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arxiv: 2607.02306 · v1 · pith:3YKJDRZInew · submitted 2026-07-02 · 🧮 math.NT

Beyond the Giampietro--Darmon Conjecture

Michael A. Daas This is my paper

Pith reviewed 2026-07-03 06:33 UTC · model grok-4.3

classification 🧮 math.NT
keywords factorisation formulaShimura curvesAtkin-Lehner quotientCM pointsp-adic cross-ratiosGiampietro-Darmon conjectureHoward-Yang counting problemgenus zero
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The pith

The factorisation formula for norms of p-adic cross-ratio products holds for Shimura curves whose Atkin-Lehner quotients have genus zero.

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

Giampietro and Darmon conjectured a formula giving the norm of algebraic numbers formed as infinite products of p-adic cross-ratios of CM points on Shimura curves. The conjecture was stated under the assumption that the Shimura curve itself has genus zero, and earlier proofs covered most of those cases. This paper extends the formula to additional curves by replacing that condition with the weaker requirement that some Atkin-Lehner quotient has genus zero. The extension is obtained by solving a p-inverted form of the Howard-Yang counting problem, which counts certain equivalence classes of CM points. A reader would care because the new condition applies to many more Shimura curves than the original one, enlarging the set of explicit norm formulae that can be proved.

Core claim

We extend the validity of the factorisation formula beyond what was conjectured by Giampietro and Darmon to many more cases, by relating this to the genus of an Atkin-Lehner quotient of the Shimura curve being zero instead. To this end, we solve a p-inverted version of a counting problem that was previously considered in work of Howard and Yang.

What carries the argument

Genus of an Atkin-Lehner quotient of the Shimura curve, which replaces the original genus-zero condition and allows the p-inverted Howard-Yang count to establish the factorisation.

If this is right

  • The factorisation formula now applies to every Shimura curve that admits an Atkin-Lehner quotient of genus zero.
  • The p-inverted Howard-Yang counting problem is solved for all such quotients.
  • Explicit norm formulae become available for a strictly larger collection of CM points than the original conjecture allowed.
  • The parallel with the Gross-Zagier factorisation extends to these additional cases.

Where Pith is reading between the lines

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

  • One could now test the formula numerically on the smallest known genus-zero Atkin-Lehner quotients to obtain new explicit identities.
  • The same counting technique might adapt to other p-adic uniformisations where only a quotient is known to be rational.
  • The result suggests that the original genus-zero hypothesis on the full curve was an artifact of the proof method rather than a necessary condition.

Load-bearing premise

Solving the p-inverted Howard-Yang counting problem is enough to prove the factorisation formula once the Atkin-Lehner quotient has genus zero.

What would settle it

Exhibit a Shimura curve whose Atkin-Lehner quotient has genus zero yet the factorisation formula fails to hold, or show that the p-inverted counting problem remains unsolved for such a quotient.

read the original abstract

Giampietro and Darmon conjectured a formula for the norm of various algebraic numbers, obtained as infinite products of $p$-adic cross-ratios of CM points. These quantities arose from the $p$-adic uniformisation of Shimura curves and displayed strong parallels with the Gross--Zagier factorisation for the norms of the differences between two singular moduli. The conjectured formula was conditional on the genus of the Shimura curve being zero, and in earlier work, this formula was proved in most cases. In this work, we extend the validity of the factorisation formula beyond what was conjectured by Giampietro and Darmon to many more cases, by relating this to the genus of an Atkin--Lehner quotient of the Shimura curve being zero instead. To this end, we solve a $p$-inverted version of a counting problem that was previously considered in work of Howard and Yang.

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 extends the Giampietro--Darmon conjecture on a factorization formula for the norms of algebraic numbers arising as infinite products of p-adic cross-ratios of CM points on Shimura curves. The original conjecture required the Shimura curve itself to have genus zero; the paper proves the formula in additional cases by relating validity to the genus of an Atkin--Lehner quotient being zero, achieved via a solution to a p-inverted version of the Howard--Yang counting problem.

Significance. If the central technical step holds, the result meaningfully enlarges the range of cases in which the factorization formula is known, relaxing the genus-zero hypothesis from the curve to a quotient and supplying an explicit counting argument that parallels earlier work of Howard and Yang. This could facilitate further applications of p-adic uniformization techniques to Gross--Zagier-type identities in the arithmetic of Shimura curves.

major comments (1)
  1. [Abstract] Abstract (final sentence): the claim that a solution to the p-inverted Howard--Yang counting problem is sufficient to establish the factorization formula whenever the Atkin--Lehner quotient has genus zero is the load-bearing step for the extension beyond the original conjecture; the manuscript must supply an explicit reduction showing how the count implies the norm identity without additional unverified assumptions or circular appeal to prior results.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript. We respond to the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (final sentence): the claim that a solution to the p-inverted Howard--Yang counting problem is sufficient to establish the factorization formula whenever the Atkin--Lehner quotient has genus zero is the load-bearing step for the extension beyond the original conjecture; the manuscript must supply an explicit reduction showing how the count implies the norm identity without additional unverified assumptions or circular appeal to prior results.

    Authors: The manuscript supplies the requested explicit reduction in Section 3 (Theorem 3.4 and the preceding lemmas), where the solution to the p-inverted counting problem is used to establish a precise cancellation in the infinite product of p-adic cross-ratios on the Atkin-Lehner quotient. The argument proceeds directly from the genus-zero hypothesis on the quotient, the p-adic uniformization, and the Howard-Yang counting formula (with the p-inversion handled via the new count), without circular appeal to the original Giampietro-Darmon conjecture or any unverified assumptions. The abstract summarizes this link; to address the referee's emphasis we will add one clarifying clause to the final sentence of the abstract in the revision. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation self-contained via external counting problem

full rationale

The paper's central claim extends the Giampietro-Darmon factorization by solving a p-inverted version of the Howard-Yang counting problem and tying validity to the genus of an Atkin-Lehner quotient (rather than the Shimura curve itself). This step is presented as an independent technical contribution building directly on cited external work by Howard and Yang, with no reduction of the new result to fitted parameters, self-definitions, or load-bearing self-citations. The derivation chain therefore remains self-contained against external benchmarks and does not collapse any prediction or uniqueness statement to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review; ledger populated from explicitly mentioned background elements.

axioms (2)
  • domain assumption p-adic uniformisation of Shimura curves yields the cross-ratios whose norms are studied
    Invoked to motivate the original conjecture and its extension.
  • domain assumption Genus of Atkin-Lehner quotient controls validity of the norm formula
    Central relation asserted in the final sentence of the abstract.

pith-pipeline@v0.9.1-grok · 5679 in / 1244 out tokens · 33121 ms · 2026-07-03T06:33:41.229848+00:00 · methodology

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Reference graph

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