Counting number fields of bounded discriminant
by
Jordan Ellenberg
Princeton University
Coauthors: Akshay Venkatesh

An old conjecture holds that the number N_n(X) of degree n number fields with discriminant less than X is asymptotic to c_n X when X grows and n is fixed. This conjecture was proved by Davenport and Heilbronn for n = 3, and recently for n = 4,5 by Bhargava. For general n, however, the best known upper bound, due to Schmidt, was N_n(X) << X^{(n+2)/4}. We prove the much stronger bound N_n(X) << X^{exp(C sqrt(log n))}. While the theorem appears number-theoretic, the arithmetic input is minimal, and the method is purely algebro-geometric; the main idea is to relate the problem of counting number fields to the problem of counting integral points on certain carefully chosen varieties related to Hilbert schemes of points on affine space. We will also describe function-field analogues of the main theorem, and speculate about relations between the theorems proved here and the Batyrev-Manin heuristics for rational points on Fano varieties.

Date received: May 18, 2004