The solvability of the inverse volcano problem over non-prime finite fields

For a finite field $\mathbf{F}_{p^k}$ and a prime $\ell \neq p$, consider the graph $G$ of $\ell$-isogenies between ordinary elliptic curves over $\mathbf{F}_{p^k}$. Kohel proved that the connected components of $G$ have a remarkable structure, now called an $\ell$-volcano graph. Bambury, Campagna, and Pazuki investigated the inverse volcano problem: given a volcano graph $V$, can one find it as a connected component of $G$ over $\mathbf{F}_{p^k}$? They gave a complete positive answer over $\mathbf{F}_p$, and described a specific counterexample over $\mathbf{F}_{p^2}$. In this paper, we generalise the results of Bambury-Campagna-Pazuki by providing a precise framework for the inverse volcano problem over $\mathbf{F}_{p^k}$. The solvability of the problem for an $\ell$-volcano graph $V$ of depth $d$ is typically determined by the relation between $d$ and the $\ell$-valuation $r$ of $k$. When $r$ is small in comparison to $d$, we prove that there are infinitely many primes $p$ solving the inverse problem for $V$. The situation where $r$ is large in comparison to $d$ is more delicate: in many cases we prove that the inverse problem for $V$ is unsolvable; in a few other cases the problem appears to be solvable, but our proof of this is conditional on a variant of the Cohen-Lenstra heuristics for class groups of imaginary quadratic fields. We provide some computational evidence in support of these modified heuristics.