# Algebra/Number Theory Seminar

## Generic Newton polygon for exponential sums in n-variable with parallelotope base

Rufei Ren, University of Rochester

Wednesday, September 13th, 2017
12:00 PM - 1:00 PM
Hylan 1106A

Let p be a prime number. Every n-variable polynomial f(x1,…,xn) over a finite field of characteristic p defines an Artin–Schreier–Witt tower of varieties whose Galois group is isomorphic to Zp. Our goal of this paper is to study the Newton polygon of the L-functions associated to a finite character of Zp and a generic polynomial whose convex hull is an n-paralleltope D. We denote this polygon by GNP(D). We prove a lower bound of GNP(D), which is stronger than the usual Hodge polygon and is called the improved Hodge polygon IHP(D). We conjecture that GNP(D) and IHP(D) are the same. Indeed, when p is larger than a fixed number determined by D, we prove that GNP(D) and IHP(D) coincide at infinitely many points. As a corollary, we deduce that the slopes of GNP(D) roughly form an arithmetic progression with increasing multiplicities.

Event contact: dinesh dot thakur at rochester dot edu