  The classical theory of $p$-adic (elliptic) modular forms arose in the 1970's
from the work of
  J.-P.\ Serre \cite{se1} who took $p$-adic limits of the $q$-expansions of
these forms. It was soon expanded by N.\ Katz \cite{ka1} with a more functorial
approach. Since then the theory has grown in a variety of directions. In the
late 1970's, the theory of modular forms associated to Drinfeld modules was
born in analogy with elliptic modular forms \cite{go1}, \cite{go2}. The
associated expansions at $\infty$ are quite complicated and no obvious limits
at finite primes ${\mathfrak v}$ were apparent. Recently, however, there has
been progress in the $\mathfrak v$-adic theory, \cite{vi1}. Also recently, A.\
Petrov \cite{pe1}, building on previous work of \cite{lo1}, showed that there
is an intermediate expansion at $\infty$ called the "$A$-expansion," and he
constructed families of cusp forms with such expansions. It is our purpose in
this note to show that Petrov's results also lead to interesting ${\mathfrak
v}$-adic cusp forms \`a la Serre. Moreover the existence of these forms allows
us to readily conclude a mysterious decomposition of the associated Hecke
action.
