  The fluctuation determinant, the preexponential factor for the instanton
transition, has been computed several years ago in the Abelian Higgs model,
using a method based on integrating the Euclidean Green' function. A more
elegant method for computing functional determinants, using the Gel'fand-Yaglom
theorem, has been applied recently to a variety of systems. This method runs
into difficulties if the background field has nontrivial topology, as is the
case for the instanton in the Abelian Higgs model. A shift in thre effective
centrifugal barriers makes the s-wave contribution infinite, an infinity that
is compensated by the summation over the other partial waves. This requires
some modifications of the Gel'fand-Yaglom method which are the main subject of
this work. We present here both, the Green' s function and the Gel'fand-Yaglom
method and compare the numerical results in detail.
