(Joint with M. Bhargava, T. Taniguchi, F. Thorne, J. Tsimerman, Y. Zhao)

Given a number field K of fixed degree n over Q,classical theorem of Brauer–Siegel asserts that the size of the class group of K is bounded by cD^(1/2+epsilon), where c is constant depending on epsilon and D is the absolute value of the discriminant of K. However, It is conjectured that the p-torsion subgroup of the class group of K is bounded by a similar bound where we drop 1/2. Only the case n=p=2 of this conjecture in known. In fact, for most pairs (n,p), the best known bounds come from the “convex” Brauer–Siegel bound.

In this talk, we will prove a subconvex bound on the size of the 2-torsion in the class groups of number fields, for all degrees n. We will also discuss an application of this result towards improved bounds on the ranks of elliptic curves.