Christopher Hooley Lecture Series 201226 Feb 2016, by Sponsored events in
10 – 19 January 2012
Organisers: Oliver Johnson
Exponential sums over finite fields and allied topics from a slightly personal point of view
Exponential sums over finite fields (and especially those involving summations over residues, modulo a prime number, or their extensions to general moduli) both historically and currently play an important part in the theory of numbers.
In this lecture series we shall develop some of the more interesting and important aspects of exponential sums including the special case involving the number of solutions of equations over finite fields. Beginning with a classical bias, we shall bear in mind how our results presage the results of Dwork and Deligne, which it will be the object of the later part of the series to explain and exploit.
In the first tranche of lectures, after an introduction, we shall amongst other things discuss quadratic and cubic congruences and their treatment through exponential sums and geometrical methods. We shall also consider the applications of the results obtained and their relevance to laws of higher reciprocity.
The prerequisites of the course are modest. For the most part, only a knowledge of elementary number theory and some acquaintance with finite fields will be needed.
Six lectures in this series: January 10, 11, 12, 17, 18, 19