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On exponential sums of xd + λxe with p ≡ e(mod d)
Finite Fields and Their Applications ( IF 1.2 ) Pub Date : 2021-08-10 , DOI: 10.1016/j.ffa.2021.101907
Qingjie Zhang 1 , Chuanze Niu 1
Affiliation  

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 pe(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(p1)-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=d1 and pd1(modd), we conjecture Pe,d coincides with (ψ(1)1)a(p1)-adic Newton polygon of Lf,ψ(s) for all e if pe(modd) is large enough.



中文翻译:

在 xd + λxe 的指数和与 p ≡ e(mod d)

ψ是一个字符Z 按顺序 , 和 F(X)=Xd+λX电子d 次的二项式,其中(d,电子)=1. L函数的牛顿斜率的确定F,ψ() 很有趣,仍然对一般人开放 d,电子那个互质。如果电子(模组d) 足够大,算术多边形 电子,d 被定义并显示为经典的下界 (ψ(1)-1)一种(-1)-adic Newton 多边形 F,ψ(). 此外,我们证明它们重合时电子=2对于大p,因此牛顿斜率F,ψ()被确定。结合欧阳张的结果电子=d-1d-1(模组d),我们推测 电子,d(ψ(1)-1)一种(-1)-adic Newton 多边形 F,ψ()对于所有e if电子(模组d) 足够大。

更新日期:2021-08-10
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