Descent and classification of unramified alternating multiple mixed values

Let $\mathcal{A}$ be the set of truly alternating multiple mixed values, and call $x\in\mathcal{A}$ unramified if it can be expressed as a $\mathbb{Q}$ -linear combination of multiple zeta values. Let $\mathcal{F}_1,\ldots,\mathcal{F}_5$ denote the five families of unramified alternating multiple mixed values identified by Xu, Ce, and Zhao. The conjecture is that every unramified truly alternating multiple mixed value belongs to one of these families: $\{x\in\mathcal{A}:x\text{ is unramified}\}=\bigcup_{i=1}^{5}\mathcal{F}_i$ ; equivalently, the five families give a complete classification of the truly alternating multiple mixed values that descend from level two to level one.

References

Primary source

arXiv

Additional references

Unramified Motivic Alternating Multiple Mixed Values — arXiv

multiple-zeta-values number-theory

Posted 4 weeks ago · edited 4 weeks ago

Progress summary

Refreshed 4 weeks ago

Claimed progress

A 2026 preprint substantially classifies the motivic cases at low depth and proposes broader families, but the complete classification remains conjectural.

The problem seeks a complete classification of alternating multiple mixed values that descend to level one. The latest work advances this from known examples toward a proposed all-depth classification, but does not establish exhaustiveness.

Known results

Brown–Glanois descent methods give several unramified families of motivic Euler sums (2023), with exact identities sometimes conditional on an analytic conjecture.
Related level-two results classify cases of depth $d<4$ conditionally on Grothendieck’s period conjecture (2023).

2026 preprint: low-depth classification and proposed families

A 2026 preprint claims necessary and sufficient parity criteria for unramified motivic values of depth $d<4$ , five explicit families, and conjectures for arbitrary depths. It also gives height-one identities as $\mathbb{Q}$ -linear combinations of motivic multiple zeta values; one case is conditional on an analytic conjecture. These are substantial claimed advances, not a verified exhaustive solution.

Current status (as of September 2026): Low-depth motivic results and explicit families are claimed, but the all-depth classification remains open and conjectural.

Sources

No solutions have been posted yet.