The 3-adic regulators and wild kernels

For any number field, J.-F. Jaulent introduced a new invariant called the group of logarithmic classes in 1994. This invariant is proved to be closely related to the wild kernels of number fields. In this paper, we show how to compute the kernel of the natural homomorphism from the group of logarithmic classes to the group of p-ideal classes by computing the p-adic regulator which is a classical invariant in number theory. As an application, we prove Ganglʹs conjecture on 9-rank of the tame kernel of imaginary quadratic field .