On a new type of $\ell$-adic regulator for algebraic number fields. II

Abstract:

In a preceding paper of the author, a new type of an $\ell$-adic regulator $\mathfrak R_\ell (K)$ was introduced for an algebraic number field $K$ such that the prime $\ell$ splits completely in $K$. Nevertheless, the element $\mathfrak R_\ell (K)\in \mathbb Z_\ell$ is defined only up to an arbitrary factor in $(\mathbb Z_\ell ^\times )^2$. In the present paper, under the assumption of the validity of the Shanuel conjecture (both Archimedean and $\ell$-adic), the definition of $\mathfrak R_\ell (K)$ as a certain number in $\mathbb Z_\ell$ is given. For a real quadratic field $K$, such a defition can be obtained without using any additional conjectures.