Orthogonal decompositions of the space of algebraic numbers modulo torsion

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Title: Orthogonal decompositions of the space of algebraic numbers modulo torsion
Author: Fili, Paul Arthur
Abstract: 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 .
URI: http : / /hdl .handle .net /2152 /ETD -UT -2010 -05 -1416
Date: 2010-10-20

Citation

Orthogonal decompositions of the space of algebraic numbers modulo torsion. Doctoral dissertation, University of Texas at Austin. Available electronically from http : / /hdl .handle .net /2152 /ETD -UT -2010 -05 -1416 .

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