# Counting points, counting fields, and heights on stacks

### Counting points, counting fields, and heights on stacks

Series: Algebra and Number Theory Seminar
Location: ENR2 S395
Presenter: Jordan Ellenberg, University of Wisconsin

The basic objects of algebraic number theory are number fields, and the basic invariant of a number field is its discriminant, which in some sense measures its arithmetic complexity.  A basic finiteness result is that there are only finitely many degree-d number fields of discriminant at most X; more generally, for any fixed global field K, there are only finitely many degree-d extensions L/K whose discriminant has norm at most X.  (The classical case is where K = Q.)

When a set is finite, we greedily ask if we can compute its cardinality.  Write N_d(K,X) for the number of degree-d extensions of K with discriminant at most d.  A folklore conjecture holds that N_d(K,X) is on order c_d X.  In the case K = 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.

I’ll talk about two recent results on this old problem:

1) (joint with TriThang Tran and Craig Westerland)  We prove that N_d(F_q(t),X)) < c_eps X^{1+eps} for all d, and similarly prove Malle’s conjecture “up to epsilon” — this is much more than is known in the number field case, and relies on a new upper bound for the cohomology of Hurwitz spaces coming from quantum shuffle algebras:  https://arxiv.org/abs/1701.04541

2) (joint with Matt Satriano and David Zureick-Brown) The form of Malle’s conjecture is very 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 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.