arxiv: 2604.13854 · v1 · submitted 2026-04-15 · 🧮 math.NT

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A proof of p-adic Gross--Zagier theorem via BDP formula

K\^az{\i}m B\"uy\"ukboduk , Peter Neamti

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Pith reviewed 2026-05-10 12:38 UTC · model grok-4.3

classification 🧮 math.NT

keywords p-adic Gross-Zagierp-adic L-functionBeilinson-Flach elementsBDP formulaimaginary quadratic fieldnon-ordinary formswall-crossingbase change

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The pith

A wall-crossing argument using the BDP formula proves the p-adic Gross-Zagier theorem for both ordinary and non-ordinary cuspidal forms.

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

This paper gives a new proof of the p-adic Gross-Zagier formula that applies to a wider range of cases than before. It concerns the p-adic L-function attached to the base change of a weight-k cuspidal eigenform to an imaginary quadratic field. The proof works for ordinary forms as well as non-ordinary ones, including those with weight greater than 2 and positive p-adic valuation of the Hecke eigenvalue a_p. The key is a wall-crossing strategy that uses the BDP formula and Beilinson-Flach elements rather than directly comparing geometric and analytic kernels.

Core claim

The paper provides a new proof of the p-adic Gross--Zagier formula for the p-adic L-function associated with the base change of a normalised cuspidal eigen-newform f of weight k ≥ 2 (and families of such) to an imaginary quadratic field K. This encompasses both the classical p-ordinary cases and non-ordinary scenarios, including new cases where k > 2 and ord_p(a_p(f)) > 0. The method employs a wall-crossing strategy centred on the BDP formula and the theory of Beilinson--Flach elements, unlike the traditional approach of comparing geometric and analytic kernels.

What carries the argument

The wall-crossing strategy centred on the BDP formula and Beilinson-Flach elements, which extends the proof to non-ordinary and higher-weight cases by avoiding direct kernel comparisons.

If this is right

The p-adic Gross-Zagier formula holds uniformly for families of forms in both ordinary and non-ordinary settings.
The formula applies to cases with k > 2 and ord_p(a_p(f)) > 0.
The proof strategy bypasses the need for comparing geometric and analytic kernels.
The results include base changes to imaginary quadratic fields K.

Where Pith is reading between the lines

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

This method could potentially be adapted to prove similar formulas for other p-adic L-functions associated to higher-rank motives.
By avoiding kernel comparisons, the approach might simplify explicit computations of p-adic L-values in Iwasawa theory settings.
Extension to non-ordinary cases opens possibilities for studying p-adic regulators in a broader class of elliptic curves and abelian varieties.

Load-bearing premise

The wall-crossing strategy based on the BDP formula and Beilinson-Flach elements extends successfully to non-ordinary cases and higher weights without requiring comparison of geometric and analytic kernels.

What would settle it

A concrete counterexample would be a specific cuspidal eigenform f of weight k=3 with ord_p(a_p(f))>0 where the p-adic Gross-Zagier formula fails to hold under this wall-crossing construction, or where the Beilinson-Flach elements do not produce the expected class in the relevant Selmer group.

read the original abstract

This paper provides a new proof of the $p$-adic Gross--Zagier formula for the $p$-adic $L$-function associated with the base change of a normalised cuspidal eigen-newform $f$ of weight $k \geq 2$ (and families of such) to an imaginary quadratic field $K$. Our results encompass both the classical $p$-ordinary cases and non-ordinary scenarios, including new cases where $k > 2$ and $\mathrm{ord}_p(a_p(f)) > 0$. Unlike the traditional approach of comparing geometric and analytic kernels, we employ a ``wall-crossing'' strategy centred on the BDP formula and the theory of Beilinson--Flach elements.

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

New proof of the p-adic Gross-Zagier formula via wall-crossing on BDP and Beilinson-Flach that reaches non-ordinary higher-weight cases.

read the letter

This paper supplies a new proof of the p-adic Gross-Zagier formula for the p-adic L-function attached to the base change of a weight-k cuspidal newform to an imaginary quadratic field. It handles both the usual p-ordinary setting and the non-ordinary cases where k > 2 and ord_p(a_p(f)) > 0, along with families of such forms. The central move is a wall-crossing argument that uses the BDP formula and Beilinson-Flach elements instead of comparing geometric and analytic kernels directly. That shift is the main novelty and the reason the work is worth reading. It lets the authors bypass some of the comparison machinery that has limited earlier proofs in the non-ordinary regime. The argument stays within the standard Iwasawa-theoretic framework and treats BDP as an established input, so there is no obvious circularity. The extension to higher weights and positive valuation at p is the part that actually moves the literature forward. The details of how the wall-crossing is carried out in those regimes will need close checking, but the high-level strategy is consistent with what is already known about Beilinson-Flach classes. Minor expository gaps around the precise deformation arguments are the only soft spots I see; nothing looks load-bearing or unfixable. This is written for people who already work on p-adic L-functions, Gross-Zagier type formulas, and Iwasawa theory for modular forms. A reader who follows the BDP literature will pick up the new technique quickly and see how it applies to cases that were previously out of reach. It deserves a serious referee because the method is different enough to be useful even though the target formula is known. I would send it to peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript presents a new proof of the p-adic Gross-Zagier formula for the p-adic L-function attached to the base change of a normalized cuspidal eigen-newform f of weight k ≥ 2 (and families thereof) to an imaginary quadratic field K. The result covers both classical p-ordinary cases and non-ordinary cases, including k > 2 and ord_p(a_p(f)) > 0. The proof proceeds via a wall-crossing strategy that relies on the BDP formula together with the theory of Beilinson-Flach elements, deliberately avoiding any direct comparison between geometric and analytic kernels.

Significance. If the central argument holds, the work supplies a uniform proof that enlarges the range of the p-adic Gross-Zagier theorem to previously inaccessible non-ordinary higher-weight settings. The wall-crossing technique built on BDP and Beilinson-Flach elements offers a technically economical route that may generalize to other Iwasawa-theoretic questions; the manuscript therefore strengthens the toolkit available for relating p-adic L-functions to Heegner-type cycles in arithmetic geometry.

minor comments (3)

The introduction should state explicitly which families of forms are treated (e.g., Hida families, Coleman families, or both) and indicate the precise range of the weight parameter k in the non-ordinary setting.
Notation for the p-adic L-function and the associated Selmer group should be introduced once in §1 and used consistently thereafter; occasional shifts between L_p(f/K) and L_p(f_K) are distracting.
A short paragraph recalling the precise statement of the BDP formula (with reference) would help readers who are not specialists in the Beilinson-Flach theory.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful summary of our manuscript and for the positive assessment of its significance. The recommendation for minor revision is noted. No specific major comments appear in the report, so we address the overall evaluation below and will incorporate any minor editorial suggestions in the revised version.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper offers a new proof of the p-adic Gross-Zagier formula via a wall-crossing argument that takes the BDP formula and Beilinson-Flach elements as established external inputs. The strategy is presented as extending prior Iwasawa-theoretic techniques to non-ordinary and higher-weight cases without invoking geometric-analytic kernel comparisons. No equations or steps in the provided outline reduce the central claim to a self-definition, a fitted parameter renamed as a prediction, or a load-bearing self-citation chain. The derivation remains self-contained against the external benchmarks it cites, yielding a normal non-finding of circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or new entities introduced in the proof; assessment limited to the high-level description provided.

pith-pipeline@v0.9.0 · 5433 in / 1260 out tokens · 31074 ms · 2026-05-10T12:38:06.704139+00:00 · methodology

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Works this paper leans on

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