A case of the Rodriguez Villegas conjecture

Abstract

Let L be a number field and let E be any subgroup of the units O_L^* of L. If rank(E) = 1, Lehmer's conjecture predicts that the height of any non-torsion element of E is bounded below by an absolute positive constant. If rank(E) = rank(O_L^*), Zimmert proved a lower bound on the regulator of E which grows exponentially with [L:Q]. Fernando Rodriguez Villegas made a conjecture in 2002 that "interpolates" between these two extremes of rank. Here we prove a high-rank case of this conjecture. Namely, it holds if L contains a subfield K for which [L:K] >> [K:Q] and E contains the kernel of the norm map from O_L^* to O_K^*.