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On functional equations for Chow polylogarithms

Vasily Bolbachan

TL;DR

The paper develops a structural framework linking Chow polylogarithms to classical polylogarithms by formulating a Bloch-like complex $Λ(X,m)$ and proving a Beilinson-Soule type vanishing for $Λ(K,m)$. It then shows that the functional equations for Chow polylogarithms formally generate all functional equations for the classical polylogarithms via a transfer map to a regulator-type group $\mathcal R_m(F)$. The work uses alterations, filtrations, and Galois descent to establish vanishing and to relate boundary conditions to reciprocity laws, ultimately connecting to a Kontsevich–Zagier style viewpoint on periods. This provides a conceptual route to derive and understand functional equations in higher motivic settings and clarifies the link between regulator maps and Bloch-type groups for polylogarithms.

Abstract

Chow polylogarithms are some special functions arising in explicit description of the Beilinson regulator map. The most interesting functional equation for this function reflects its vanishing on the boundary in the Bloch's cycle complex. We show that this functional equation formally follows from more simple ones, namely skew-symmetry, functoriality and multiplicativity. To prove this, we study some analogue of Bloch's cycle complex and establish for this complex an analogue Beilinson-Soule vanishing conjecture. A. Goncharov defined a group of functional equations for classical polylogarithms. We show that any such functional equation formally follows from functional equations for Chow polylogarithms stated above.

On functional equations for Chow polylogarithms

TL;DR

The paper develops a structural framework linking Chow polylogarithms to classical polylogarithms by formulating a Bloch-like complex and proving a Beilinson-Soule type vanishing for . It then shows that the functional equations for Chow polylogarithms formally generate all functional equations for the classical polylogarithms via a transfer map to a regulator-type group . The work uses alterations, filtrations, and Galois descent to establish vanishing and to relate boundary conditions to reciprocity laws, ultimately connecting to a Kontsevich–Zagier style viewpoint on periods. This provides a conceptual route to derive and understand functional equations in higher motivic settings and clarifies the link between regulator maps and Bloch-type groups for polylogarithms.

Abstract

Chow polylogarithms are some special functions arising in explicit description of the Beilinson regulator map. The most interesting functional equation for this function reflects its vanishing on the boundary in the Bloch's cycle complex. We show that this functional equation formally follows from more simple ones, namely skew-symmetry, functoriality and multiplicativity. To prove this, we study some analogue of Bloch's cycle complex and establish for this complex an analogue Beilinson-Soule vanishing conjecture. A. Goncharov defined a group of functional equations for classical polylogarithms. We show that any such functional equation formally follows from functional equations for Chow polylogarithms stated above.

Paper Structure

This paper contains 11 sections, 17 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. The complex $\Lambda$ and Beilinson-Soule vanishing conjecture
  3. Application to classical polylogarithms
  4. Beilinson-Soule vanishing for the complex $\Lambda(\mathbb K,m)$
  5. Preliminaries lemmas
  6. The filtration $\mathcal{F}$
  7. The proof of Theorem \ref{['th:BS_Lambda']}
  8. Functional equations for classical polylogarithms
  9. The group $\mathcal{R}_m(F)$
  10. The proof of Theorem \ref{['th:functional_equations_classical_polylogarithms']}
  11. The proof of Lemma \ref{['lemma:differential_of_polylogarithms']}

Key Result

Proposition 1.1

Let $k\geq 1$. There is a unique map $\mathop{\mathrm{\partial}}\nolimits_\nu\colon\Lambda^k F^\times\to \Lambda^{k-1}\overline{F}_\nu^{\times}$ such that the following property holds. Let $a_1,\dots, a_k\in F^\times$ and assume that $\nu(a_2)=\dots=\nu(a_k)$. Then $\mathop{\mathrm{\partial}}\nolimi

Theorems & Definitions (31)

  • Proposition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Lemma 2.1
  • proof
  • ...and 21 more