Take the tour ×  Let $q$ and $q'$ be complex numbers with
$0<|q|,|q'|<1$, and let $m$ and $n$ be positive integers.
Suppose that $q^m={q'}^n$. Then the map $$
f:\mathbb{C}^\times/q^{\mathbb{Z}} \to
\mathbb{C}^\times/{q'}^{\mathbb{Z}}\qquad \text{defined
by}\qquad u\mapsto u^m $$ gives an isogeny of (analytic)
elliptic curves over $\mathbb{C}$.  The Tate curve
$\mathrm{Tate}(q)$ is an (algebraic) elliptic curve over the
Laurent series ring $\mathbb{Z}((q))$ which can be used to give
a uniformization of the curve $\mathbb{C}^\times/q^\mathbb{Z}$
by means of certain well known explicit formulae.  My question
is:  Does there exist an isogeny $\mathrm{Tate}(q)\to
\mathrm{Tate}(q')$ of elliptic curves defined over
$\mathbb{Z}((q,q'))/(q^m-{q'}^n)$ which "lifts" the map $f$
above, and if so, how do you prove it exists?  It should suffice
to construct such an isogeny for $(m,n)=(m,1)$, and use dual
isogenies and composition to get the general case.  (I'm being
vague about "lifts", because one has to worry about conve