Asymptotic charges of $p-$forms and their dualities in any $D$

We compute the surface charges associated to $p-$form gauge fields in arbitrary spacetime dimension for large values of the radial coordinate. In the critical dimension where radiation and Coulomb falloff coincide we find asymptotic charges involving asymptotic parameters, i.e. parameters with a component of order zero in the radial coordinate. However, in different dimensions we still find nontrivial asymptotic charges now involving parameters that are not asymptotic times the radiation-order fields. For $p$=1 and $D>4$, our charges thus differ from those presented in the literature. We then show that under Hodge duality electric charges for $p-$forms are mapped to magnetic charges for the dual $q-$forms, with $q = D-p-2$. For charges involving fields with radiation falloffs the duality relates charges that are finite and nonvanishing. For the case of Coulomb falloffs, above or below the critical dimension, Hodge duality exchanges overleading charges in one theory with subleading ones in its dual counterpart.