7 problems

For $x>0$, let ${\mathcal{F}}_x$ be the set of real quadratic fields with discriminant at most $x$. For $K\in{\mathcal{F}}_x$, let $A_K$ be its $p$-class group and let $G_K$ be the…

Let $F$) be a number field and $p$ a prime. Write $\ell_p{F}$ for the length of the $p$-class tower, namely the derived length $\operatorname{dl}(\mathrm{G}_p^\infty{F})$ of its…

Let $F=\mathbb{Q}(\sqrt{d})$ be a real quadratic field with fundamental discriminant $d>1$, $3$-class group $\operatorname{Cl}_3(F)\simeq C_3\times C_3$, shallow transfer kernel ty…

An IPAD is the invariant used in the source to record the abelianization and related index-subgroup data of a Schur $\sigma$-$3$-group. Call an IPAD “large” in the informal sense i…

An IPAD is the invariant displayed in the source in the form of an abelianization together with the corresponding abelianizations of selected index subgroups. The notation…

Let $G$ be a finitely generated pro-$p$ group. Let $\operatorname{Freq}(G)$ denote its limiting frequency among imaginary quadratic fields, and let $\operatorname{Meas}(G)$ denote…

Existence-of-frequency conjecture. The limit defining $\operatorname{Freq}(G)$ exists, and hence gives the frequency with which $G$ occurs as a $p$-class tower group among imaginar…