In this paper, we study various ramifications arising from division points
of Drinfeld modules, abelian $T$-modules, formal modules, etc.
A motivation for this is to know how many isogeny classes and
isomorphism classes of Drinfeld $A$-modules exist over a finite extension
of the fraction field of $A$.
We will see that, modulo the isogeny conjecture, an isogeny class can
contain infinitely many isomorphism classes and,
without any restriction on ramification at the infinite places,
there can be infinitely many isogeny classes.
[ See [13] for a correction on this point.]

To explane some of the results, let
$F$ be a function field in one variable over a finite field,
$\infty$ a fixed place of $F$,
$A$ the ring of elements of $F$ which are regular outside $\infty$, and
$K$ a finite extension of $F$.
Given a Drinfeld $A$-module $\phi$ over $K$ and a prime $v$ of $A$,
we denote by $K(\phi; v^n)$ the field of $v^n$-division points of $\phi$.
Then it turns out that the ramification at various primes in the tower
$( K(\phi; v^n)/K )_{n \geq 1\}$
is bounded at the places over $\infty$ by a divisor depending only on $\phi$,
and at the finite places, it is controlled in a fairly precise way
in terms of the "discriminant'' $\Delta(\phi)$.
Roughly speaking, $\Delta(\phi)$ is the coefficient of the
leading term of the defining equation of $\phi$.
For finite places, this result is analogous to the case of
abelian varieties over number fields.
[ ... ]
But at infinite places, there occur new phenomena,
which we describe by example in Section 2.