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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 $\phi$?

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    $\begingroup$ The group of isogenies $C \to C'$ (including the zero isogeny) is free $\mathbb Z$-module of rank four equipped with a quadratic form, given by taking the degre of an isogeny. Whether there is an isogeny of given degree then depends on whether this quadratic form represents that degree or not. This is part of the classical subject of Brandt matrices. It is discussed in Ribet's Inventiones 100 article, 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 $\endgroup$
    – Emerton
    Commented Jan 23, 2012 at 15:53
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    $\begingroup$ A reference for the Brandt matrix point of view which might be more comprehensive and better suited for the level of generality you're exploring is given in Chapters 1 & 2 of Gross' Canadian Journal article ( wstein.org/papers/bib/… ). Generally there's a lot you can say about isogenies of supersingular elliptic curves when you want to say something like "There exists some supersingular elliptic curve such that _." If you want to start specifying the supersingular elliptic curves, the theory gets much more fussy. $\endgroup$
    – stankewicz
    Commented Jan 23, 2012 at 17:05
  • $\begingroup$ Thanks a lot for the references, they should be very helpful. I was also pointed to David Kohel's thesis, where (among other results) he provides bounds on the number of elliptic curves isogenous to a given curve $C$ via an isogeny of a given degree; he also uses Brandt matrices. $\endgroup$
    – Vesna Stojanoska
    Commented Jan 23, 2012 at 20:42

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