n,kr,nXr,nkr,n with ∏n=1d−1hr,n,kr,n≠0 if and only if GNP(A(s,d),F¯p) has its breaking points after the origin at((n,n(n+1)2d+(1−sd)kr,np−1))n=1,2,…,d−1. If a≠0 then for any f=xd+axs∈A(s,d)(Q¯) and for any prime p≡rmodd large enough we have that NP(f¯)=GNP(A(s,d),F¯p) andlimp→∞NP(f¯)=HP(A(s,d)).Our method applies to compute the generic Newton polygon of Artin–Schreier family yp−y=xd+axs parameterized by a for p large enough.