### An Introduction to the Theory of Newton Polygons for L-functions of Exponential Sums, by Daqing Wan

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. In addition to
giving a thorough treatment of the basic elementary local theory, we
also describe the deeper global theory and include some explicit examples
to illustrate how to use the main theorems.
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.

Daqing Wan <dwan@math.uci.edu>