  Jupiter Trojans are are characterized by dark photometric colors, high
inclinations and an asymmetry in number of bodies between the two swarms.
Different models have been proposed to explain the high inclination of the
Trojans and to interpret their relation with the TNOs, but none of them can
also satisfactorily explain the asymmetry ratio. Recently it has been found
that the asymmetry can arise if Jupiter has migrated inwards by at least a few
au during its growth. The asymmetry and the dark colors of the Trojans are
natural outcomes of this model, but simulations with massless unperturbed disc
particles led to a flat distribution of the Trojan inclinations and a final
total mass that was 3-4 orders of magnitude larger than the current one. In our
work, we investigate the possible origin of the peculiar inclination
distribution of the Trojans in the scenario where Jupiter migrates inwards. We
analyze different possibilities: (a) the secular evolution of an initially flat
Trojan population, (b) the presence of planetary embryos among the Trojans and
(c) capture of the Trojans from a pre-stirred planetesimal population. We find
that the secular evolution of the Trojans and Saturn do not affect the Trojan
inclinations appreciably, nor is there any significant mass depletion. Embryos
embedded in the swarms, in contrast, can stir the Trojan inclinations and can
also deplete the swarms efficiently, but it turns out that it is very difficult
to get rid of all of the massive bodies. We propose that the disc where
Jupiter's core was forming was already excited by the presence of other embryos
competing in Jupiter's core's feeding zone. We show that the trapped Trojans
preserve their high inclination through the gas phase of the disc and that
Saturn's perturbations are more effective on highly inclined Trojans, leading
to a lower capture efficiency and to a substantial depletion of the swarms.
