Open Collections

Description

A folklore conjecture is that the number $N_d(K,X)$ of degree-$d$ extensions of $K$ with discriminant at most $d$ is on order $c_d X$. In the case $K = \mathbb{Q}$, this is easy for $d=2$, a theorem of Davenport and Heilbronn for $d=3$, a much harder theorem of Bhargava for $d=4$ and $5$, and completely out of reach for $d > 5$. More generally, one can ask about extensions with a specified Galois group $G$; in this case, a conjecture of Malle holds that the asymptotic growth is on order $X^a (log X)^b$ for specified constants $a,b$. The form of Malle's conjecture is reminiscent of the Batyrev--Manin conjecture, which says that the number of rational points of height at most $X$ on a Batyrev-Manin variety also grows like $X^a (log X)^b$ for specified constants $a,b$. What's more, an extension of $\mathbb{Q}$ with Galois group $G$ is a rational point on a Deligne-Mumford stack called $BG$, the classifying stack of $G$. A natural reaction is to say "the two conjectures is the same; to count number fields is just to count points on the stack $BG$ with bounded height" The problem: there is no definition of the height of a rational point on a stack. I'll explain what we think the right definition is, and explain how it suggests a heuristic which has both the Malle conjecture and the Batyrev--Manin conjecture as special cases. This is joint with Jordan Ellenberg and Matt Satriano.