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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, proved by Hermite at the end of the 19th century, is that there are only finitely many degree-d number fields of discriminant at most X. It thus makes sense to put all the number fields in order of their discriminant, and ask if we can say how many youâve encountered by the time you get to discriminant X.
This is an old problem, governed by a conjecture of Narkiewicz. Interest in this area was revitalized by the work of Bhargava; the first step in his program was to count number fields of degree 4 and 5. (Degree 6 remains completely out of reach!) Iâll talk about the long history of this problem and its variants, and discuss two recent results:
1) (joint with TriThang Tran and Craig Westerland) We prove that the upper bound conjectured by Narkiewicz is true âup to epsilon" when Q is replaced by a rational function field F_q(t) â 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) Another much-studied counting problem in number theory is the Batyrev-Manin conjecture, which asks about the number of rational points on a variety of bounded height, or, in more concrete terms, questions like:
âHow many solutions does an equation like x^3 + y^3 + z^3 + w^3 = 0 have in integers of absolute value at most X?â
It turns out thereâs a way to synthesize the Narkiewicz conjecture and the Batyrev-Manin conjecture into a unified heuristic which includes both of those conjectures as special cases, and which says much more in general. This involves defining âthe height of a rational point on an algebraic stackâ and I will say as much about what this means as thereâs time to!