Distribution questions for isogeny graphs over finite fields

In the first part of the paper, we fix a non-CM elliptic curve $E/\mathbb{Q}$ and an odd prime $\ell$ and investigate the distribution of invariants associated to the $\ell$-volcano containing the reduction $E_p$, as $p$ ranges over primes of good ordinary reduction. Let $H(p)$ be the height of the volcano and let $d'(p)$ denote the relative position of $j(E_p)$ above the floor, and let $r\ge 0$ be an integer. Assuming that the $\ell$-adic Galois representation attached to $E$ is surjective, we derive an explicit formula for the natural density of primes $p$ for which $H(p)=r$ (resp.\ $d'(p)=r$). In the non-surjective case, we show that all sufficiently large heights occur with positive density. In the second part of the paper, we analyze the distribution of $\ell$-volcano heights over a finite field $\mathbb{F}_q$ and consider the limit as $q\to\infty$. Using analytic estimates for sums of Hurwitz class numbers in arithmetic progressions, we compute exact limiting densities for ordinary elliptic curves whose $\ell$-isogeny graph has a prescribed height $r$.