Heuristics and conjectures in the direction of a $p$-adic Brauer-Siegel Theorem

Let $p$ be a fixed prime number. Let $K$ be a totally real number field of discriminant $D_K$, and let ${\mathcal T}_K$ be the torsion group of the Galois group of the maximal abelian $p$-ramified pro-$p$-extension of $K$. We conjecture the existence of a constant ${\mathcal C}_p$ such that ${\rm log}({\raise 1.5pt \hbox {${\scriptscriptstyle \char93 }$}} {\mathcal T}_K) \leq {\mathcal C}_p \cdot {\rm log}(\sqrt {D_K})$ when $K$ varies in some specified families (e.g., fields of fixed degree). In some sense, we suggest the existence of a $p$-adic analogue, of the classical Brauer-Siegel Theorem, depending here on the valuation of the residue at $s=1$ (essentially equal to ${\scriptscriptstyle \char93 } {\mathcal T}_K$) of the $p$-adic zeta-function $\zeta _p(s)$ of $K$. We shall use different definitions from that of Washington, given in the 1980s, and approach this question via the arithmetic study of ${\mathcal T}_K$ since $p$-adic analysis seems to fail because of possible abundant ``Siegel zeros'' of $\zeta _p(s)$, contrary to the classical framework. We give extensive numerical verifications for quadratic and cubic fields (cyclic or not) and publish the PARI/GP programs directly usable by the reader for numerical improvements. We give some examples of families of number fields where ${\mathcal C}_p$ exists. Such a conjecture (if exact) reinforces our conjecture that any fixed number field $K$ is $p$-rational (i.e., ${\mathcal T}_K=1$) for all $p \gg 0$.