On exponential sums of xd + λxe with p ≡ e(mod d)

Abstract

Let ψ be a character of Zp of order pm, and f(x)=xd+λxe be a binomial of degree d with (d,e)=1. The determination of the Newton slopes of the L-functions Lf,ψ(s) is interesting and still open for general d,e that coprime. If p≡e(modd) is large enough, an arithmetic polygon Pe,d is defined and shown to be the lower bound for the classical (ψ(1)−1)a(p−1)-adic Newton polygon of Lf,ψ(s). In addition, we show they coincide when e=2 for large p, hence the Newton slopes of Lf,ψ(s) are determined. Combining Ouyang-Zhang's results on e=d−1 and p≡d−1(modd), we conjecture Pe,d coincides with (ψ(1)−1)a(p−1)-adic Newton polygon of Lf,ψ(s) for all e if p≡e(modd) is large enough.