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

TL;DR

This paper generalizes the inverse volcano problem over finite fields, establishing conditions for its solvability based on the relation between the volcano's depth and the field extension degree, with results supported by heuristics and computations.

Contribution

It provides a precise framework for the inverse volcano problem over finite fields $ extbf{F}_{p^k}$, extending prior results and analyzing solvability conditions related to field extensions.

Findings

Infinite solutions for small $ extit{r}$ relative to depth $d$

Many cases where the inverse problem is unsolvable when $ extit{r}$ is large

Conditional solutions based on Cohen-Lenstra heuristics

Abstract

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.