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arxiv: 2604.20662 · v1 · submitted 2026-04-22 · 🧮 math.NT · math.AG

p-adic elliptic polylogarithms and cubic Chabauty

Jennifer S. Balakrishnan , Francesca Bianchi , Netan Dogra This is my paper

Pith reviewed 2026-05-09 23:01 UTC · model grok-4.3

classification 🧮 math.NT math.AG
keywords Chabauty-Coleman-Kim methodp-adic elliptic polylogarithmselliptic curvesintegral pointsKim's conjecturep-adic L-functionsMordell-Weil rank
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The pith

p-adic elliptic polylogarithms give an explicit containing set for the depth 3 Chabauty-Coleman-Kim points on rank at most 2 elliptic curves

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

The paper extends the Chabauty-Coleman-Kim method to depth three for elliptic curves of Mordell-Weil rank at most two. It produces an explicit polynomial in the p-adic elliptic polylogarithms whose zeros contain the full depth-three Kim set, assuming a special value of the p-adic L-function is nonzero. This gives a concrete, finite list of p-adic points that must include every integral point on the curve. The construction is used to check Kim's conjecture on new examples by testing which candidate zeros are actually integral. A reader cares because the formula turns an otherwise abstract description of the Kim set into something that can be written down and evaluated.

Core claim

Under the assumption that a special value of the p-adic L-function is non-zero, the Chabauty-Coleman-Kim set in depth 3 for an elliptic curve of rank at most 2 is contained in the zero set of a polynomial whose terms are p-adic elliptic polylogarithms. The authors derive this formula and use it to confirm new instances of Kim's conjecture.

What carries the argument

The p-adic elliptic polylogarithm, a p-adic analytic function on the elliptic curve that generalizes the p-adic logarithm and supplies the higher-depth invariants needed for the depth-three Kim method.

If this is right

  • The integral points of the curve must lie among the finitely many zeros of this explicit polynomial.
  • Kim's conjecture can be tested on additional elliptic curves by checking which of these zeros are integral.
  • The method upgrades the quadratic formula known for rank-one curves to a higher-degree polynomial for rank-two curves.
  • Explicit computation of the polylogarithms becomes a practical step toward determining all integral points.

Where Pith is reading between the lines

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

  • If the non-vanishing condition holds broadly, the approach could supply an effective algorithm for locating all integral points on rank-two elliptic curves.
  • The link to p-adic L-functions suggests that analytic properties of the curve control the size of the depth-three Kim set.
  • Comparable polynomial descriptions may exist at greater depth provided similar non-vanishing hypotheses are available.

Load-bearing premise

A special value of the p-adic L-function attached to the elliptic curve is non-zero.

What would settle it

An explicit elliptic curve of rank two whose p-adic L-function special value is nonzero, yet one of its integral points lies outside the zero set of the constructed polynomial in the p-adic elliptic polylogarithms.

read the original abstract

The Chabauty--Coleman--Kim method, under favourable circumstances, describes the set of integral points of a hyperelliptic curve inside the $p$-adic zeroes of certain transcendental functions. For an elliptic curve of Mordell--Weil rank one, the Chabauty--Coleman--Kim set in depth 2 is given by the zeroes of a (finite union of) quadratic polynomial(s) in the $p$-adic logarithm of the elliptic curve and the local $p$-adic height at $p$. Here, we give an explicit formula for a finite set containing the Chabauty--Coleman--Kim set in depth 3 for an elliptic curve of rank at most 2 under an assumption on non-vanishing of a special value of a $p$-adic $L$-function. The finite set is given by the zeroes of a polynomial in $p$-adic elliptic polylogarithms. We use these formulas to verify new instances of Kim's conjecture.

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

2 major / 2 minor

Summary. The manuscript gives an explicit formula for a finite set containing the depth-3 Chabauty–Coleman–Kim set on an elliptic curve of Mordell–Weil rank at most 2, conditional on the non-vanishing of a special value of a p-adic L-function. The set is realized as the zero locus of a polynomial whose coefficients are p-adic elliptic polylogarithms; the formula is then used to verify additional instances of Kim’s conjecture.

Significance. If the derivation is correct, the work supplies the first explicit, computable description of the cubic-level CCK set for rank ≤2 elliptic curves. This extends the range of curves for which the method yields effective bounds on integral points and furnishes new, unconditional verifications of Kim’s conjecture under a single, explicitly stated arithmetic hypothesis.

major comments (2)
  1. The non-vanishing hypothesis on the p-adic L-function special value is load-bearing for both finiteness and the explicit polynomial description. The manuscript should state precisely for which primes p and which curves the hypothesis is known or expected to hold, with references to existing results on p-adic L-functions of elliptic curves.
  2. The construction of the polynomial in the p-adic elliptic polylogarithms is asserted to contain the depth-3 CCK set, but the precise relation between the polylogarithm values and the unipotent fundamental group filtration (or the corresponding Selmer variety) is not made fully explicit in the main theorem statement.
minor comments (2)
  1. Notation for the p-adic elliptic polylogarithms should be introduced with a brief reminder of the normalization used (e.g., which Coleman function or which branch of the logarithm).
  2. The statement of Kim’s conjecture in the introduction would benefit from a one-sentence reminder of its precise formulation for elliptic curves.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive recommendation for minor revision. We address the two major comments point by point below and will incorporate the suggested clarifications into the revised version.

read point-by-point responses

  1. Referee: The non-vanishing hypothesis on the p-adic L-function special value is load-bearing for both finiteness and the explicit polynomial description. The manuscript should state precisely for which primes p and which curves the hypothesis is known or expected to hold, with references to existing results on p-adic L-functions of elliptic curves.

    Authors: We agree that the non-vanishing hypothesis plays a central role. In the revised manuscript we will add a dedicated paragraph, placed after the statement of the main theorem, that summarizes the known and expected cases. This will include: (i) the ordinary good-reduction case at p where non-vanishing follows from the p-adic Gross–Zagier formula and results of Perrin-Riou; (ii) the supersingular case under the assumption that the p-adic L-function is non-zero at the central point, with references to the work of Kato, Pollack, and Stevens; and (iii) a brief discussion of the expected density of such primes for a fixed curve of rank at most 2. Appropriate citations to the literature on p-adic L-functions of elliptic curves will be included. revision: yes

  2. Referee: The construction of the polynomial in the p-adic elliptic polylogarithms is asserted to contain the depth-3 CCK set, but the precise relation between the polylogarithm values and the unipotent fundamental group filtration (or the corresponding Selmer variety) is not made fully explicit in the main theorem statement.

    Authors: We thank the referee for highlighting this point. We will revise the statement of the principal theorem to include a short explanatory sentence (or footnote) that explicitly identifies the polynomial as the image, under the natural map from the depth-3 unipotent fundamental group to the graded pieces of the Selmer variety, of the p-adic elliptic polylogarithms evaluated at the relevant points. This will make the link between the polylogarithm values and the filtration on the unipotent group transparent without changing the mathematical content of the result. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper states an explicit formula for a finite set containing the depth-3 Chabauty-Coleman-Kim set for elliptic curves of rank at most 2, conditional on an explicit non-vanishing hypothesis for a p-adic L-function special value. This hypothesis is presented as an external assumption rather than derived from the paper's own constructions. The p-adic elliptic polylogarithms are drawn from prior literature as standard tools, with no indication that their definitions reduce to the target result by construction or that fitted parameters are relabeled as predictions. The central derivation applies existing p-adic methods to higher depth without load-bearing self-citations that collapse the claim to its inputs, and the verifications of Kim's conjecture are independent applications. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard p-adic analytic constructions plus one explicit non-vanishing assumption; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • standard math Standard properties of p-adic logarithms, heights, and polylogarithms on elliptic curves
    Invoked throughout the construction of the Chabauty-Coleman-Kim sets and the polynomial.
  • ad hoc to paper Non-vanishing of a special value of a p-adic L-function
    Explicitly required for the finite set to contain the depth-3 Chabauty-Coleman-Kim set.

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