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Zagier-Hoffman's conjectures in positive characteristic II

Bo-Hae Im, Hojin Kim, Khac Nhuan Le, Tuan Ngo Dac, Lan Huong Pham

Abstract

Zagier-Hoffman's conjectures predict the dimension and a basis for the $\mathbb Q$-vector spaces spanned by $N$th cyclotomic multiple zeta values (MZV's) of fixed weight where $N$ is a natural number. For $N=1$ (MZV's case), half of these conjectures have been solved by the work of Terasoma, Deligne-Goncharov and Brown with the help of Zagier's identity. The other half are completely open. For $N=2$ (alternating MZV's case) and $N=3,4,8$, Deligne-Goncharov and Deligne solved the same half of these conjectures for $N$th-cyclotomic MZV's. For other values of $N$, no sharp upper bound on the dimension is known. In this paper we completely establish, for all $N$, Zagier-Hoffman's conjectures for $N$th cyclotomic multiple zeta values in positive characteristic. By working with the tower of all cyclotomic extensions, we present a proof that is uniform on $N$ and give an effective algorithm to express any cyclotomic multiple zeta value in the chosen basis. This generalizes all previous work on these conjectures for MZV's and alternating MZV's in positive characteristic.

Zagier-Hoffman's conjectures in positive characteristic II

Abstract

Zagier-Hoffman's conjectures predict the dimension and a basis for the -vector spaces spanned by th cyclotomic multiple zeta values (MZV's) of fixed weight where is a natural number. For (MZV's case), half of these conjectures have been solved by the work of Terasoma, Deligne-Goncharov and Brown with the help of Zagier's identity. The other half are completely open. For (alternating MZV's case) and , Deligne-Goncharov and Deligne solved the same half of these conjectures for th-cyclotomic MZV's. For other values of , no sharp upper bound on the dimension is known. In this paper we completely establish, for all , Zagier-Hoffman's conjectures for th cyclotomic multiple zeta values in positive characteristic. By working with the tower of all cyclotomic extensions, we present a proof that is uniform on and give an effective algorithm to express any cyclotomic multiple zeta value in the chosen basis. This generalizes all previous work on these conjectures for MZV's and alternating MZV's in positive characteristic.
Paper Structure (44 sections, 27 theorems, 182 equations)

This paper contains 44 sections, 27 theorems, 182 equations.

Table of Contents

  1. Classical multiple zeta values
  2. $N=1$ (the case of MZV's)
  3. $N=2$ (the case of alternating MZV's) and $N=3,4,8$
  4. Other values of $N$
  5. Multiple zeta values in positive characteristic
  6. Statement of the main theorems
  7. Main results
  8. Known results
  9. Outline of the proof and plan of the paper
  10. Part A: an analogue of Brown-Deligne's theorems
  11. Part B: dual $t$-motives connected to CMPL's at roots of unity
  12. Plan of the paper
  13. Cyclotomic MZV's and CMPL's at roots of unity
  14. Notation
  15. Multiple zeta values in positive characteristic
  16. ...and 29 more sections

Key Result

Theorem 3

For $k \in \mathbb{N}$ we have $\dim_{\mathbb Q} \mathcal{Z}_k \leq d_k$.

Theorems & Definitions (48)

  • Conjecture 1: Zagier's conjecture
  • Conjecture 2: Hoffman's conjecture
  • Theorem 3: Deligne-Goncharov, Terasoma
  • Theorem 4: Brown
  • Theorem A: Hoffman's conjecture in positive characteristic
  • Theorem B: Zagier's conjecture in positive characteristic
  • Remark 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Remark 1.4
  • ...and 38 more