The t-motivic mixed Carlitz zeta category and Carlitz-Thakur multi-zeta values

TL;DR

This paper constructs a canonical t-motivic category over $_q(t)$ that encompasses all mixed t-motives with Carlitz-Thakur multi-zeta values, establishing foundational structures and exploring extensions over finite fields.

Contribution

It introduces a canonical, Tannakian t-motivic category containing all mixed t-motives with Carlitz-Thakur multi-zeta values and proves the existence of related categories over finite extensions independently of conjectures.

Findings

Constructed the t-motivic mixed Carlitz zeta category over $_q(t)$.

Proved the category is Tannakian, neutral, with a weight filtration.

Established existence of categories over finite extensions, assuming a conjecture.

Abstract

We construct the t-motivic mixed Carlitz zeta category over $\F_q(t)$ and show that it contains all the (mixed) t-motives with Carlitz-Thakur multi-zeta values as periods constructed by Anderson and Thakur. Our construction is canonical and our category is Tannakian and neutral and every object is equipped with a weight filtration whose graded pieces are Carlitz motives over $\F_q(t)$. For any finite separable extension L/\F_q(t) we show that existence of a similar category over $L$ is a consequence of a version of a conjecture of L. Taelman. Along the way we also prove the existence of the category of mixed $t$-motives and the category of mixed Carlitz motives over any $L$ (these two existence results are independent of any conjectures).