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.