Sanity check on existence of isogenies Say I have an elliptic curve $E$ over a base scheme $S$, and a locally constant function $d\colon S\to \mathbb{N}$. I'm guessing that in general there is no way to cook up an isogeny $\phi\colon E \to E'$ over $S$ such that the degree of $\phi$ agrees with $d$. Is this correct? 
In characteristic zero I imagine there's no problem, and similarly if $d$ is appropriately coprime to the characteristic otherwise. But I feel there should be some serious issues when, say, in characteristic $p$ where $d$ takes values not coprime to $p$.
The problem reduces to knowing that there is a (flat, I think) sub-group-scheme of $E$ whose order is described by $d$, and this seems hard
 A: There is a very famous paper of Mazur, called "Rational isogenies of prime degree" (EuDML), where he determines explicitly a finite set of primes $P$ with the property that if $E$ is an elliptic curve over $\mathbb Q$ with a rational isogeny of prime degree $p$, then $p\in P$ (and conversely, for every $p\in P$ there exists a rational elliptic curve with an isogeny of degree $p$). Therefore, in general your question has a negative answer.
Moreover, Serre's open image theorem states that if $E$ is a non-CM elliptic curve over a number field $K$, then the Galois representation $G_K\to GL_2(\widehat{\mathbb Z})$ induced by the torsion points has open image, and therefore the image has finite index. This implies, in particular, that for a prime $p$ big enough, $E$ cannot have a rational isogeny of degree $p$. But more seems to be true: Serre conjectured that the implied constant in my previous sentence depends only on $K$, and not on the curve itself. Of course this would imply that for every number field $K$ there are only finitely many primes $p$ such that a non-CM elliptic curve $E$ over $K$ has a rational isogeny of degree $p$. Caveat: I'm not an expert, so it might be that this particular result is known to be true via another argument! Also, I don't know the state of the art for CM elliptic curves.
