On quaternionic ordinary families of modular forms and $p$-adic $L$-functions

Eris Rocha Walchek; Matteo Longo; Paola Magrone

arxiv: 2510.04301 · v3 · submitted 2025-10-05 · 🧮 math.NT

On quaternionic ordinary families of modular forms and p-adic L-functions

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

Pith reviewed 2026-05-18 10:29 UTC · model grok-4.3

classification 🧮 math.NT

keywords quaternionic modular formsordinary familiesp-adic L-functionsSerre-Tate expansionsOhta control theoremsinterpolation

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The pith

Quaternionic ordinary families of modular forms yield a big p-adic L-function that interpolates known ones at classical points.

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

The authors construct power series attached to quaternionic ordinary families of modular forms by means of Serre-Tate expansions. They then produce a big p-adic L-function from these power series. The big L-function is designed so that it recovers the p-adic L-functions of Burungale and Magrone upon specialization to classical modular forms. The construction rests on extending Ohta's control theorems from the elliptic to the quaternionic setting. A reader would care because this supplies a p-adic interpolation device in a non-split context where such objects had not previously been available.

Core claim

We use Serre--Tate expansions of modular forms to construct power series attached to quaternionic ordinary families of modular forms. We associate to these power series a big p-adic L-function interpolating the p-adic L-functions constructed by Burungale and Magrone at classical specializations. A crucial ingredient is the generalization of some results of Ohta to the quaternionic setting.

What carries the argument

Serre-Tate expansions of modular forms, which produce power series for the quaternionic ordinary families and, together with generalized Ohta control theorems, allow the construction of the big p-adic L-function.

If this is right

The big p-adic L-function recovers the classical p-adic L-functions of Burungale and Magrone at ordinary specializations.
The construction extends the ordinary-family framework from the split to the quaternionic case.
Power series attached to the families become available for further p-adic study.

Where Pith is reading between the lines

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

The big L-function could serve as input for main-conjecture type statements in the quaternionic setting.
The same Serre-Tate and control-theorem approach may adapt to other indefinite quaternion algebras or higher-weight families.
Comparison with Iwasawa-theoretic constructions over quaternion algebras becomes feasible.

Load-bearing premise

The generalization of Ohta's results supplies the necessary control theorems in the quaternionic setting.

What would settle it

An explicit check at a classical specialization where the constructed big p-adic L-function differs from the Burungale-Magrone p-adic L-function would show the interpolation property fails.

read the original abstract

We use Serre--Tate expansions of modular forms to construct power series attached to quaternionic ordinary families of modular forms. We associate to these power series a big $p$-adic $L$-function interpolating the $p$-adic $L$-functions constructed by Burungale and Magrone at classical specializations. A crucial ingredient is the generalization of some results of Ohta to the 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

They generalize Ohta's control theorems to quaternionic ordinary families and attach a big p-adic L-function via Serre-Tate expansions that interpolates Burungale-Magrone specializations.

read the letter

The main point is that this paper generalizes some of Ohta's results on ordinary families to the quaternionic setting and uses Serre-Tate expansions to build power series, then attaches a big p-adic L-function that interpolates the ones constructed by Burungale and Magrone at classical points. The abstract flags the Ohta generalization as the key step, which makes sense because the ordinary projector and Hecke algebra need to behave well over the Iwasawa algebra for the power series to be defined properly. They do a reasonable job explaining why Serre-Tate expansions are a good tool here for handling the ordinary part without relying on the usual GL(2) Hida theory directly. That approach looks adapted to the quaternion algebra case and gives a concrete way to interpolate the earlier L-functions. The soft spot is the control theorems themselves. Ohta's original work uses specific freeness and rank properties that depend on the local behavior at p and unramified places. In the quaternionic setting the algebra can be ramified at other finite places, which changes how the Atkin-Lehner operators act and how the Hecke algebra sits over the Iwasawa algebra. The abstract states that the generalization exists but does not show the explicit rank statements or how the ramified places are handled. If those properties do not carry over with the same multiplicity or freeness, the power series attachment and the interpolation map would not go through as claimed. This is a specialized paper aimed at people already working on p-adic L-functions and Iwasawa theory for quaternion algebras. A reader who knows the Burungale-Magrone constructions and Ohta's control theorems would see the most value, as it tries to push those tools into a non-split setting. It deserves a serious referee because the construction targets a genuine gap and the methods are laid out clearly enough to check. I recommend sending it out for review, with the referee asked to verify the freeness and rank claims in the quaternionic case.

Referee Report

2 major / 1 minor

Summary. The manuscript constructs power series attached to quaternionic ordinary families of modular forms using Serre-Tate expansions. It associates to these a big p-adic L-function that interpolates the p-adic L-functions of Burungale and Magrone at classical specializations. The construction depends on generalizing certain results of Ohta to the quaternionic setting.

Significance. If the central claims hold, the work would extend the theory of ordinary families and p-adic L-functions to the quaternionic case, supplying new interpolation tools that build directly on Burungale-Magrone constructions and potentially enabling applications to Iwasawa theory for quaternion algebras.

major comments (2)

[Abstract] Abstract and introduction: the manuscript asserts that a generalization of Ohta's control theorems supplies the required freeness of the Hecke algebra over the Iwasawa algebra and commutation of the ordinary projector with quaternionic Atkin-Lehner operators, yet no explicit statements, rank computations, or verifications of these properties are given for the quaternion algebra case (where local conditions at ramified places differ from the GL(2) setting). This is load-bearing for the definition of the power series and the interpolation map.
[Construction of the power series (likely §3 or §4)] The step attaching power series via Serre-Tate expansions and constructing the big L-function assumes the ordinary family satisfies the expected multiplicity-one and freeness properties; without a detailed proof or reference to a verified control theorem in the quaternionic setting, the interpolation claim cannot be checked.

minor comments (1)

[Notation and preliminaries] Clarify the precise definition of the weight space and the action of the quaternionic Atkin-Lehner operators on the ordinary projector to make the comparison with the classical Ohta setting explicit.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and insightful comments on our manuscript. We address each major comment below and will make revisions to improve the clarity and completeness of the presentation regarding the generalization of Ohta's results to the quaternionic setting.

read point-by-point responses

Referee: [Abstract] Abstract and introduction: the manuscript asserts that a generalization of Ohta's control theorems supplies the required freeness of the Hecke algebra over the Iwasawa algebra and commutation of the ordinary projector with quaternionic Atkin-Lehner operators, yet no explicit statements, rank computations, or verifications of these properties are given for the quaternion algebra case (where local conditions at ramified places differ from the GL(2) setting). This is load-bearing for the definition of the power series and the interpolation map.

Authors: We agree with the referee that the abstract and introduction would benefit from more explicit statements and verifications of the generalized control theorems. In the revised version, we will include precise statements of the freeness of the Hecke algebra over the Iwasawa algebra, the commutation of the ordinary projector with the quaternionic Atkin-Lehner operators, and rank computations that take into account the local conditions at ramified places, which differ from the GL(2) case. This will make the load-bearing properties clearer. revision: yes
Referee: [Construction of the power series (likely §3 or §4)] The step attaching power series via Serre-Tate expansions and constructing the big L-function assumes the ordinary family satisfies the expected multiplicity-one and freeness properties; without a detailed proof or reference to a verified control theorem in the quaternionic setting, the interpolation claim cannot be checked.

Authors: We acknowledge that the construction section assumes these properties based on the generalization of Ohta's results. To address this, we will revise the manuscript to provide a more detailed sketch of the proof of the control theorem in the quaternionic setting or add a reference to the specific verification. This will allow readers to check the interpolation claim more readily. revision: yes

Circularity Check

0 steps flagged

No significant circularity; construction relies on external Ohta generalization and prior interpolation targets

full rationale

The paper constructs power series attached to quaternionic ordinary families via Serre-Tate expansions and associates to them a big p-adic L-function that interpolates the p-adic L-functions of Burungale and Magrone at classical specializations. The abstract explicitly identifies the generalization of Ohta results to the quaternionic setting as the crucial independent ingredient supplying the needed control theorems. No equation or step in the provided description reduces a claimed prediction or first-principles result to a fitted input by construction, nor does any load-bearing premise rest solely on a self-citation chain whose authors overlap with the present work. The reference to Burungale-Magrone is to the external objects being interpolated rather than to a uniqueness theorem or ansatz that would render the new construction tautological. The derivation chain therefore remains self-contained against the cited external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed on abstract only. No free parameters or new entities are named. The work rests on standard properties of modular forms and the unproven-in-abstract generalization of Ohta results.

axioms (2)

domain assumption Serre-Tate expansions exist and produce power series for quaternionic ordinary families of modular forms
Used as the starting point for constructing the attached power series.
domain assumption Results of Ohta generalize to the quaternionic setting
Explicitly identified as the crucial ingredient enabling the rest of the construction.

pith-pipeline@v0.9.0 · 5596 in / 1435 out tokens · 50283 ms · 2026-05-18T10:29:00.901679+00:00 · methodology

discussion (0)

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We use Serre–Tate expansions of modular forms to construct power series attached to quaternionic ordinary families of modular forms. ... A crucial ingredient is the generalization of some results of Ohta to the quaternionic setting.

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