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

This paper constructs a big $p$ -adic $L$ -function for quaternionic ordinary families of modular forms by utilizing Serre--Tate expansions and generalizing Ohta's results, which interpolates the classical $p$ -adic $L$ -functions defined by Burungale and Magrone.

Original authors: Matteo Longo, Paola Magrone, Eris Rocha Walchek

Published 2026-05-05

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CC BY 4.0

Original authors: Matteo Longo, Paola Magrone, Eris Rocha Walchek

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to listen to a specific, very quiet radio station (a mathematical object called a modular form) that only plays at certain frequencies. Usually, mathematicians can tune into these stations easily when they are "classical" (like standard radio waves). But this paper is about tuning into a much more complex, "quaternionic" version of these stations, which are harder to catch and don't behave like normal radio waves.

The authors, Longo, Magrone, and Walchek, have built a new kind of super-tuner (a "big p-adic L-function") that can listen to a whole family of these stations at once, rather than just one at a time.

Here is how they did it, using simple analogies:

1. The Problem: The "Ghost" Signals

In the world of numbers, there are special shapes and patterns called modular forms. Think of them as musical notes. Some notes are easy to write down on a sheet of music (classical forms). But there are "quaternionic" notes that live in a more abstract, 4-dimensional space.

The problem is that these quaternionic notes are hard to pin down. If you try to write them down using standard methods, the "sheet music" gets messy or disappears. The authors needed a way to capture these notes clearly so they could study them.

2. The Tool: The "Serre-Tate Microscope"

To solve this, the authors use a tool called a Serre-Tate expansion.

The Analogy: Imagine you have a blurry photo of a distant mountain. You can't see the details. But if you use a special microscope that looks at how the mountain deforms or changes shape under a lens, you can reconstruct the mountain's true shape mathematically.
In the paper: They use this "microscope" to look at how these quaternionic shapes change. Instead of writing the music as a standard song, they write it as a power series—a long, infinite list of numbers that acts like a unique fingerprint for each shape. This allows them to see the "integral" (whole number) details that usually get lost.

3. The Construction: Building the "Big Tuner"

The main goal of the paper is to build a Big p-adic L-function.

The Analogy: Imagine you have a radio that can only play one song at a time. The authors built a "Big Radio" that can play a whole playlist of songs simultaneously.
How it works: They take a family of these mathematical shapes (a "Hida family," which is like a continuous stream of related songs) and use their "microscope" to turn each one into a power series. Then, they weave all these series together into one giant mathematical object.
The Magic Trick: This "Big Radio" has a special feature: if you tune it to a specific, classical frequency (a "classical specialization"), it perfectly reproduces the sound of the specific p-adic L-function that mathematicians already knew how to make for that single note. It connects the new, complex world to the old, familiar world.

4. The Secret Ingredient: Borrowing from the "GL2" Neighbors

The authors admit that building this for quaternionic shapes is very hard. They couldn't just invent a new theory from scratch.

The Analogy: It's like trying to build a bridge across a wide, dangerous river. They didn't have the blueprints for a bridge in this specific location, so they looked at a similar, safer bridge built by a neighbor (mathematicians Ohta and Hida) for a different type of river (the "GL2" case).
The Adaptation: They took the neighbor's blueprints and tried to adapt them to their own, more difficult river. They successfully generalized some of the neighbor's results, but they also hit a wall. They couldn't prove every part of the neighbor's theory works in their new setting. They had to stop short of a "perfect" bridge, acknowledging that some parts of their structure are still a bit shaky (specifically, they couldn't prove the "freeness" of certain modules, which is a technical way of saying they couldn't guarantee the bridge is perfectly rigid in every direction).

5. The Result: A Map to the "Greenberg Selmer Group"

Why build this "Big Radio"?

The Goal: The authors want to study a specific mathematical object called the Greenberg Selmer group. Think of this as a "security system" for a quadratic field (a specific type of number system).
The Connection: By using their "Big Radio" to listen to these quaternionic shapes, they can track "Heegner points" (special landmarks on the map). These landmarks are crucial for understanding the security system.
The Limitation: The paper stops short of fully solving the security system's mysteries. It provides the tool (the Big L-function) and the map (the interpolation of values), but the final proof of the security system's behavior is a job for future work.

Summary

In short, the authors have created a powerful new mathematical instrument. They used a special "microscope" (Serre-Tate expansions) to turn difficult, abstract shapes into readable lists of numbers. They stitched these lists together to create a "Big Tuner" that can listen to a whole family of these shapes at once. While they couldn't fix every wobble in their construction (unlike their neighbors who built a similar device for a simpler case), they successfully built a device that connects the complex, abstract world of quaternionic numbers to the familiar world of classical number theory, paving the way for future discoveries about how these number systems behave.

Title: On Quaternionic Ordinary Families of Modular Forms and p-adic L-functions
Authors: M. Longo, P. Magrone, E. R. Walchek
Date: May 5, 2026

1. Problem Statement

The paper addresses the construction of a "big" $p$ -adic $L$ -function for indefinite rational quaternion algebras, analogous to the theory developed for $\text{GL}_2$ by Castella [Cas20]. The primary goal is to interpolate $p$ -adic $L$ -functions associated with classical specializations of a Hida family of modular forms on a quaternion algebra.

A central difficulty identified by the authors is the lack of a canonical normalization for the Jacquet–Langlands lift of an elliptic modular form to a quaternion algebra. Unlike the $\text{GL}_2$ case, where eigenforms can be normalized by setting the first Fourier coefficient to 1, the quaternionic setting lacks a direct Fourier expansion. Consequently, defining the "correct" lift and establishing integral properties for families of such forms is non-trivial. The paper seeks to resolve this by combining Serre–Tate expansions with an extension of Ohta's theory of ordinary families to the quaternionic setting.

2. Methodology

The authors employ a geometric and arithmetic approach centered on Shimura curves associated with indefinite quaternion algebras of discriminant $D > 1$ .

Serre–Tate Expansions: Following the work of Mori [Mor95, Mor11, Mor22] and others, the authors utilize Serre–Tate coordinates to define power series expansions for quaternionic modular forms. This replaces the classical Fourier expansion, which is not directly available in the quaternionic context.
Integral Models and Igusa Towers: The construction relies on the geometry of Shimura curves $X_m$ and their ordinary loci. The authors work with the Igusa tower over the special fiber of these curves to define arithmetic trivializations of $p$ -divisible groups associated with quaternionic multiplication (QM) abelian surfaces.
Generalization of Ohta's Theory: The paper extends results of Ohta [Oht93, Oht95, Oht99, Oht00] regarding ordinary families of modular forms to the quaternionic case. This involves:
- Defining modules of inverse limits of ordinary modular forms ( $S^*_\infty$ ) using Serre–Tate expansions.
- Establishing the relationship between these modules and the cotangent spaces of specific quotients of Jacobians of Shimura curves ( $A_m, B_m, C_m$ ).
- Utilizing the Jacquet–Langlands correspondence to relate these quaternionic objects to their $\text{GL}_2$ counterparts, thereby leveraging known results on semistable reduction and $p$ -divisible groups.
Eichler–Shimura Isomorphism: The authors construct an injective map from the inverse limit of cohomology groups (specifically the "ramified" quotient $B^*_\infty$ ) to the module of power series $J[[T_x]]$ . This map is constructed via the Serre–Tate expansion of differential forms on the universal deformation of QM abelian surfaces.

3. Key Contributions and Results

Construction of the Big $p$ -adic $L$ -function:
The main result (Theorem 6.3) establishes the existence of a big $p$ -adic $L$ -function $L_{I, \xi} \in \tilde{I}[[\Gamma_\infty]]$ , where:

$I$ is a primitive branch of a Hida family attached to a residual representation.
$\xi$ is a $p$ -adic family of Hecke characters.
$\Gamma_\infty$ is the Galois group of the anticyclotomic extension of an imaginary quadratic field $K$ .

The function $L_{I, \xi}$ interpolates the $p$ -adic $L$ -functions $L_{f_\kappa, \xi_\kappa}$ constructed by Burungale and Magrone [Bur17, Mag22] at classical arithmetic specializations $\kappa$ of weight $k \geq 2$ . Specifically, for an arithmetic morphism $\kappa$ , the specialization $L_{I, \xi}(\kappa)$ coincides with $L_{f_\kappa, \xi_\kappa}$ (up to explicit factors) as ideals in the appropriate ring.

Integral Theory of Quaternionic Modular Forms:
The paper develops an integral theory of modular forms on Shimura curves using Serre–Tate expansions. It defines submodules $S^*_\infty$ of inverse limits of modular forms and proves that these modules embed injectively into power series rings over the Iwasawa algebra. This provides a framework for studying integrality conditions in the quaternionic setting, analogous to the theory of $\Lambda$ -adic modular forms in the $\text{GL}_2$ case.

Control Theorem and Isomorphism:
The authors prove a control theorem (Theorem 5.3) relating the inverse limit of cohomology groups to the space of classical modular forms at a specific weight. Furthermore, they establish an exact sequence of $J^*$ -modules (Theorem 5.7) decomposing the cohomology into parts corresponding to the ordinary and non-ordinary components of the Galois representation, showing that the relevant quotient is free of rank 1.

Interpolation Formula:
Theorem 6.3 provides the explicit interpolation formula:
$(L_{I, \xi}(\kappa)) = (L_{f_\kappa, \xi_\kappa})$
as ideals in $\tilde{I}[[\Gamma_\infty]]$ . Corollary 6.4 further details the interpolation properties for anticyclotomic Hecke characters, relating the square of the $p$ -adic $L$ -value to the central critical value of the complex $L$ -function.

4. Limitations and Modesty of Claims

The authors explicitly acknowledge that their work does not constitute a full analogue of Wiles' theory of $\Lambda$ -adic modular forms or a complete quaternionic counterpart to Ohta's [Oht95, Theorem 2.3.6].

Missing Control Theorem: The paper notes the inability to prove a full control theorem for the modules $S^*_\infty$ (as in [Hid93] and [Oht95]). This prevents the demonstration that $S^*_\infty$ is a free module of finite rank over the Iwasawa algebra $\Lambda_O$ .
Obstructions: The lack of a clean connection between Hecke actions and $T_x$ -expansions (Serre–Tate expansions) is cited as a primary obstruction. While explicit descriptions of Hecke operators exist, the interaction with the Serre–Tate expansion at different points in the Igusa tower complicates the analysis of integrality under the action of $U_p$ and $U_p^*$ .
Scope: The theory is restricted to the ordinary case. The authors note that while perfectoid techniques (e.g., [CHJ17]) have extended Ohta's theory to finite slope families, adapting these to the specific integrality conditions required for this construction remains difficult.

5. Significance

The paper claims significance primarily in its methodological approach and its potential application to the study of Greenberg Selmer groups.

Direct Comparison with Weight 2: Unlike other recent constructions of $p$ -adic families of $L$ -functions (e.g., [JLZ21, LRZ25, Sev25]) which avoid direct comparison with weight 2 specializations, this approach builds directly on weight 2 specializations following the original Ohta–Hida philosophy.
Heegner Points and Selmer Groups: The ultimate goal stated is the study of the Greenberg Selmer group $\text{Sel}(K, T)$ for the Hida big Galois representation $T$ over an imaginary quadratic field $K$ . The construction using Shimura curves (with $D > 1$ ) allows for the use of Heegner points (or Heegner cycles) attached to towers of Shimura curves. The authors argue that Heegner points on modular curves (the $D=1$ case) are not suitable for studying $\text{Sel}(K, T)$ when $D > 1$ , making the quaternionic construction necessary for this specific arithmetic application.
Complementary Framework: The work is presented as complementary to existing methods, offering a direct path to interpolating Heegner points in weight 2, which could lead to a quaternionic analogue of results by Castella [Cas13, Cas20].

In summary, the paper successfully constructs a big $p$ -adic $L$ -function for quaternionic ordinary families by generalizing Ohta's theory via Serre–Tate expansions, while candidly admitting that a full structural theory of quaternionic $\Lambda$ -adic forms remains an open and difficult problem.