Special L-values and log-algebraicity on Drinfeld modules


Matt Papanikolas

13:30:00 - 14:30:00

103 , Mathematics Research Center Building (ori. New Math. Bldg.)

We will explore the theory of Goss L-functions, which are defined by Dirichlet series over function fields in positive characteristic and which take values in the function field. Much like one finds over number fields, Goss L-functions arise naturally from Galois representations associated to Drinfeld modules and more generally to higher dimensional Drinfeld modules of Anderson. For Goss L-series for Dirichlet characters, Anderson introduced the idea of realizing special L-values via specializations of certain \'log-algebraic\' power series identities on rank one Drinfeld modules. His identities lead to formulas for evaluations L(1,chi) in terms of torsion points on Drinfeld modules. Subsequently we have generalized Anderson\'s identities to tensor powers of the Carlitz module, and multivariable versions have been obtained by Anglès, Pellarin, and Tavares Ribeiro.  In the present talk, using work of Taelman as a starting point, we will investigate log-algebraicity identities for Drinfeld modules of rank greater than one, leading to identities between special values of Goss L-functions, Drinfeld logarithms, and special points.  Joint work with C.-Y. Chang and A. El-Guindy.