Stark--type Conjectures "over Z" and their Refined Versions

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ABSTRACT Cristian Popescu University of Texas at Austin 98 01267 Professor Popescu will study connections between Stark's conjecture for L-functions and the construction of Euler systems. The investigator will study the functorial behavior of the conjectures concerning higher order of vanishing, looking for refined versions of these conjectures, and consider their connections with Euler systems. One of the most important mathematical ideas of the second half of the century, is that analytic formula often encodes discrete information. For example, one might want to count the number of solutions of a particular equation, but discover that the solutions are very hard to find. On the other hand, you might want to count the number of solutions to a sequence of equations and have the answers as a sequence. Mathematicians call these kind of problems 'discrete'. The functions that appear in calculus, and for which calculus works so well, are not 'discrete', but 'analytic.' Amazingly, the right kind of analytic function can contain the answer or answers to the discrete examples about. This project involves L-functions which were among the first examples of this kind of analytic function. In the past thirty years, a number of people including H. Stark have found new ways L-functions encode discrete information. Because many of these connections are still only conjecture, L-functions are very important in current mathematics.