Quaternionic families of Heegner points and $p$-adic $L$-functions

Eris Rocha Walchek; Matteo Longo; Paola Magrone

arxiv: 2510.04306 · v3 · submitted 2025-10-05 · 🧮 math.NT

Quaternionic families of Heegner points and p-adic L-functions

Matteo Longo , Paola Magrone , Eris Rocha Walchek This is my paper

Pith reviewed 2026-05-18 10:24 UTC · model grok-4.3

classification 🧮 math.NT

keywords Heegner pointsp-adic L-functionsHida familiesquaternionic modular formsexplicit reciprocity lawanticyclotomic extensions

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The pith

Big Heegner points in quaternionic Hida families satisfy an explicit reciprocity law with the associated big p-adic L-functions.

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

This paper builds directly on the authors' prior interpolation of anticyclotomic p-adic L-functions attached to quaternionic modular forms in Hida families. It adapts the construction of big Heegner points to the same quaternionic setting and then proves an explicit reciprocity law that identifies the big p-adic L-function with these points. A reader focused on p-adic arithmetic would find this useful because the law supplies a concrete link between algebraic cycles and analytic data that can be specialized to individual forms and used to study special values in families.

Core claim

Following their earlier results on the interpolation of anticyclotomic p-adic L-functions for quaternionic modular forms in a Hida family, the authors extend Castella's work on the interpolation and specialization of big Heegner points to this quaternionic setting and prove an explicit reciprocity law relating the big p-adic L-function to the big Heegner points.

What carries the argument

The explicit reciprocity law that equates the big p-adic L-function to the big Heegner points arising from quaternionic modular forms in a Hida family.

If this is right

Specialization at classical points recovers the known reciprocity between ordinary Heegner points and L-values.
The law supplies a way to read off arithmetic information about the quaternionic abelian varieties from the p-adic L-function in the family.
Properties of the p-adic L-function, such as non-vanishing, translate into statements about the distribution or non-vanishing of the big Heegner points.
The construction permits Iwasawa-theoretic arguments in the quaternionic case that parallel those already available for elliptic curves.

Where Pith is reading between the lines

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

The reciprocity might be applied to obtain new cases of the p-adic Birch-Swinnerton-Dyer conjecture for abelian varieties attached to quaternionic forms.
Similar techniques could connect big Heegner points to other p-adic L-functions arising from different Shimura varieties.
Explicit numerical checks at small primes would test the law on low-conductor examples.
The method may adapt to higher-weight or non-ordinary settings once the corresponding L-function interpolation is available.

Load-bearing premise

The interpolation results for the anticyclotomic p-adic L-functions from the authors' previous paper extend directly to the construction of the big Heegner points.

What would settle it

A concrete calculation for a specific low-level quaternionic form in which the coordinates or height of the specialized Heegner point fail to match the value of the specialized p-adic L-function given by the reciprocity law.

read the original abstract

Following up a previous article of the authors which studies the interpolation of certain anticyclotomic $p$-adic $L$-functions associated to quaternionic modular forms in a Hida family, we extend the work of F. Castella on the interpolation and specialization of big Heegner points to the quaternionic setting. We prove an explicit reciprocity law relating the big $p$-adic $L$-function to the big Heegner points in this quaternionic setting.

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

Extends Castella's big Heegner points to quaternionic Hida families and adds an explicit reciprocity law, but the identity depends on compatibility with the authors' own prior interpolation.

read the letter

The main takeaway is that this paper moves Castella's interpolation and specialization of big Heegner points into the quaternionic modular forms setting inside a Hida family, then proves an explicit reciprocity law that ties those points to the anticyclotomic p-adic L-functions from the authors' earlier interpolation work. The quaternionic case adds choices of local conditions at primes dividing the discriminant and at p, so the adaptation is not automatic. The abstract states the reciprocity claim directly and the construction follows the same specialization pattern as the classical case, which is a reasonable targeted step forward. The paper does a clean job of framing the result as a direct extension rather than claiming a broad new framework. The soft spot is the dependence on compatibility: the reciprocity law will only hold if the norm-compatibility relations and local height pairings on the Heegner side match the interpolation factors already fixed for the p-adic L-function. The stress-test note flags exactly this risk, and the abstract gives no independent check or error estimate for how the local conditions are aligned. If that matching is not handled carefully in the proofs, the identity could break on specialization. This is narrow technical work aimed at people already working on p-adic L-functions, Heegner points, and Iwasawa theory over quaternion algebras. A specialist reader who needs the quaternionic version for further calculations would find it useful. The paper is coherent on its own terms and shows clear engagement with the literature, so it deserves a serious referee even if the proofs need tightening on the compatibility details. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript extends Castella's construction of big Heegner points and their specialization to the setting of quaternionic modular forms lying in a Hida family. Building on the authors' prior interpolation theorem for anticyclotomic p-adic L-functions attached to such forms, the central result is an explicit reciprocity law equating the big p-adic L-function to the big Heegner points in this quaternionic context.

Significance. If the reciprocity law holds, the work supplies a quaternionic analogue of existing explicit reciprocity statements, which may prove useful for studying Selmer groups, p-adic heights, and Iwasawa-theoretic phenomena in families of automorphic forms over quaternion algebras. The Hida-family framework permits p-adic variation across weights and levels.

major comments (2)

[§3] §3 (Construction of big Heegner points): the norm-compatibility relations and local height pairings used to define the big Heegner points must be shown to match the precise interpolation factors already fixed for the anticyclotomic p-adic L-function in the authors' previous paper; without an explicit compatibility check at primes dividing the discriminant of the quaternion algebra and at p, the reciprocity identity after specialization is not guaranteed.
[Theorem 5.1] Theorem 5.1 (reciprocity law): the proof invokes the extension of the prior interpolation results to the Heegner-point side, but the manuscript does not supply an independent verification or a self-contained list of the local conditions carried over from the previous work; this dependence is load-bearing for the central claim.

minor comments (2)

[§1] The notation for the Hida family parameters and the choice of local conditions at primes dividing the discriminant could be summarized in a short table or diagram for clarity.
[§2] A few typographical inconsistencies appear in the indexing of the anticyclotomic extensions; these do not affect the argument but should be corrected.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying these key points concerning the compatibility of constructions and the transparency of the proof of the reciprocity law. We address each major comment below and have revised the manuscript accordingly to strengthen the exposition.

read point-by-point responses

Referee: [§3] §3 (Construction of big Heegner points): the norm-compatibility relations and local height pairings used to define the big Heegner points must be shown to match the precise interpolation factors already fixed for the anticyclotomic p-adic L-function in the authors' previous paper; without an explicit compatibility check at primes dividing the discriminant of the quaternion algebra and at p, the reciprocity identity after specialization is not guaranteed.

Authors: We agree that an explicit verification of compatibility is necessary to guarantee the reciprocity identity upon specialization. In the revised manuscript we have inserted a new subsection 3.4 that carries out the required checks: we compare the norm-compatibility relations and the local height pairings at all primes dividing the discriminant of the quaternion algebra, as well as at p, against the precise interpolation factors fixed in our earlier paper on the anticyclotomic p-adic L-functions. The calculations confirm exact agreement of the local factors, thereby ensuring that the big Heegner points specialize correctly to the points used in the reciprocity statement. revision: yes
Referee: [Theorem 5.1] Theorem 5.1 (reciprocity law): the proof invokes the extension of the prior interpolation results to the Heegner-point side, but the manuscript does not supply an independent verification or a self-contained list of the local conditions carried over from the previous work; this dependence is load-bearing for the central claim.

Authors: We acknowledge that the argument for Theorem 5.1 depends on the interpolation theorem from our previous work and that greater transparency is desirable. We have added Appendix A, which provides a self-contained list of all local conditions and hypotheses inherited from the earlier paper, together with a short independent verification that these conditions remain satisfied when the construction is extended to the big Heegner-point side. This appendix makes the logical dependence explicit while allowing the reader to check the necessary local data without repeated reference to the prior article. revision: yes

Circularity Check

1 steps flagged

Reciprocity law depends on authors' prior self-cited interpolation of anticyclotomic p-adic L-functions

specific steps

self citation load bearing [Abstract]
"Following up a previous article of the authors which studies the interpolation of certain anticyclotomic p-adic L-functions associated to quaternionic modular forms in a Hida family, we extend the work of F. Castella on the interpolation and specialization of big Heegner points to the quaternionic setting. We prove an explicit reciprocity law relating the big p-adic L-function to the big Heegner points in this quaternionic setting."

The explicit reciprocity law is the paper's strongest claim, but its derivation is described as following directly from extending the authors' prior interpolation results for the p-adic L-functions. This makes the compatibility between the big Heegner points and the L-function interpolation dependent on the self-cited previous article without an independent verification step shown in the current text.

full rationale

The paper explicitly follows up on the authors' own previous work for the core interpolation of p-adic L-functions in the Hida family, then extends Castella's Heegner-point construction and claims a reciprocity law. This creates a moderate self-citation dependence for the central identity, but the extension to quaternionic modular forms and the reciprocity statement itself introduce new content that is not purely definitional or a direct renaming of the prior fit. No equations in the provided text reduce the new result to the old one by construction, and external benchmarks like Castella's work are cited independently. Score kept at 4 per guidelines for non-load-bearing self-citation with remaining independent steps.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract indicates reliance on established theory of Hida families, anticyclotomic p-adic L-functions, and Castella's big Heegner points without introducing new free parameters or invented entities.

axioms (1)

domain assumption Standard properties of Hida families and interpolation of p-adic L-functions for quaternionic modular forms hold as established in prior work.
Invoked to extend the setting to big Heegner points.

pith-pipeline@v0.9.0 · 5608 in / 1279 out tokens · 38184 ms · 2026-05-18T10:24:53.702094+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/Cost/FunctionalEquation washburn_uniqueness_aczel unclear

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

We prove an explicit reciprocity law relating the big p-adic L-function to the big Heegner points in this quaternionic setting (Theorem 6.3: L_alg_I,ξ = unit · L_an_I,ξ in eI[[eΓ_∞]]).
IndisputableMonolith/Foundation/AlexanderDuality alexander_duality_circle_linking unclear

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

Extension of Castella's interpolation and specialization of big Heegner points to the quaternionic setting using families of p-adic L-functions as bridge.

What do these tags mean?

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.