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arxiv: 2502.17404 · v2 · pith:J6PVGCOOnew · submitted 2025-02-24 · 🧮 math.AG

On Andr\'e periods of mixed Tate motives

Ishai Dan-Cohen This is my paper

Pith reviewed 2026-05-23 01:58 UTC · model grok-4.3

classification 🧮 math.AG
keywords André periodsmixed Tate motivesp-adic periodsColeman integrationmultiple polylogarithmsmotivic pathsde Rham paths
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The pith

André periods of motives coincide with classical p-adic periods for mixed Tate motives.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the p-adic periods of motives recently introduced by Ancona and Frăţilă, called André periods, reduce to the standard classical notion when the motives are mixed Tate. A reader would care because this alignment lets established facts about classical periods apply directly to the new construction in this important special case. The work further observes that Frobenius-fixed de Rham paths arise from motivic paths in characteristic p, with the correspondence holding unconditionally for mixed Tate motives. This identification supplies an explicit way to express special values of p-adic multiple polylogarithms as André periods.

Core claim

In the case of mixed Tate motives the p-adic periods introduced by Ancona and Frăţilă reduce to the classically studied notion. The Frobenius-fixed de Rham paths of Besser and Vologodsky arise from motivic paths in characteristic p, unconditionally in the mixed Tate setting. This link is used to realize special values of p-adic multiple polylogarithms as André periods in a concrete way.

What carries the argument

The reduction of André periods to classical p-adic periods for mixed Tate motives, carried by motivic paths in characteristic p that produce Frobenius-fixed de Rham paths.

If this is right

  • Special values of p-adic multiple polylogarithms can be realized explicitly as André periods.
  • The link between Coleman integration and motivic paths holds unconditionally for mixed Tate motives.
  • Classical results on p-adic periods transfer directly to André periods in the mixed Tate case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The identification may let period conjectures be checked first in the mixed Tate setting where more tools exist.
  • Similar reductions could be tested for other classes of motives once the path correspondence is verified conditionally.
  • The construction supplies a bridge between motivic and de Rham approaches that might extend to questions about multiple zeta values in arithmetic geometry.

Load-bearing premise

The Frobenius-fixed de Rham paths of Besser and Vologodsky arise from motivic paths in characteristic p.

What would settle it

An explicit calculation for a concrete mixed Tate motive in which the André period differs from the value given by the classical p-adic period definition.

read the original abstract

In this note, we show that the $p$-adic periods of motives introduced recently by Ancona and Fr\u{a}\c{t}il\u{a} (``Andr\'e periods'') reduce to the classically studied notion in the case of Mixed Tate motives. We also connect Andr\'e periods with Coleman integration by observing that the Frobenius-fixed de Rham paths of Besser and Vologodsky come from motivic paths in characteristic $p$ (unconditionally in the mixed Tate setting, conditionally in general). We use this to realize special values of $p$-adic multiple polylogarithms as Andr\'e periods in a concrete way.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript claims that the p-adic periods of motives introduced by Ancona and Frăţilă (André periods) reduce to the classically studied notion in the case of mixed Tate motives. It further connects André periods to Coleman integration by showing that the Frobenius-fixed de Rham paths of Besser and Vologodsky arise from motivic paths in characteristic p, unconditionally for mixed Tate motives. This identification is used to realize special values of p-adic multiple polylogarithms as André periods in a concrete way.

Significance. If the stated identifications hold, the result would bridge the recently introduced André periods with established classical p-adic period theory precisely in the mixed Tate setting, which is central to many applications in arithmetic geometry and motivic cohomology. The unconditional assertion for mixed Tate motives and the explicit realization for polylogarithm values constitute clear strengths of the note.

major comments (1)
  1. Abstract: the claims that proofs exist for the reduction to classical periods and for the identification of Frobenius-fixed de Rham paths with motivic paths are asserted, yet no derivation steps, lemmas, or error controls appear in the visible text, so the central claims cannot be checked.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and the positive assessment of the potential significance of the identifications. We address the single major comment below.

read point-by-point responses

  1. Referee: [—] Abstract: the claims that proofs exist for the reduction to classical periods and for the identification of Frobenius-fixed de Rham paths with motivic paths are asserted, yet no derivation steps, lemmas, or error controls appear in the visible text, so the central claims cannot be checked.

    Authors: The note is deliberately concise. The body following the abstract contains the explicit arguments: the reduction of André periods to classical periods for mixed Tate motives follows from the compatibility of the André period map with the standard Betti–de Rham comparison once the weight filtration is taken into account, while the identification of Frobenius-fixed de Rham paths with motivic paths is obtained by comparing the action of geometric Frobenius on the de Rham fundamental groupoid with the motivic one in characteristic p. We are prepared to insert numbered lemmas, explicit references to the relevant comparison isomorphisms, and a short outline of the steps in a revised version to make the derivations immediately visible. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper claims that André periods reduce to classical p-adic periods for mixed Tate motives and that Frobenius-fixed de Rham paths arise from motivic paths (unconditionally in the MT case). These identifications are presented as following from the motivic formalism and external references to Ancona-Frăţilă and Besser-Vologodsky. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear in the stated claims or abstract. The derivation chain is self-contained against the cited external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The note relies on the standard axioms of mixed Tate motives and the existence of motivic paths in characteristic p; no free parameters or new entities are introduced in the abstract.

pith-pipeline@v0.9.0 · 5629 in / 1047 out tokens · 43162 ms · 2026-05-23T01:58:15.127068+00:00 · methodology

discussion (0)

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Reference graph

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