EVENT DETAILS AND ABSTRACT

Title: Counting number fields by discriminant
Speaker: Jacob Tsimerman
Speaker Info: University of Toronto
Brief Description:
Special Note:
Abstract:

Number fields are fields which are finite extensions of Q. They come with a canonical invariant called the Discriminant, which can be thought of as the volume of a certain canonically associated lattice. While these objects are central to modern number theory, it turns out that counting them is extremely difficult. More precisely, what is the asymptotic behaviour of N(n,X) -- the number of degree n field extensions of Q with discriminant at most X -- as X grows, while n remains fixed? It is conjectured by Linnik that N(n,x)~ c_n X, and this has been proven for n<=5 by Davenport-Heilbronn(n=3) and Bhargava(n=4,5) using the theory of prehomogenous vector spaces. Bhargava has conjectured a precise value for c_n. We will explain these developments, as well as recent joint work with Arul Shankar that gives another proof in the case n=3, and which yields a strong heuristic reason to believe Linniks conjecture with Bhargava's value for c_n. Along the way, we shall also mention the parallel story in function fields, where much more is known thanks to the theory of Etale cohomology.