Special L-values for Drinfeld modules over elliptic curves


Matt Papanikolas

14:00:00 - 15:00:00

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

Over the past several decades there has been significant progress in understanding special values of zeta and L-functions associated to Drinfeld modules over a polynomial ring in one variable over a finite field. The early work of Carlitz and then later work of Anderson, Gekeler, Goss, Thakur, and Yu provided a firm footing about power sums and reciprocal sums of polynomials, which led to results on special values of function field L-series of Goss. More recently work of Pellarin and Taelman took these ideas in new directions and derived additional special value formulas for other L-series.

In this talk we will present results on special values of Goss and Pellarin L-series for Drinfeld modules over coordinate rings of elliptic curves over finite fields, and we will discuss the challenges of transitioning to higher genus cases. The main tool we will use is the shtuka function for the elliptic curve, which allow us to interpolate reciprocal sum formulas of Thakur and to provide a new proof of a 1994 log-algebraicity theorem of Anderson. Furthermore we prove special value identities for Pellarin L-series, which interpolate values of Dirichlet L-functions at s=1 as well as zeta functions at certain negative integers. Joint with N. Green.