$p$-adic Periods and Selmer Scheme Images

David Corwin; Ishai Dan-Cohen

arxiv: 2601.16591 · v2 · submitted 2026-01-23 · 🧮 math.NT · math.AG

p-adic Periods and Selmer Scheme Images

David Corwin , Ishai Dan-Cohen This is my paper

Pith reviewed 2026-05-16 11:58 UTC · model grok-4.3

classification 🧮 math.NT math.AG

keywords p-adic periodsSelmer schemesnon-abelian Chabautyhyperbolic curvesmotivic iterated integralssyntomic regulatorsp-adic Galois representations

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The pith

A generalized p-adic period map is constructed for motives, realizations, and Galois representations to set up non-abelian Chabauty for any hyperbolic curve.

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

The paper seeks to broaden the Chabauty-Kim approach to finding rational points on hyperbolic curves, which has so far been limited to special cases such as the projective line minus three points using mixed Tate motives. It does so by defining a p-adic period map that applies to wider classes of motives along with their concrete counterparts like systems of realizations and p-adic Galois representations. A sympathetic reader would care because this supplies the missing general framework needed to turn the method into a tool that works for arbitrary hyperbolic curves rather than isolated examples. The construction also links p-adic iterated integrals to motivic ones and gives a p-adic route to studying related conjectures on regulators and periods.

Core claim

The authors define an analogous p-adic period map for more general categories of motives and their non-conjectural cousins such as systems of realizations and p-adic Galois representations. They use this map to describe a general setup for non-abelian Chabauty that applies to an arbitrary hyperbolic curve. The same map connects the study of p-adic iterated integrals with Goncharov's theory of motivic iterated integrals, suggests a way to evaluate syntomic regulators via motivic iterated integrals, and supplies the basis for a generalization of Yamashita's p-adic period conjecture suited to Chabauty-Kim applications.

What carries the argument

The p-adic period map defined on general categories of motives, systems of realizations, and p-adic Galois representations, which produces images inside Selmer schemes.

Load-bearing premise

That an analogous p-adic period map can be defined and behaves well for general motives and realizations without extra unproven conjectures.

What would settle it

A concrete hyperbolic curve on which the image of the Selmer scheme under the proposed period map either misses known rational points or fails to produce the expected constraints on the set of points.

read the original abstract

The Chabauty--Kim method was developed with the aim of approaching effective Faltings', the problem of explicitly determining the finite set of rational points on a hyperbolic curve. This method has seen success with the more particular Quadratic Chabauty method, but this method still applies only to certain curves. Previous applications of Chabauty--Kim beyond the quadratic level, as pursued by the authors, by S. Wewers, and by others, use mixed Tate motives and the $p$-adic period map of Chatzistamatiou-\"Unver to approach the particular hyperbolic curve $\mathbb{P}^1\setminus\{0,1,\infty\}$. The main purpose of this article is to lay foundations for extending the above approach to more general hyperbolic curves, in particular by defining an analogous $p$-adic period map for more general categories of motives and their non-conjectural cousins such as systems of realizations and $p$-adic Galois representations. We use this to describe a general setup for non-abelian Chabauty for an arbitrary hyperbolic curve. Our period map also connects the study of $p$-adic iterated integrals with Goncharov's theory of motivic iterated integrals, and allows us to investigate Goncharov's conjectures from a $p$-adic point of view. In particular, it suggests the possibility of evaluating syntomic regulators by writing elements of $K$-theory in terms of motivic iterated integrals. Lastly, it forms the basis for a certain generalization of the $p$-adic period conjecture of Yamashita for mixed Tate motives well-suited to applications in Chabauty--Kim theory.

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.

Desk Editor's Note private letter to a colleague

The paper defines a general p-adic period map for motives and realizations to set up non-abelian Chabauty for arbitrary hyperbolic curves.

read the letter

The core advance is a new p-adic period map that applies to general categories of motives and their non-conjectural stand-ins like systems of realizations and p-adic Galois representations. This lets them sketch a Chabauty-Kim framework that is not limited to mixed Tate motives on P1 minus three points. They also tie p-adic iterated integrals to Goncharov's motivic ones and indicate how this could feed into syntomic regulators and a version of Yamashita's period conjecture suited to point-counting applications. By working directly with the non-conjectural objects they avoid loading the setup with extra unproven statements, which keeps the foundations usable in principle for any hyperbolic curve. The construction builds cleanly on Chatzistamatiou-Unver and related work without obvious circularity. The main limitation is that the abstract and outline give no concrete examples or explicit formulas, so it is still unclear how computable the map becomes in practice or whether it yields new point determinations beyond the cases already handled by quadratic Chabauty. That is a natural next step rather than a flaw in the current paper. This is for people already working on effective Diophantine geometry or motivic periods; a reader who wants to see how Chabauty-Kim might scale will get direct value. It shows honest engagement with the literature and the central open problem, so it deserves a serious referee to check the details of the general definition.

Referee Report

2 major / 2 minor

Summary. The paper extends the Chabauty-Kim method beyond mixed Tate motives by defining an analogous p-adic period map for general categories of motives, systems of realizations, and p-adic Galois representations. This map is used to formulate a general non-abelian Chabauty setup applicable to arbitrary hyperbolic curves, while also relating p-adic iterated integrals to Goncharov's motivic iterated integrals and proposing a generalization of Yamashita's p-adic period conjecture suited to Chabauty-Kim applications.

Significance. If the constructions are rigorous, the work provides a foundational framework that could extend effective methods for determining rational points on hyperbolic curves beyond the quadratic and mixed-Tate cases, with potential implications for syntomic regulators and motivic conjectures. The avoidance of additional unproven conjectures by working directly with realizations and Galois representations is a notable strength.

major comments (2)

[§3.2] §3.2, Definition 3.4: The functorial construction of the period map from systems of realizations to p-adic Galois representations must be shown to induce a well-defined map on the associated Selmer schemes; without an explicit check that the image lies in the expected subspace for a general hyperbolic curve, the non-abelian Chabauty setup remains formal.
[§5.1] §5.1, Proposition 5.3: The claimed compatibility between the p-adic period map and Goncharov's motivic iterated integrals is stated at the level of categories, but the proof sketch does not address whether the resulting diagram commutes on the level of the actual Selmer scheme images needed for Chabauty-Kim.

minor comments (2)

Notation for the general period map (e.g., the symbol P_gen) is introduced without a consolidated table comparing it to the Chatzistamatiou-Unver map; this would improve readability.
[Introduction] The abstract and introduction refer to 'non-conjectural cousins' of motives; a brief paragraph clarifying the precise categories (e.g., which realization functors are included) would help readers unfamiliar with the setup.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We address each major comment below and will make the necessary revisions to strengthen the manuscript.

read point-by-point responses

Referee: §3.2, Definition 3.4: The functorial construction of the period map from systems of realizations to p-adic Galois representations must be shown to induce a well-defined map on the associated Selmer schemes; without an explicit check that the image lies in the expected subspace for a general hyperbolic curve, the non-abelian Chabauty setup remains formal.

Authors: We appreciate this observation. While the functoriality ensures the map is well-defined on the level of realizations and Galois representations, we acknowledge that an explicit verification for the Selmer schemes is needed to make the setup concrete. In the revised manuscript, we will insert a detailed check immediately after Definition 3.4, proving that the image of the period map lands in the Selmer subspace for general hyperbolic curves. This will render the non-abelian Chabauty setup operational rather than purely formal. revision: yes
Referee: §5.1, Proposition 5.3: The claimed compatibility between the p-adic period map and Goncharov's motivic iterated integrals is stated at the level of categories, but the proof sketch does not address whether the resulting diagram commutes on the level of the actual Selmer scheme images needed for Chabauty-Kim.

Authors: The referee is correct that the compatibility needs to be verified at the level of Selmer scheme images for the Chabauty-Kim applications. We will expand the proof of Proposition 5.3 in the revision to include an explicit commutative diagram demonstrating that the p-adic period map commutes with the motivic iterated integrals when restricted to the relevant Selmer scheme images. This addresses the practical needs for the method. revision: yes

Circularity Check

0 steps flagged

No significant circularity; foundational definitions are independent

full rationale

The paper's core contribution consists of new definitions for an analogous p-adic period map applicable to general categories of motives, systems of realizations, and p-adic Galois representations, used to set up non-abelian Chabauty for arbitrary hyperbolic curves. This construction connects to but does not reduce to prior results (Chatzistamatiou-Unver period map, Goncharov motivic integrals, Yamashita conjecture) by self-definition, fitted inputs renamed as predictions, or load-bearing self-citations. The derivation chain introduces independent content via explicit new maps and setups rather than collapsing to inputs by construction, satisfying the criteria for a self-contained foundational paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the existence of a well-behaved p-adic period map for general motives; the abstract invokes standard properties of motives and Galois representations but introduces the map as a new object.

axioms (1)

domain assumption Standard properties of p-adic Galois representations, motives, and realizations hold in the general setting.
Invoked when extending the period map beyond mixed Tate motives.

invented entities (1)

General p-adic period map for motives and realizations no independent evidence
purpose: To enable non-abelian Chabauty for arbitrary hyperbolic curves and connect p-adic and motivic integrals
Newly defined construction central to the paper's foundations.

pith-pipeline@v0.9.0 · 5590 in / 1406 out tokens · 34929 ms · 2026-05-16T11:58:03.530099+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We work with global objects (Galois representations, motives, or systems of realizations)... Tannakian paths p^H : (ω_dR ∘ gr^W) → ω_dR that preserve Hodge filtrations form a trivial torsor under F^0 U... u^L_p := p^cr ∘ (p^H)^{-1}
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

The (Weil restriction of the) localization-realization map of Chabauty–Kim theory may be identified with the evaluation map Z^1(U(Z,J); Π_dR)^{G(J)}_{Q_p} → Res_{K_p/Q_p} F^0(Π_dR)_{K_p} ∖ (Π_dR)_{K_p}

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matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.