On successive minima and the Absolute Siegel's Lemma

We prove a new bound on the product of the nth successive minimum of an automorphism of GL(N,kA) and the (N−n+1)th successive minimum of its dual, extending a classical inequality of K. Mahler for polar lattices to the geometry of numbers over the adèles. As a corollary we derive the Absolute Siegel's Lemma in as stated by D. Roy and J.L. Thunder.