Reflection Principles and Bounds for Class Group Torsion

We introduce a new method to bound ℓ-torsion in class groups, combining analytic ideas with reflection principles. This gives, in particular, new bounds for the 3-torsion part of class groups in quadratic, cubic and quartic number fields, as well as bounds for certain families of higher degree fields and for higher ℓ. Conditionally on GRH, we obtain a nontrivial bound for ℓ-torsion in the class group of a general number field.