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The p-adic period conjecture for motivic iterated integrals

Let ZZ be an open integer scheme, let \pf\pf be a closed point of ZZ with Z\pf≅Spec⁡(Zp)Z_{\pf}\cong\operatorname{Spec}(\mathbb{Z}_p), and let A(Z)A(Z) be the ring of motivic iterated integrals equipped with the \pf\pf-adic period map

per⁡\pf ⁣:A(Z)→Qp.\operatorname{per}_{\pf}\colon A(Z)\to\mathbb{Q}_p.

The p-adic period conjecture. For every open integer scheme ZZ, the period map per⁡\pf ⁣:A(Z)→Qp\operatorname{per}_{\pf}\colon A(Z)\to\mathbb{Q}_p is injective. This is presented as a pp-adic analogue of a portion of the Kontsevich–Zagier period conjecture; the source notes an abelian-number-field case in the literature but does not state a general proof.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The p-adic period conjecture for motivic iterated integrals

    Let ZZ be an integer scheme with fraction field KK, and let p\mathfrak{p} be a closed point of ZZ. Let

    per⁡p:A(Z)⟶Kp\operatorname{per}_\mathfrak{p}:A(Z)\longrightarrow K_\mathfrak{p}

    be the pp-adic period map associated with Besser–Coleman integration. The pp-adic period conjecture. The map per⁡p\operatorname{per}_\mathfrak{p} is injective.

    Injectivity would say that motivic iterated-integral functions are detected by their pp-adic periods. The source states this conjecture and cites Yamashita for it, but gives no resolution here.

    source: Ishai Dan-Cohen, “Mixed Tate motives and the unit equation II”, arXiv:1510.01362 (2019).

Sources & referencesView supporting material

Primary source

David Corwin and Ishai Dan-Cohen, “The polylog quotient and the Goncharov quotient in computational Chabauty-Kim theory I”, arXiv:1812.05707 (2020).

chabauty-kimmixed-tate-motivesmotivic-cohomologyp-adic-cohomologyp-adic-periodsperiod-conjectures

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