Inverse realization for prescribed primes

Determine whether, for any fixed prime ell, there exist infinitely many primes p for which the conclusions of Theorems 1 and 2 hold, and whether, for any fixed prime p, there exist infinitely many primes ell for which those conclusions hold; equivalently, establish the proposed strengthening of the inverse results for tectonic and directed-cycle craters.

Background

The appendix proves inverse results showing that every abstract tectonic crater, and every directed cycle of a prescribed length, can arise as the crater of a connected component of an isogeny graph for infinitely many choices of the primes p and ell, with suitable level parameters. The authors then identify a stronger unresolved question: whether one of the two primes can be fixed in advance while retaining infinitude of choices for the other. They explain that proving this would require constructing an imaginary quadratic field and auxiliary integers satisfying all the conditions used in the proof of Theorem 1, and explicitly note that their method does not establish the stronger claim.

References

One might ask whether it would be possible to improve Theorems~\ref{thm:inverse1} and \ref{thm:A2} to prove that for any given $l$ (resp. $p$), there are infinitely many $p$ (resp. $l$) such that the conclusions of the theorems hold. To do so, we would need to find an imaginary quadratic field $K$ and integers $N',M',Q$ that satisfy all the conditions used in the proof of Theorem~\ref{thm:inverse1}. However, our method does not work in such generality.

— Isogeny graphs with level structures arrising from the Verschiebung map  (2501.03846 - Lei et al., 7 Jan 2025) in Appendix, final remark following Theorem A2