Newton polygon of L function of xd + λxd−1 + μx

Abstract

Let f(x)=xd+λxd−1+μx be a trinomial in Fq[x] and χm be a character of order pm. If p>2d and p≡−1(modd), we obtain the q-adic Newton polygon of L⁎(f,χm,s). For μ=0, we give the q-adic Newton polygon of L⁎(f,χm,s) if p>2d under some conditions. We prove that Zp-tower given by xd+λxd−1+∑pnfn(x) is slope stable under certain conditions.