University of Georgia
Mathematics Department Colloquium
Cristian D. Popescu
University of Texas at Austin
January 13, 2000
Stark--type Conjectures ``over {\bf Z}"
Abstract:
In the 1970s and 1980s Stark developed a remarkable
conjecture aimed at interpreting the first non--vanishing
derivative of an Artin $L$--function $L_{K/k, S}(s,\chi)$ at
$s=0$ in terms of the arithmetic properties of the Galois
extension of global fields $K/k$. Work of Tate, Chinburg, and
Stark himself has revealed far reaching applications of
Stark's Conjecture to Hilbert's 12--th Problem and the theory
of Galois module structure of groups of units and ideal--class
groups.
In his search for new examples of Euler systems, Rubin has
formulated in 1994 a strong version (``over {\bf Z}'', in
Tate's terminology) of Stark's Conjecture for {\it abelian}
$L$--functions of {\it arbitrary order of vanishing at $s=0$}.
Our recent study of the functorial base--change behavior of
Rubin's Conjecture led us to formulating a seemingly more
natural Stark--type conjecture ``over {\bf Z}''. We will
discuss and provide evidence for this new statement, as well as
briefly describe the main goals of the conjectural program
initiated by Stark.