Open Access Publications from the University of California

## Generic Newton polygon for exponential sums in two variables with triangular base

• Author(s): Ren, Rufei
• Xiao, Liang
• et al.
Abstract

Let \$p\$ be a prime number. Every two-variable polynomial \$f(x_1, x_2)\$ over a finite field of characteristic \$p\$ defines an Artin--Schreier--Witt tower of surfaces whose Galois group is isomorphic to \$\mathscr{Z}_p\$.

Our goal of this paper is to study the Newton polygon of the \$L\$-functions associated to a finite character of \$\mathscr{Z}_p\$ and a generic polynomial whose convex hull is a fixed triangle \$\Delta\$. We denote this polygon by \$\GNP(\Delta)\$. We prove a lower bound of \$\GNP(\Delta)\$, which we call the improved Hodge polygon \$\IHP(\Delta)\$, and we conjecture that \$\GNP(\Delta)\$ and \$\IHP(\Delta)\$ are the same. We show that if \$\GNP(\Delta)\$ and \$\IHP(\Delta)\$ coincide at a certain point, then they coincide at infinitely many points.

When \$\Delta\$ is an isosceles right triangle with vertices \$(0,0)\$, \$(0, d)\$ and \$(d, 0)\$ such that \$d\$ is not divisible by \$p\$ and that the residue of \$p\$ modulo \$d\$ is small relative to \$d\$, we prove that \$\GNP(\Delta)\$ and \$\IHP(\Delta)\$ coincide at infinitely many points. As a corollary, we deduce that the slopes of \$\GNP(\Delta)\$ roughly form an arithmetic progression with increasing multiplicities.