The number of $D_4$-extensions of $\mathbb{Q}$

Abstract: We will begin with a summary of how Malle's conjecture and Bhargava's heuristics can be used to develop the "Malle--Bhargava heuristics", predicting the asymptotics in families of number fields, ordered by a general class of invariants.

We will then specialize in the case of $D_4$-number fields. Even in this (fairly simple) case, where the fields can be parametrized quite explicitly, the question of determining asymptotics can get quite complicated. We will discuss joint work with Altug, Varma, and Wilson, in which we recover asymptotics when quartic $D_4$ fields are ordered by conductor. And we will finally discuss joint work with Varma, in which we recover Malle's conjecture for octic $D_4$-fields.

number theory

Audience: researchers in the topic

Heilbronn number theory seminar

Series comments: This is part of the University of Bristol's Heilbronn number theory seminar. If you wish to attend the talk (and are not a Bristolian), please register using this form or email us at bristolhnts@gmail.com with your name and affiliation (if any).

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