TITLE: What is isogeny?
QUESTION [23 upvotes]: Can it be explained in a concrete fashion what isogeny is? Apparently some of the proofs of Fermat's last theorem are essentially this but it is not clear to me what that actually is. (It would be good to see examples which aren't Fermat's theorem though)

REPLY [17 votes]: Regarding the motivation for your question, it might be that you've been reading
statements such as "any elliptic curve is isogenous to an elliptic curve contained in the Jacobian of a modular curve", or statements related to Faltings's theorem/Tate's conjecture: that two elliptic curves with the same $L$-function are isogenous.
Firstly: over the complex numbers, an elliptic curve $E$ is of the form $\mathbb C/\Lambda$, for a lattice  $\Lambda$ in $\mathbb C$.  The curves isogenous to $E$ are precisely those of the form $\mathbb C/\Lambda'$, where $\Lambda'$ is another lattice which is contained in $\mathbb Q\otimes \Lambda$ (i.e. $\Lambda'$ is commensurable with $\Lambda$).  
Why does this come up?  Well,
in some sense, isogenous elliptic curves over $\mathbb Q$
have almost the same number-theoretic properties.  For example, they have good reduction at the same set of primes $p$, and if $p$ is a prime of good reduction, then the number of solutions mod $p$ of the two curves will be the same.  This is why they have the same $L$-function.(The $L$-function is constructed using as input the number of solutions to the elliptic curve modulo primes $p$.)  
Since the relationship between elliptic curves and modular forms (the
one that Wiles proved) is mediated via
their $L$-functions, isogenous elliptic curves come from the same modular
form.  This is why the concept of isogeny comes up all the time in discussion
of FLT and Wiles's proof.