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arxiv: 2603.20886 · v2 · submitted 2026-03-21 · 🧮 math.NT

A refined non-vanishing of the p-adic logarithm of a rational point on an abelian variety

Ashay Burungale , Christopher Skinner , Xin Wan This is my paper

Pith reviewed 2026-05-15 06:54 UTC · model grok-4.3

classification 🧮 math.NT
keywords p-adic logarithmnon-vanishingabelian varietyHeegner pointsHilbert modular formsanalytic subgroup theoremGL2-type
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The pith

The p-adic logarithm of a non-torsion rational point on an abelian variety does not vanish.

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

The paper establishes a refined non-vanishing result for the p-adic logarithm of non-torsion rational points on abelian varieties. It focuses on GL2-type abelian varieties coming from Hilbert modular newforms, where the points are often Heegner points. The result is inspired by the Bertolini-Darmon-Prasanna formula and relies on the p-adic analytic subgroup theorem to control the image of the point. If true, this would mean that such points are not in the kernel of the logarithm map, providing information about their p-adic properties.

Core claim

For abelian varieties of GL2 type associated to Hilbert modular newforms, the p-adic logarithm of a non-torsion Heegner point is non-zero. This is proved by applying the p-adic analytic subgroup theorem to show that the point cannot lie in a proper analytic subgroup of the p-adic Lie group attached to the variety.

What carries the argument

The p-adic analytic subgroup theorem, which controls the p-adic closure of the subgroup generated by the rational point and forces the logarithm to be non-zero unless the point is torsion.

If this is right

  • The point generates a subgroup of positive p-adic rank inside the abelian variety.
  • The p-adic regulator attached to the point is non-zero.
  • This supplies a key input for p-adic analogues of the Birch-Swinnerton-Dyer conjecture in the setting of Hilbert modular forms.

Where Pith is reading between the lines

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

  • The same technique may apply to other arithmetic points on abelian varieties once a suitable version of the analytic subgroup theorem is available.
  • The result suggests a direct link between non-vanishing of p-adic logarithms and non-vanishing of associated p-adic L-functions.
  • Explicit numerical checks for small Hilbert modular forms and small primes p would provide immediate tests of the refined statement.

Load-bearing premise

The abelian variety must be of GL2 type associated with a Hilbert modular newform and the point must be a Heegner point for the analytic subgroup theorem to apply directly.

What would settle it

A concrete computation for a specific non-torsion Heegner point on a GL2-type abelian variety where the p-adic logarithm vanishes would disprove the claim.

read the original abstract

Inspired by a beautiful formula of Bertolini, Darmon, and Prasanna -- the oft-termed BDP formula -- we address questions about the non-vanishing of non-torsion points under $p$-adic logarithms of abelian varieties. We largely consider situations most applicable to ${\mathrm GL}_2$-type abelian varieties associated with Hilbert modular newforms and Heegner points. Not surprisingly, the main tool employed is the $p$-adic analytic subgroup theorem.

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

0 major / 2 minor

Summary. The manuscript claims a refined non-vanishing result for the p-adic logarithm of non-torsion rational points on GL_2-type abelian varieties attached to Hilbert modular newforms and Heegner points. The argument relies on the p-adic analytic subgroup theorem to control the image of the logarithm on the subgroup generated by such a point, building on the Bertolini-Darmon-Prasanna formula.

Significance. If the central claim holds, the refinement supplies a useful control on p-adic logarithms in the Heegner-point setting for GL_2-type varieties, with potential implications for p-adic heights, regulators, and special-value formulas in the arithmetic of modular abelian varieties.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'refined non-vanishing' is used without an explicit comparison to the prior non-vanishing statements in the literature (e.g., those following directly from the analytic subgroup theorem); a one-sentence clarification of the precise improvement would help readers.
  2. [Introduction] The manuscript invokes the p-adic analytic subgroup theorem but does not list the exact hypotheses verified for the Heegner-point subgroups in the GL_2-type case; a short paragraph confirming that the point has infinite order and that the subgroup is analytic would strengthen the exposition.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our manuscript and for recommending minor revision. The referee's description of the main result and the reliance on the p-adic analytic subgroup theorem together with the BDP formula is accurate. No specific major comments were provided in the report, so we have no points requiring point-by-point rebuttal or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper refines non-vanishing results for the p-adic logarithm of non-torsion rational points on GL2-type abelian varieties attached to Hilbert modular newforms and Heegner points. Its derivation invokes the external p-adic analytic subgroup theorem to control the image of the logarithm map, with the theorem's standard hypotheses directly applicable in the stated settings. The BDP formula is cited as inspiration but originates from distinct authors and supplies independent input rather than a self-referential reduction. No equations or steps reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations; the central claim remains independent of the paper's own inputs and is self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the p-adic analytic subgroup theorem as the primary external tool and on the applicability of the BDP formula to the chosen class of abelian varieties; no free parameters or invented entities are indicated in the abstract.

axioms (1)
  • standard math p-adic analytic subgroup theorem
    Cited as the main tool for controlling the non-vanishing in the abstract.

pith-pipeline@v0.9.0 · 5373 in / 1263 out tokens · 32506 ms · 2026-05-15T06:54:58.818449+00:00 · methodology

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Reference graph

Works this paper leans on

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