Unramified Euler sums and Hoffman $\star$ basis

Abstract

When looking at how periods of π m 1 (P 1 {0, 1, ∞}), i.e. multiple zeta values, embeds into periods of π m 1 (P 1 {0, ±1, ∞}), i.e. Euler sums, an explicit criteria via the coaction ∆ acting on their motivic versions I comes out. In this paper, adopting this Galois descent approach, we present a new basis for the space H 1 of motivic multiple zeta values via motivic Euler sums. Up to an analytic conjecture II , we also prove that the motivic Hoffman star basisUnder a general motivic identity that we conjecture III , these bases are identical. Other examples of unramified ES with alternating patterns of even and odds are also provided. Contents 1 Introduction 1 2 Motivic Background 5 3 Definitions, antipode and hybrid relations 13 4 Some Unramified Euler sums 21 5 Hoffman basis 24 6 Motivic Linebarger Zhao conjecture 31 Appendices 33 I Following Brown's point of view. II Similar to Zagier's one for Hoffman basis case done by F. Brown. III The motivic version of a Linebarger Zhao's identity, [32].

On motivic multiple $t$ values, Saha's basis conjecture, and generators of alternating MZV's

“…This was extended by Glanois [10,11] to the case of alternating motivic MZV's (and motivic MZV's at higher roots of unity). This again (often) allows one to recursively lift identities of real alternating MZV's to alternating motivic MZV's, by recursively verifying D <N vanishes, and using the numerical identity to fix the final unknown coefficient of ζ m (N ) via the period map.…”

“…Now substitute in the evaluation of t({2} a ) = π 2a 2 2a (2a)! from (11), and convert the last term to a classical (non-alternating) MZV, to obtain…”

“…Glanois, Corollary 2.4.5[10], Theorem 2.2[11]). The kernel of D <N on alternating motivic MZV's is 1 dimensional in weight N , and spanned by ζ m (N ), ker D <N ∩ H(2) = ζ m (N )Q .…”

On a variant of multiple zeta values of level two

“…. , k r ) ∈ Z if ∀k i ≥ 2, by using the motivic method employed in [7]. Also he could prove Conjecture 5.3 or Conjecture 5.2 except the 'only' parts.…”

On variants of the Euler sums and symmetric extensions of the Kaneko-Tsumura conjecture

“…That is, these sums are expressible in terms of MZVs. In 2021, Murakami [18,Theorem 42] proved this conjecture by using the motivic method employed in [8] (see also [14,Remark 5.6]).…”