An introduction to the theory of Newton polygons for L-functions of exponential sums

TL;DR

This paper introduces the theory of Newton polygons for L-functions of exponential sums, providing a systematic decomposition approach that simplifies complex global problems into manageable local cases, confirming conjectures in low dimensions.

Contribution

It develops a flexible collapsing decomposition theorem that advances understanding of Newton polygons and verifies the Adolphson-Sperber conjecture in low dimensions.

Findings

Confirmed the full Adolphson-Sperber conjecture for dimensions n ≤ 3.

Disproved the conjecture for dimensions n ≥ 4.

Provided new decomposition theorems simplifying the analysis of Newton polygons.

Abstract

This expository paper is based on the author's series of lectures delivered at the January 1999 Mini-course in Number Theory, held at Sogang University (Seoul). The aim is to give an elementary and self-contained introduction to the theory of Newton polygons (namely, the $p$-adic Riemann hypothesis) for L-functions of exponential sums over a finite field. The main idea of our approach is to establish a suitable local to global principal which would allowus to reduce the harder ``global'' case to various easier well understood "local" cases for which Stickelberger's theorem applies. For this purpose, a systematic decomposition method was introduced in our earlier paper and several decomposition theorems were proved, including the facial decomposition theorem, the star decomposition theorem and the hyperplane decomposition theorem. These theorems easily recover previously known results on Newton polygons of exponential sums. They are also enough for a number of further applications such as Mazur's conjecture for a generic hypersurface and a weaker form of the more general Adolphson-Sperber conjecture for a generic exponential sum. Using our earlier method, we have recently found a more flexible collapsing decomposition theorem. This theorem gives a fairly satisfactory answer to the full form of the Adolphson-Sperber conjecture. In particular, as an application, we now know that the full form of the Adolphson-Sperber conjecture is true in every low dimension $n\leq 3$ but false in every high dimension $n\geq 4$. These results are explained in these notes.