p-adic fields

4:00 pm Thursday, April 9, 2015

Ramin Takloo-Bighash (University of Illinois at Chicago)

The aim of arithmetic statistics is understanding the distribution of number theoretic objects of growing arithmetic complexity. For example, one might like to know how many number fields there are with bounded discriminant; or how many elliptic curves with rank bounded by a growing quantity X. A problem that has gained some interest in recent years is understanding the distribution of integral rings, or orders in number theoretic language. In this talk I will report on recent works on this problem. In particular, I will sketch the proof of an asymptotic formula for the number of orders of bounded discriminant in a given quintic number field, emphasizing the role played by the model theory of p-adic fields. This is joint work with Nathan Kaplan (Yale) and Jake Marcinek (undergrad, Caltech).