Riemann hypothesis for the zeta function of a function field over a finite field
Marie Brilland Yann Ranorovelonalohotsy · SUNScholar (Stellenbosch University) · 2013
Riemann Hypothesis for the Zeta Function of a Function Field over a Finite Field Marie Brilland Yann Ranorovelonalohotsy Department of Mathematical Sciences, University of Stellenbosch, Private Bag X1, Matieland 7602, South Africa. Thesis: MSc (Math) December 2013 Let K be a function eld over a nite eld. Fix a place (∞) of K, which we shall call the prime at in nity. We consider the ring A = {y ∈ K : y is regular at P for every place P 6= (∞)}, which we call the ring of integers of K with respect to (∞). There is a bijection between the set of proper ideals of A and the places of K di erent from (∞). We de ne the zeta function ζA(s) for the ring A in a way analogous to the Dedekind zeta function of the ring of integers of a number eld. The analogue of the Riemann Hypothesis for ζA(s) was rst proved by Andre Weil in 1948, and our goal is to give an exposition of a simpler proof of this theorem due to Enrico Bombieri. ii Stellenbosch University http://scholar.sun.ac.za