BSD implies a non-trivial bound on 5-torsion in class groups of imaginary quadratic fields

Abstract

We prove new conditional bounds on the the p-torsion of class groups of quadratic fields for p = 3, 5. Our methods first recast the problem in the language of Selmer groups of Galois modules, which allows us to relate these groups to Selmer groups of elliptic curves. We then obtain a global estimate using the refined BSD conjecture, in a similar way to how one normally uses the Brauer-Siegel bound.Our methods are potentially very general, but rely on the existence of motives with very special Fp-cohomology. In particular, the restriction to p = 3, 5 stems from needing an elliptic curve over Q with trivial p-torsion.