Suppose we are given two non-isomorphic supersingular elliptic curves $C$ and $C'$ (in characteristic $p$). Is there an isogeny $C\to C'$ of a given degree (say, power of a prime $l$ different from the characteristic)? In other words, I would like to know which (or at least how many) cyclic isogenies of degree $l^k$ with domain $C$ have $C$ itself as a codomain, and whether there is some uniformity among the codomains of different isogenies.

Another way to phrase the question would be: given an isogeny $\phi: C\to C'$, could we relate the $j$-invariants of $C$ and $C'$ using properties of $\phi$?

Inventiones 100article, if I remember correctly, and this quadratic forms/theta-series point of view is also explored in my paper here: math.uchicago.edu/~emerton/pdffiles/two.pdf – Emerton Jan 23 '12 at 15:53