# Special zeta values using tensor powers of Drinfeld modules

## TL;DR

This paper investigates tensor powers of rank 1 Drinfeld modules over elliptic curves, providing formulas for associated functions and proving the transcendence of Goss zeta values and related periods.

## Contribution

It introduces new formulas for logarithm and exponential coefficients of Drinfeld modules and establishes the transcendence of Goss zeta values and periods.

## Key findings

- Formulas for coefficients of logarithm and exponential functions
- Existence of vectors containing Goss zeta values with special properties
- Proof of transcendence of Goss zeta values and certain ratios

## 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 vector valued Anderson generating functions, we give formulas for the coefficients of the logarithm and exponential functions associated to these A-modules. We then 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 Goss zeta values and periods of Drinfeld modules as well as the transcendence of certain ratios of those quantities.