Cohen-Lenstra-Gerth Heuristics via Automorphism Counts

For a finite abelian 2-group $G$, we study the frequency with which quadratic imaginary number fields $K$ have 2-part of their class group $K$ isomorphic to $G$. A philosophy enunciated by Gerth extends the Cohen-Lenstra heuristics for imaginary quadratic number fields to the case $p=2$, by referencing both the 2-rank and the 4-rank of the group in question. A recent paper by Smith provides relative density statements about the $2^{k+1}$-rank of such a class group given its $2^1$- through $2^k$-ranks, for $k\ge 2$. We deduce from Smith’s results an explicit automorphism-count-theoretic statement of the Cohen-Lenstra-Gerth heuristics, also describing connections to “higher R'{e}dei matrices” introduced by Kolster to study the $2^k$-ranks of the class group of $K$.