Title: Primality tests, elliptic curves and field theory

Abstract: This talk would be of interest to anyone interested in algebraic varieties, field theory or number theory. We'll start by reviewing Pepin's primality test for Fermat numbers, which are numbers of the form F_n = 2^n+1. This test features a 'universal point' on an algebraic group (a multiplicative group) whose powers determine the primality of the given number. We will show how this is related to a field theory problem involving the splitting of rational primes for quadratic extensions. I'll also describe some directly analogous primality tests for other special types of numbers, which use elliptic curves. The field theory part of the problem is slightly elevated to the more general splitting of primes in abelian extensions. I will be working this summer on a project to develop a primality test that uses results about elliptic curves and non-abelian extensions, which would provide many more opportunities to develop tests for specific types of numbers.