Number Theory Seminar

Tuesday, Nov. 4
Doug Haessig, U of R
An introduction to Bombieri's proof of an upper bound for the degree of an L-function of exponential sums

ABSTRACT:

In 1965, Enrico Bombieri demonstrated that the L-function associated to exponential sums over a finite field is a rational function, a result already known by Bernard Dwork but never explicitly written down. An immediate consequence of this rationality is an explicit formula for the exponential sums in terms of the zeros and poles of the L-function. This is great news until one learns that information about the zeros and poles is difficult to obtain; for instance, it not even clear how many zeros and poles there are! Fortunately, a very good upper bound for the degree of the L-function was obtained by Bombieri among other things. Amazingly, his ingenious proof is quite short and requires no smoothness conditions, which are typical for degree bounds. In this mostly self-contained talk, we will gently walk through Bombieri's proof using intuition whenever possible.