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

This paper extends F. Castella's work on big Heegner points to the quaternionic setting by establishing an explicit reciprocity law that relates big $p$ -adic $L$ -functions to big Heegner points within a Hida family.

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 the world of numbers not as a static list, but as a vast, living landscape. In this landscape, there are special "mountains" called Heegner points. For a long time, mathematicians have known how to climb these mountains when the terrain is relatively flat and simple (like the world of standard elliptic curves).

This paper is about building a bridge to a much more rugged, complex terrain called quaternionic algebras. The authors (Longo, Magrone, and Walchek) are essentially saying: "We know how to navigate the simple mountains, and we know how to navigate the complex ones separately. Now, we are going to show you how to walk directly from one to the other, proving they are connected by a hidden path."

Here is a breakdown of their journey using everyday analogies:

1. The Two Maps (The Problem)

Imagine you have two different maps of the same territory, but they are drawn in different languages.

Map A (The Analytic Map): This map is drawn using formulas and patterns. It's like a weather map showing pressure systems. In math, this is the p-adic L-function. It tells you about the "shape" of the numbers based on how they behave in patterns.
Map B (The Algebraic Map): This map is drawn using physical landmarks. It's like a treasure map showing where specific rocks or trees are located. In math, these landmarks are Heegner points (special solutions to equations).

For a long time, mathematicians had a way to translate Map A to Map B, but only for the "flat" terrain (standard elliptic curves). The authors wanted to do this for the "rugged" terrain (quaternionic curves), which had been much harder to navigate.

2. The Family of Families (Hida Families)

Usually, mathematicians study one specific mountain at a time. But this paper looks at a whole family of mountains that change shape smoothly as you walk through them.

Think of a Hida family as a continuous ribbon of mountains. As you walk along the ribbon, the mountains change their height and shape (their "weight"), but they are all part of the same continuous structure.
The authors are looking at a "Big Heegner Point," which is like a giant, flexible rope that stretches across this entire ribbon of mountains, connecting all the different shapes at once.

3. The Bridge (The Reciprocity Law)

The core achievement of the paper is building a bridge between the two maps for this rugged terrain.

They prove a Reciprocity Law. Think of this as a universal translator. They show that if you take the "Analytic Map" (the p-adic L-function) and translate it, it matches perfectly with the "Algebraic Map" (the Big Heegner Point).
The Equation: They write an equation that says: The Pattern (L-function) = The Landmark (Heegner Point).
They didn't just guess this; they proved it by looking at a specific "crossing point" on the ribbon where the mountains are exactly height 2 (a special, well-understood spot). Because the mountains are connected in a family, if the maps match at that one spot, they must match everywhere along the ribbon.

4. Why It Matters (The Treasure Hunt)

Why do we care if these two maps match? Because it helps us find treasure hidden in the Selmer groups.

In this mathematical world, a Selmer group is like a vault. Inside the vault, there are "keys" (solutions to equations).
The authors prove that because the maps match, the vault is not empty. Specifically, they show that there is at least one key inside.
The Result: They prove that a specific "Big Heegner Point" is not a "ghost" (it's not zero or trivial). Because this point exists and is real, it forces the vault (the Selmer group) to have a specific size (rank 1).
This is a big deal because it confirms a long-standing guess (Conjecture 10.3 from a 2011 paper by the authors and Vigni) that these vaults should indeed contain a key.

5. The "Relaxed" Rule

One of the clever tricks in this paper is how they handle the rules of the terrain.

Usually, to find these landmarks, the terrain has to follow very strict rules (like a specific type of road must be open).
The authors relaxed these rules. They allowed for a scenario where some roads are closed (primes that are "inert") as long as an even number of them are closed.
The Analogy: Imagine you are trying to cross a river. The old rule said, "You can only cross if there are no bridges at all." The new rule says, "You can cross if there are an even number of bridges." This opens up many more places where you can cross, making the theory much more powerful.

Summary

In simple terms, this paper is a navigation guide.

It takes a complex, rugged mathematical landscape (quaternionic algebras).
It connects a theoretical map (L-functions) with a physical map (Heegner points) using a "family" of shapes.
It proves that these two maps are actually the same thing.
Because they are the same, it proves that a specific mathematical "vault" (Selmer group) contains a key, confirming that the structure of these numbers is richer and more connected than we previously knew.

They didn't invent new mountains; they just proved that the path between two different ways of looking at them is solid and walkable.

Technical Summary: Quaternionic Families of Heegner Points and p-adic L-functions

Problem Statement
This paper addresses the extension of the theory of "big" Heegner points and their relationship to $p$ -adic $L$ -functions from the classical setting of modular curves (associated with $\mathrm{GL}_2$ ) to the setting of indefinite quaternion algebras. Specifically, the authors aim to generalize the results of Castella [Cas13, Cas20], who established an explicit reciprocity law relating big Heegner points to generalized Heegner cycles via $p$ -adic $L$ -functions in the Hida family setting. The central challenge in the quaternionic setting is adapting these techniques, particularly the use of Serre–Tate expansions and the comparison of specializations at weight 2, to Shimura curves associated with indefinite quaternion algebras.

Methodology
The authors employ a strategy centered on the density of weight 2 specializations within a Hida family to bridge the gap between algebraic and analytic objects.

Quaternionic Hida Families: The work is set within the framework of a primitive branch $I$ of a Hida family of quaternionic modular forms, passing through a $p$ -stabilized newform $f$ of weight $k_0 \equiv 2 \pmod{2(p-1)}$ . The authors assume standard hypotheses regarding the residual Galois representation (irreducible and $p$ -distinguished) and the splitting behavior of primes in a quadratic imaginary field $K$ .
Big Heegner Points: Utilizing the construction of Fouquet [Fou13] and Longo–Vigni [LV11], the authors define "big" Heegner points $Z_c$ in the Iwasawa cohomology of a big Galois representation $T^\dagger_\xi$ attached to the Hida family $I$ and a family of Hecke characters $\xi$ .
Analytic $p$ -adic $L$ -functions: The analytic side relies on the family of $p$ -adic $L$ -functions $L^{\text{an}}_{I,\xi}$ constructed in the authors' previous work [LMW26]. This construction generalizes Hida–Ohta theory to the quaternionic setting, utilizing Serre–Tate expansions of modular forms on Shimura curves.
Algebraic $p$ -adic $L$ -functions: The algebraic side is defined via a big Perrin-Riou logarithmic map applied to the big Heegner point $Z_c$ , yielding $L^{\text{alg}}_{I,\xi}$ .
Coleman Integration and Specialization: A crucial technical component involves the use of Coleman integration on Shimura curves. The authors generalize results from modular curves to define Coleman primitives for overconvergent modular forms on Shimura curves. They compute the specialization of the algebraic $L$ -function at arithmetic morphisms of weight 2, relating it to the evaluation of these primitives at Heegner points.
Reciprocity Law: By comparing the weight 2 specializations of both the analytic and algebraic $L$ -functions, and leveraging the density of such specializations in the Hida family, the authors establish an equality between the two global objects.

Key Contributions and Results

Explicit Reciprocity Law: The primary result is Theorem 6.3, which proves an equality (up to units) between the analytic $p$ -adic $L$ -function $L^{\text{an}}_{I,\xi}$ and the algebraic $p$ -adic $L$ -function $L^{\text{alg}}_{I,\xi}$ in the ring $\tilde{I}[[\tilde{\Gamma}_\infty]]$ :
$L^{\text{an}}_{I,\xi} = L^{\text{alg}}_{I,\xi}$
This equality is derived by showing that their specializations at weight 2 arithmetic morphisms coincide, utilizing the explicit description of big Heegner points as limits of classical Heegner points.
Non-Torsion of Big Heegner Points: As a direct consequence of the reciprocity law and the assumption that the generic root number of the Hida family is $+1$ , the authors prove that the big Heegner point $Z_c$ is not $I$ -torsion (Corollary 6.4). This confirms Conjecture 10.3 from [LV11].
Arithmetic Applications to Selmer Groups: The non-torsion property of $Z_c$ leads to structural results regarding Selmer groups:
- The Nekovář extended Selmer group $\tilde{H}^1_f(K, T^\dagger)$ associated with the big Galois representation is a free $I$ -module of rank 1 (Corollary 6.5).
- For all but finitely many arithmetic specializations $\nu$ , the dimension of the Bloch-Kato Selmer group $H^1_f(K, T^\dagger_\nu)$ over the residue field is 1.
Generalized Heegner Hypothesis: A significant feature of these results is that they hold under a "relaxed" Heegner hypothesis. Unlike previous works restricted to cases where all primes dividing the level split in $K$ , this paper allows for an even (positive) number of primes dividing the level $N$ to be inert in $K$ (specifically, $N^-$ is a product of an even number of distinct primes).

Significance and Limitations
The paper claims to successfully adapt the techniques of Castella and Darmon–Rotger to the quaternionic setting, providing the first explicit reciprocity law relating big Heegner points to $p$ -adic $L$ -functions in this context. This generalization is significant because it extends the scope of Iwasawa theory for Hida families to include cases with inert primes in the level, broadening the applicability to a wider class of Shimura curves.

However, the authors note a specific limitation regarding the relationship between higher-weight specializations and generalized Heegner cycles. While the weight 2 specializations are explicitly related to generalized Heegner cycles (via the work of one of the authors in [Mag22]), the paper does not currently provide a complete description of the higher-weight specializations of the Perrin-Riou logarithm in terms of generalized Heegner cycles. The authors attribute this to the lack of a quaternionic analogue of the Kings–Loeffler–Zerbes result [KLZ17], which relies on Beilinson–Kato elements and Kuga–Sato varieties—ingredients not yet fully available in the quaternionic setting. Thus, the reciprocity law is established via the density of weight 2 points, but the direct geometric interpretation for higher weights remains an open problem.