Several algorithmic problems on supersingular elliptic curves are

currently under close scrutiny. When analysing algorithms or reductions

in this context, one often runs into the following type of question:

given a supersingular elliptic curve E and an object x attached to E, if

we consider a random large degree isogeny f : E -> E' and carry the

object x along f, how is the resulting f(x) distributed among the

possible objects attached to E'? We propose a general framework to

formulate this type of question precisely, and prove a general

equidistribution theorem under a condition that is easy to check in

practice. The proof goes from elliptic curves to quaternionic

automorphic forms via an augmented Deuring correspondence, and then to

classical modular forms via the Jacquet-Langlands correspondence. This

is joint work with Benjamin Wesolowski.