Averages for the l-torsion in class groups

We present new upper bounds for the average size of the l-torsion Cl_K[l] of the class group of K, as K runs through certain natural families of number fields and l is a positive integer. Our results use a refinement of a key argument,
used in most results of this type, which links upper bounds for Cl_K[l] to the existence of many primes splitting completely in K, and that are small compared to the discriminant of K. The improvements are achieved by introducing
a new family of specialised invariants of number fields to replace the discriminant in this key argument, in conjunction with new counting results for these invariants. This is joint work with Christopher Frei.