Special zeta values using tensor powers of Drinfeld modules

We study tensor powers of rank $1$ sign-normalized Drinfeld $A$-modules, where $\mathbf{A}$ is the coordinate ring of an elliptic curve over a finite field of size $q$. Using the theory of vector valued Anderson generating functions, we give formulas for the coefficients of the logarithm and exponential functions associated to these $\mathbf{A}$-modules. We then show that for $n \leq q$ there exists an $n$-dimensional vector whose bottom coordinate contains a Goss zeta value evaluated at $n$, where the evaluation of this vector under the exponential function is defined over the Hilbert class field. This allows us to prove the transcendence of these Goss zeta values and periods of Drinfeld modules as well as the transcendence of certain ratios of these quantities.

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This project was partially supported by NSF Grant DMS-1501362.

Received 12 February 2018

Accepted 10 February 2019

Published 6 March 2020