Paper
Distribution questions for isogeny graphs over finite fields
Authors
Anwesh Ray
Abstract
In the first part of the paper, we fix a non-CM elliptic curve and an odd prime and investigate the distribution of invariants associated to the -volcano containing the reduction , as ranges over primes of good ordinary reduction. Let be the height of the volcano and let denote the relative position of above the floor, and let be an integer. Assuming that the -adic Galois representation attached to is surjective, we derive an explicit formula for the natural density of primes for which (resp.\ ). 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 -volcano heights over a finite field and consider the limit as . Using analytic estimates for sums of Hurwitz class numbers in arithmetic progressions, we compute exact limiting densities for ordinary elliptic curves whose -isogeny graph has a prescribed height .