  Let $C$ be an algebraically closed field containing the finite field $F_q$
and complete with respect to an absolute value $|\;|$. We prove that under
suitable constraints on the coefficients, the series $f(z) = \sum_{n \in \Z}
a_n z^{q^n}$ converges to a surjective, open, continuous $F_q$-linear
homomorphism $C \rightarrow C$ whose kernel is locally compact. We characterize
the locally compact sub-$F_q$-vector spaces $G$ of $C$ which occur as kernels
of such series, and describe the extent to which $G$ determines the series.
  We develop a theory of Newton polygons for these series which lets us compute
the Haar measure of the set of zeros of $f$ of a given valuation, given the
valuations of the coefficients. The ``adjoint'' series $f^\ast(z) = \sum_{n \in
\Z} a_n^{1/q^n} z^{1/q^n}$ converges everywhere if and only if $f$ does, and in
this case there is a natural bilinear pairing $$ \ker f \times \ker f^\ast
\rightarrow F_q $$ which exhibits $\ker f^\ast$ as the Pontryagin dual of $\ker
f$. Many of these results extend to non-linear fractional power series. We
apply these results to construct a Drinfeld module analogue of the Weil
pairing, and to describe the topological module structure of the kernel of the
adjoint exponential of a Drinfeld module.
