Generic twisted T-adic exponential sums of binomials

Abstract

The twisted T -adic exponential sum associated with x d + λx is studied. If λ = 0, then an explicit arithmetic polygon is proved to be the Newton polygon of the C-function of the twisted T -adic exponential sum. It gives the Newton polygons of the L-functions of twisted p-power order exponential sums.

Newton polygons of $L$-functions of polynomials $x^d+ax^{d-1}$ with $p\equiv-1\bmod d$

“…For f (x) = x d + ax, Zhu, Liu-Niu and Ouyang-J. Yang obtained the slopes in [Z2, Theorem 1.1], [LN1,Theorem 1.10] and [OY,Theorem 1.1], see also R. Yang [Y,§1 Theorem] for earlier results.…”

On the Newton polygons of twisted $L$-functions of binomials

“…For arbitrary u, Liu-Niu [LN11] obtained the Newton polygons when e = 1. Zhang-Niu [ZN21] also give a conjectural description of the Newton polygons when p ≡ e mod d.…”