Ratios of regulators in extensions of number fields

Abstract:

Let $L/K$ be an extension of number fields. Then \[ \operatorname {Reg} (L)/\operatorname {Reg} (K) > {c_{[L:{\mathbf {Q}}]}}{(\log |{D_L}|)^m},\] where Reg denotes the regulator, ${D_L}$ is the absolute discriminant of $L$, and ${c_{[L:{\mathbf {Q}}]}} > 0$ depends only on the degree of $L$. The nonnegative integer $m = m(L/K)$ is positive if $L/K$ does not belong to certain precisely defined infinite families of extensions, analogous to CM fields, along which $\operatorname {Reg} (L)/\operatorname {Reg} (K)$ is constant. This generalizes some inequalities due to Remak and Silverman, who assumed that $K$ is the rational field ${\mathbf {Q}}$, and modifies those of Bergé-Martinet, who dealt with a general extension $L/K$ but used its relative discriminant where we use the absolute one.