Carlitz twists: their motivic cohomology, regulators, zeta values and polylogarithms

TL;DR

This paper explores the motivic cohomology, regulators, and zeta values associated with tensor powers of the Carlitz $t$-motive, revealing deep connections with function field arithmetic and polylogarithms.

Contribution

It explicitly determines the integral $t$-motivic cohomology and class modules for tensor powers of the Carlitz $t$-motive, linking them to polylogarithms and zeta values.

Findings

Motivic cohomology governs relations among Carlitz polylogarithms.

Torsion relates to Bernoulli-Carlitz numbers.

Regulator is an isomorphism iff $n$ is coprime to the characteristic.

Abstract

The integral $t$-motivic cohomology and the class module of a (rigid analytically trivial) Anderson $t$-motive were introduced by the first author in [Gaz22b]. This paper is devoted to their determination in the particular case of tensor powers of the Carlitz $t$-motive, namely, the function field counterpart $\underline{A}(n)$ of Tate twists $\mathbb{Z}(n)$. We find out that these modules are in relation with fundamental objects of function field arithmetic: integral $t$-motivic cohomology governs linear relations among Carlitz polylogarithms, its torsion is expressed in terms of the denominator of Bernoulli-Carlitz numbers and the Fitting ideal of class modules is a special zeta value. We also express the regulator of $\underline{A}(n)$ for positive $n$ in terms of generalized Carlitz polylogarithms; after establishing their algebraic relations using difference Galois theory together with the Anderson-Brownawell-Papanikolas criterion, we prove that the regulator is an isomorphism if, and only if, $n$ is prime to the characteristic.