# Open Collections

## BIRS Workshop Lecture Videos

### Counting points, counting fields, and heights on stacks Zureick-Brown, David

#### 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.