Orthogonal decompositions of the space of algebraic numbers modulo torsion

We introduce decompositions determined by Galois field and degree of the space of algebraic numbers modulo torsion and the space of algebraic points on an elliptic curve over a number field. These decompositions are orthogonal with respect to the natural inner product associated to the L² Weil height recently introduced by Allcock and Vaaler in the case of algebraic numbers and the inner product naturally associated to the Néron-Tate canonical height on an elliptic curve. Using these decompositions, we then introduce vector space norms associated to the Mahler measure. For algebraic numbers, we formulate L[superscript p] Lehmer conjectures involving lower bounds on these norms and prove that these new conjectures are equivalent to their classical counterparts, specifically, the classical Lehmer conjecture in the p=1 case and the Schinzel-Zassenhaus conjecture in the p=[infinity] case.