WENTS Seminar: An equivariant class number formula by more elementary methods

In 2009, Büyükboduk published a result in Crelle which expressed the order of the $\chi$-part of the class-group of a number field ($\chi$ a $p$-adic character of a Galois group) in terms of $L$-functions and regulators. The proof used Kolyvagin systems, a deep theory that is seeing many applications. However, the key assumption in that theorem is currently known to hold in only a very small number of situations. We will do three things:

(i) Replace the aforementioned assumption by one which is known to hold more often.

(ii) Prove the result by more elementary methods, without using Kolyvagin systems.

(iii) Discuss why another hypothesis can be weakened as well, allowing more primes $p$ to be considered for any given Galois group.