Interpolation of numbers of Catalan type in a local field of positive characteristic

Let $k$ be a local field of positive characteristic. Let $L$ be a cocompact discrete subgroup of $k$. Let $U$ be an open compact subgroup of $k$. Let $\ell$, $u$ and $a$ be elements of $k$, with $a$ nonzero. We study the behavior of the product $\prod_{0\neq x\in (\ell+L)\cap a(u+U)}x$ as $a$ varies. Our main result, which blends harmonic analysis and local class field theory, provides local confirmation of a conjecture concerning global function fields recently made by the author (math.NT/0407535).