Vaaler, Jeffrey D.2010-10-202010-10-202017-05-112010-10-202010-10-202017-05-112010-05May 2010http://hdl.handle.net/2152/ETD-UT-2010-05-1416textWe 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.application/pdfengAlgebraic numbersWeil heightMahler measureLehmer's problemthesis2010-10-20