### Separability, multi-valued operators, and zeroes of L-functions, by David Goss

Let $\k$ be a global function field in $1$-variable over
a finite extension of $\Fp$, $p$ prime, $\infty$ a fixed place of
$\k$, and $\A$ the ring of functions of $\k$ regular outside of
$\infty$. Let $E$ be a Drinfeld module or $T$-module. Then,
as in \cite{go1}, one can construct associated characteristic $p$
$L$-functions based on the classical model of abelian
varieties {\it once} certain auxiliary choices are made. Our purpose
in this paper is to show how the well-known concept of
``maximal separable (over the completion $\k_\infty$) subfield''
allows one to construct from such $L$-functions
certain separable extensions which are independent of these choices.
These fields will then depend only on the isogeny class
of the original $T$-module or Drinfeld module and $y\in \Zp$,
and should presumably
be describable in these terms. Moreover, they give a very useful framework
in which to view the ``Riemann hypothesis'' evidence of
\cite{w1}, \cite{dv1}, \cite{sh1}.
We also establish that an element which is
{\it separably} algebraic over $\k_\infty$ can be realized as
a ``multi-valued operator'' on general $T$-modules.
This is very similar to realizing $1/2$ as the
multi-valued operator $x\mapsto \sqrt{x}$ on $\C^\ast$. Simple examples
show that this result is false for non-separable elements. This result may
eventually allow a ``two $T$'s'' interpretation of the above extensions in
terms of multi-valued operators on $E$ and certain tensor twists.

David Goss <goss@math.ohio-state.edu>