Jordan Ellenberg

Abstract: 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: arXiv.

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.

Here are some directions to Northeastern University. Lake Hall can be best accessed from the entrance on the corner of Greenleaf Street and Leon Street.

There is free parking available for people coming to the Colloquium at Northeastern's visitor parking (Rennaisance Garage). The entrance is from Columbus Avenue. If coming by car, you should park there and take the parking talon. After the lecture, you may pick up the payment coupon from one of NEU colloquium organizers.