Zagier’s polylogarithm conjecture for primitive elements of the motivic Lie coalgebra

Prove that for any number field F and integer n ≥ 1, the primitive subspace of the degree-n component of the motivic Lie coalgebra C(F) is spanned by Q-linear combinations of motivic polylogarithms Li_n^C(x) with x ∈ F \ {0,1}; equivalently, show that these elements generate K_{2n−1}(F)_Q via the identification K_{2n−1}(F)_Q ≅ ker(δ: C_n(F) → (Λ^2 C(F))_n).

Background

The motivic Hopf algebra H(F) of mixed Tate motives over a number field F has indecomposables forming the motivic Lie coalgebra C(F). Motivic polylogarithms Li_nC(x) arise from polylogarithm motives and have an explicit cobracket compatible with polylogarithm differential equations.

This conjecture asserts that these motivic polylogarithms suffice to span the primitive part in each weight, thereby capturing K_{2n−1}(F)_Q. Combined with the identification of primitives with K-theory, it would give a conceptual and explicit description of algebraic K-groups of number fields.

References

Zagier's polylogarithm conjecture predicts that motivic polylogarithms are enough to capture all primitive elements in the motivic Lie coalgebra, i.e., to span the algebraic K-theory of number fields by eq: K theory inside Lie coalgebra. Conjecture. The space of primitive elements in $\mathcal{C}_n(F)$ is spanned by $Q$-linear combinations of elements $Li_n{\mathcal{C}(x)$, for $x\in F\setminus {0,1}$.

eq: K theory inside Lie coalgebra:

K2n−1(F)Q≃ker⁡(δ:Cn(F)⟶(Λ2C(F))n).K_{2n-1}(F)_Q \simeq \ker\left(\delta : \mathcal{C}_n(F)\longrightarrow (\Lambda^2\mathcal{C}(F))_n \right).

— An introduction to mixed Tate motives (2404.03770 - Dupont, 4 Apr 2024) in Section 9.1 (Motivic polylogarithms)