| Beilinson's Conjecture on special values of L-functions and its refinement due to Bloch and Kato predict the value (or leading coefficient) of any "motivic" L-function at integral arguments. It unites Dirichlet's class number formula, Lichtenbaum's Conjecture on values of Dedekind zeta functions, the Birch and Swinnerton-Dyer Conjecture, and many more theorems and conjectures that lie at the heart of modern Number Theory and Arithmetic Geometry. The conjecture relates special values to periods of mixed motives. Today it is backed by an impressive record of numerical verification, and in some special cases it has even been proven to be true, yet, in general the conjecture is wide open and is expected to remain so in the near future. Given this state of affairs in the world of number fields it is very natural to look for inspiration in the function field universe, where things are often much better understood. Analogues to motivic L-functions are given by Goss L-functions of t-motives, and it is natural to ask what can be said about their values at integral arguments. Surprisingly, until recently almost nothing was known, not even conjecturally. A conjecture that I have put forward based on extensive numerical evidence indicates that special values of t-motives are indeed related to periods. I propose to study this relation, building on groundbreaking recent work of V. Lafforgue, and to start uncovering the subtle arithmetical properties of Drinfeld modules and t-motives that are reflected by their special values. A successful completion of this project will not only significantly advance our understanding of function field arithmetic, but ultimately will bring fresh insight to number field arithmetic as well. |
