A Variation on Leopoldt's Conjecture: Some Local Units instead of All Local Units

2013/08/21 by Dawn Nelson, Dawn C. Nelson, Nelson, Dawn C.
Mathematics · #Algebraic Geometry and Number Theory #Meromorphic and Entire Functions #advanced mathematical theories #math.NT #msc:11R27 #msc:11R99

paper · pdf · doi:10.48550/arxiv.1308.4637

23 pages, submitted

arxiv created 2013/08/21 · arxiv updated 2013/08/22

Abstract

Leopoldt's Conjecture is a statement about the relationship between the global and local units of a number field. Approximately the conjecture states that the Z_p-rank of the diagonal embedding of the global units into the product of all local units equals the Z-rank of the global units. The variation we consider asks: Can we say anything about the Z_p-rank of the diagonal embedding of the global units into the product of some local units? We use the p-adic Schanuel Conjecture to answer the question in the affirmative and moreover we give a value for the Z_p-rank (of the diagonal embedding of the global units into the product of some local units) in terms of the Z-rank of the global units and a property of the the local units included in the product.

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