Tensor Powers of Drinfeld Modules and Zeta Values
Abstract
We study tensor powers of rank 1 sign-normalized Drinfeld A-modules, where A is the coordinate
ring of an elliptic curve over a finite field. Using the theory of A-motives, we find explicit
formulas for the A-action of these modules. Then, by developing the theory of vector-valued Anderson
generating functions, we give formulas for the period lattice of the associated exponential
function. We then give formulas for the coefficients of the logarithm and exponential functions
associated to these A-modules. Finally, we show that there exists a vector whose bottom coordinate
contains a Goss zeta value, whose evaluation under the exponential function is defined over
the Hilbert class field. This allows us to prove the transcendence of certain Goss zeta values and
periods of Drinfeld modules as well as the transcendence of certain ratios of those quantities.
Citation
Green, Nathan Eric (2018). Tensor Powers of Drinfeld Modules and Zeta Values. Doctoral dissertation, Texas A & M University. Available electronically from https : / /hdl .handle .net /1969 .1 /173665.