On $u$-Multiple Zeta Values in Positive Characteristic

1. Problem Statement and Context

The paper addresses the development of a function field analog to the classical theory of $q$-analogs of Multiple Zeta Values (MZVs).

• Classical Context: In classical number theory, $q$-analogs deform standard formulas (like factorials and zeta values) such that the original theory is recovered as $q \to 1$. Recent work by Bachmann, Takeyama, and Tasaka (2018) established that "limits" of finite multiple harmonic $q$-series evaluated at roots of unity yield both finite multiple zeta values (algebraic limit) and symmetric multiple zeta values (analytic limit), providing evidence for the Kaneko–Zagier conjecture.
• Function Field Context: In the arithmetic of function fields over finite fields $\mathbb{F}_r(\theta)$, the role of integers is played by polynomials $A = \mathbb{F}_r[\theta]$, and the role of the multiplicative group is played by the Carlitz module. Thakur defined multiple zeta values ($\zeta_A(s)$) in this setting.
• The Gap: While Thakur's MZVs and finite MZVs over function fields are well-studied, a unified framework analogous to the classical $q$-analog theory (specifically involving a deformation parameter that connects analytic and algebraic limits) was missing.

Objective: To introduce $u$-multiple zeta values and finite multiple harmonic $u$-series via the Carlitz module, serving as the function field counterparts to $q$-analogs, and to investigate their properties, limits, and algebraic relations.

2. Methodology and Definitions

The author constructs the theory using the Carlitz module $C$, which acts on the additive group of the completion of the algebraic closure of $\mathbb{F}_r(\theta)$, denoted $C_\infty$.

Key Definitions:

1. The $u$-Bracket: For a formal variable $u$ and a polynomial $a \in A$, the $u$-bracket is defined as:
   $$[a]_u := \frac{C_a(u)}{u} \in C_\infty[[u]]$$
   where $C_a(u)$ is the Carlitz action. As $u \to 0$, $[a]_u \to a$, recovering the classical polynomial analog.
2. Finite Multiple Harmonic $u$-Series ($H_{<d}(s; u)$): Defined as a sum over monic polynomials with degree constraints, replacing the standard polynomial denominator $a^s$ with the $u$-bracket $[a]_u^s$:
   $$H_{<d}(s; u) = \sum_{d > \deg a_1 > \dots > \deg a_m \ge 0} \frac{1}{[a_1]_u^{s_1} \cdots [a_m]_u^{s_m}}$$
3. $u$-Multiple Zeta Values ($\zeta_u(s)$): Defined as the limit of the finite series as the degree bound $d \to \infty$:
   $$\zeta_u(s) = \lim_{d \to \infty} H_{<d}(s; u)$$
   These are viewed as formal power series in $u$ or rigid analytic functions on the Drinfeld upper-half plane.

Analytical Tools:

• Carlitz Torsion Points: The author evaluates these series at $\lambda_n = \exp_C(\tilde{\pi}/n)$, where $\tilde{\pi}$ is the Carlitz period. These points play the role of roots of unity in the classical theory.
• $r$-Shuffle Relations: The paper utilizes an inductive definition of the $r$-shuffle product ($*$) on the free monoid generated by symbols $x_k$, adapted to the positive characteristic setting (involving coefficients $\Delta_{i,j}$ derived from binomial coefficients modulo $p$).

3. Key Contributions and Results

A. Connection to Classical Limits (Theorems 1.6 & 1.7)

The paper proves that the $u$-series act as a bridge between Thakur's MZVs and finite MZVs over the function field, mirroring the classical $q$-analog results:

1. Analytic Limit: As the degree of the torsion point $n$ goes to infinity, the evaluation of the finite $u$-series at $\lambda_n$ converges to Thakur's multiple zeta value $\zeta_A(s)$.
   $$\lim_{\deg n \to \infty} H_{<\deg n}(s; \lambda_n) = \zeta_A(s)$$
2. Algebraic Limit: Taking the reduction modulo the torsion point $\lambda_v$ (for an irreducible polynomial $v$) yields the finite multiple zeta value over the function field $\zeta_{A_K}(s)$.
   $$\zeta_{A_K}(s) = \left( H_{<\deg v}(s; \lambda_v) \pmod{\lambda_v} \right)_v$$

B. Euler–Carlitz Type Formulas

The author derives an Euler–Carlitz formula for the finite $u$-series at Carlitz torsion points. For $r$-even integers $s$ (where $(r-1) \mid s$):
$$\frac{H_{<\deg n}(s; \lambda_n)}{(n\lambda_n)^s} = \frac{dBC_s(n)}{\Gamma_{s+1}}$$
where $dBC_s(n)$ are degenerate Bernoulli–Carlitz numbers. Taking limits recovers the classical Euler–Carlitz formula for $\zeta_A(s)$ and proves the vanishing of finite MZVs at $r$-even integers.

C. Formal Power Series Expansion (Theorem 1.8)

The paper provides an explicit expansion of $\zeta_u(s)$ as a formal power series in $u$:
$$\zeta_u(s) = \sum_{N=0}^\infty \gamma_N(s) u^{N(r-1)}$$
Crucially, the coefficients $\gamma_N(s)$ are shown to be $K$-linear combinations of Thakur's multiple zeta values (including those at non-positive indices). This establishes a direct link between the deformation parameter and the underlying arithmetic values.

D. Hasse–Schmidt Derivations and New Relations (Theorem 1.9)

By analyzing the coefficients of the formal expansion, the author defines a family of operators $(D_N)_{N \ge 0}$ acting on the algebra of indices.

• Result: These operators form a Hasse–Schmidt derivation over the realization map of Thakur's MZVs.
• Significance: This provides a systematic method to generate explicit algebraic relations among Thakur's MZVs at both positive and non-positive indices.
  • Example: The paper derives specific identities relating products of MZVs to sums involving shifted indices (e.g., $s_i - (r-1)$), generalizing known shuffle relations to include non-positive arguments.

4. Significance and Impact

1. Unification of Limits: The work successfully establishes the function field analog of the Bachmann–Takeyama–Tasaka (BTT) results, showing that a single object (the $u$-series) unifies the analytic (Thakur's MZVs) and algebraic (finite MZVs) perspectives in positive characteristic.
2. Extension to Non-Positive Indices: A major contribution is the ability to handle non-positive indices systematically. The derived Hasse–Schmidt derivations allow for the generation of relations involving MZVs at non-positive integers, which were previously difficult to access or relate to positive indices in a structured way.
3. Evidence for Conjectures: The results provide structural evidence for a function field version of the Kaneko–Zagier conjecture, suggesting that the algebraic relations satisfied by finite MZVs over function fields are closely tied to those of Thakur's MZVs via the "binary relations" framework.
4. Rigid Analytic Geometry: The paper demonstrates that these $u$-MZVs can be viewed as rigid analytic functions on the Drinfeld upper-half plane, opening new avenues for studying their analytic properties (e.g., zeros, poles, and $t$-expansions).

Conclusion

Hung-Chun Tsui's paper introduces a robust deformation theory for Multiple Zeta Values in positive characteristic. By defining $u$-brackets and $u$-series via the Carlitz module, the author creates a framework that not only recovers known results (Thakur's MZVs, finite MZVs, Euler–Carlitz formulas) as limits but also generates a rich new set of algebraic relations involving non-positive indices through Hasse–Schmidt derivations. This work significantly advances the understanding of the arithmetic structure of function field zeta values and their connection to finite characteristic arithmetic.