Frobenius equidistribution in families of number fields

Abstract: I will describe the notion of Frobenius equidistribution (also called Sato--Tate equidistribution) in families of $S_n$-number fields. In the fundamental case of families of -number fields, this equidistribution is only known in the case $n=3$, due to Davenport--Heilbronn, as well as $n=4$ and $n=5$, due to Bhargava. Moreover, assuming this equidistribution, Bhargava uses the Serre mass formula to develop heuristics for the asymptotics of $S_n$-number fields, when they are ordered by discriminant. I will then discuss results in two different directions. In the first direction, we consider the following question: what do we expect to happen when we order fields by natural invariants other than the discriminant? To shed some light on this, I will describe joint work with Frank Thorne in which we give a complete answer in the case of cubic fields. Second, I will describe joint work with Jacob Tsimerman, in which we develop heuristics which give evidence for Frobenius equidistribution in families of all number fields.

number theory

Audience: researchers in the topic

Around Frobenius Distributions and Related Topics IV

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