We study lateral tunneling through a quantum box including electron-el
ectron interactions in the presence of a magnetic field which breaks s
ingle particle degeneracies. The conductance at zero temperature as a
function of the Fermi energy in the leads consists of a set of peaks r
elated to changing by one the electron occupancy in the box. We find t
hat the position and heights of the peaks are controlled by many-body
effects. We compute the conductance up to 8 electrons for several case
s where correlation effects dominate. In the range of intermediate fie
lds spin selection rules quench some peaks. At low and high fields the
behavior of the conductance as a function of the number of electrons
is very different due to big changes in the many-body ground state wav
efunctions.