Seminar

There are many questions in number theory and arithmetic geometry of the sort “Does the following situation ever occur?” For instance, the inverse Galois problem asks whether every finite group occurs as the Galois group of an extension of the rationals. Similarly, one might ask whether one expects the rank of elliptic curves to be unbounded. 

Arithmetic statistics, broadly speaking, pursues the more quantitative question of how often such situations occur. The extension of the inverse Galois problem to this setting is a conjecture of Malle’s, which predicts an asymptotic formula for the number of occurrences of a given finite group G as the Galois group of a number field, as a function of the discriminant.  There are analogous statistical conjectures regarding the distribution of class groups ordered by discriminant (e.g., the Cohen-Lenstra heuristics), or the rank of elliptic curves ordered by height (Katz-Sarnak).

In this talk, we will give an introduction to these sort of questions, focusing on Malle’s conjecture. Additionally, we will explain how to formulate function field analogues of this conjecture and transform this conjecture into a problem in algebraic topology (about the homology of certain moduli spaces of branched covers of P^1). In joint work with Ellenberg and Tran, we partially solved this problem, giving the upper bound in Malle’s conjecture.