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Analytic continuation of Kochubei multiple polylogarithms and its applications

Yen-Tsung Chen

TL;DR

This work develops an analytic continuation for Kochubei polylogarithms in positive characteristic using Furusho's framework, extending to KMPLs with a corresponding monodromy module description. It shows that the continued functions satisfy the same difference equations and provides a cohomological interpretation of their algebraic-value relations via Ext^1/Frobenius-module structures, yielding lower bounds on independence and a lifting principle for relations. The Ext^1 analysis identifies the extension classes with a concrete $\overline{K}[t]$-module and a vector-space $\mathcal{V}_n$, enabling explicit series expressions and $\overline{K}$-linear relations among continued KMPL values at algebraic points. The results bear on connections to ABP-type criteria and suggest links to Thakur-type hypergeometric functions, with potential applications to the structure of Frobenius modules and hypergeometric analogues in positive characteristic.

Abstract

In the present paper, we propose an analytic continuation of Kochubei multiple polylogarithms using the techniques developed by Furusho. Moreover, we produce a family of linear relations and a linear independence result for values of our analytically continued Kochubei polylogarithms at algebraic elements from a cohomological aspect.

Analytic continuation of Kochubei multiple polylogarithms and its applications

TL;DR

This work develops an analytic continuation for Kochubei polylogarithms in positive characteristic using Furusho's framework, extending to KMPLs with a corresponding monodromy module description. It shows that the continued functions satisfy the same difference equations and provides a cohomological interpretation of their algebraic-value relations via Ext^1/Frobenius-module structures, yielding lower bounds on independence and a lifting principle for relations. The Ext^1 analysis identifies the extension classes with a concrete -module and a vector-space , enabling explicit series expressions and -linear relations among continued KMPL values at algebraic points. The results bear on connections to ABP-type criteria and suggest links to Thakur-type hypergeometric functions, with potential applications to the structure of Frobenius modules and hypergeometric analogues in positive characteristic.

Abstract

In the present paper, we propose an analytic continuation of Kochubei multiple polylogarithms using the techniques developed by Furusho. Moreover, we produce a family of linear relations and a linear independence result for values of our analytically continued Kochubei polylogarithms at algebraic elements from a cohomological aspect.
Paper Structure (14 sections, 15 theorems, 168 equations)

This paper contains 14 sections, 15 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Classical multiple polylogarithms
  3. Main results
  4. Strategy and organization
  5. Preliminaries
  6. Notation
  7. Hyperderivatives and expansions of polynomials
  8. Frobenius modules and rigid analytic trivializations
  9. Furusho's analytic continuation
  10. Analytic continuation of Kochubei polylogarithms
  11. Kochubei multiple polylogarithms
  12. Applications
  13. Relations with the structure of $\mathop{\mathrm{Ext}}\nolimits^1_{\mathscr{F}}(M_{C^{\otimes n}},M_{C^{\otimes n}})$
  14. Linear relations among KPLs at algebraic points

Key Result

Theorem 1.2.8

Let $\mathfrak{s}=(s_1,\Tilde{\mathfrak{s}})\in\mathbb{Z}_{\geq 0}^r$ with $\Tilde{\mathfrak{s}}=(s_2,\dots,s_r)\in\mathbb{Z}_{\geq 0}^{r-1}$ and $\Tilde{\mathbf{u}}=(u_2,\dots,u_r)\in\mathop{\mathrm{Mat}}\nolimits_{1\times(r-1)}(\mathbb{C}_\infty)$. There exist a discrete $A$-submodule $\mathcal{M} such that the following assertions hold.

Theorems & Definitions (36)

  • Theorem 1.2.8
  • Theorem 1.2.9
  • Remark 1.2.10
  • Lemma 2.2.1
  • proof
  • Lemma 2.2.2
  • proof
  • Definition 2.3.1
  • Example 2.3.2
  • Definition 2.3.3
  • ...and 26 more