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Hirzebruch-Zagier cycles in $p$ -adic families and adjoint $L$ -values

This paper demonstrates that generalized Hirzebruch-Zagier cycles associated with Hilbert modular varieties can be organized into $p$ -adic families, which are then used via base change to construct a multivariable $p$ -adic adjoint $L$ -function twisted by the Hecke character of a quadratic extension $E/F$ for Hida families of Hilbert modular forms.

Original authors: Antonio Cauchi, Marc-Hubert Nicole, Giovanni Rosso

Published 2026-05-26

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CC BY 4.0

Original authors: Antonio Cauchi, Marc-Hubert Nicole, Giovanni Rosso

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 a mathematician trying to understand the hidden patterns of numbers. For a long time, mathematicians have known how to study these patterns using "families" of shapes that change slightly as you tweak a knob. This paper is about building a new, very complex family of shapes and using it to measure a specific, elusive number that appears in the theory of modular forms (which are like musical notes in the world of numbers).

Here is a breakdown of what the authors, Antonio Cauchi, Marc-Hubert Nicole, and Giovanni Rosso, have done, using simple analogies.

1. The Setting: Two Worlds of Numbers

Imagine two different countries, F and E.

F is a "totally real" country (think of it as a flat, predictable landscape).
E is a slightly more complex country built on top of F (a "quadratic extension," like adding a second layer of terrain).

In these countries, there are special geometric landscapes called Hilbert modular varieties. Think of these as vast, multi-dimensional gardens.

The garden for F is called $Y_H$ .
The garden for E is called $Y_G$ .

Because E is built on F, there is a natural path (an embedding) that connects the smaller garden $Y_H$ into the larger garden $Y_G$ .

2. The Special Objects: Hirzebruch–Zagier Cycles

Inside the larger garden ( $Y_G$ ), the authors are interested in specific "sculptures" or "paths" that come from the smaller garden ( $Y_H$ ).

In the past, mathematicians Hirzebruch and Zagier discovered that on a specific type of 2D garden, these paths (called Hirzebruch–Zagier cycles) were actually the "notes" of a famous song (a modular form).
This paper takes that idea and tries to do it in much higher dimensions (for any number of dimensions $d$ ).

The Problem: These sculptures are rigid. They exist at specific "levels" (like specific zoom levels on a map). The authors wanted to know: Can we make these sculptures flow smoothly into a continuous family, so we can study them as we turn a dial?

3. The Breakthrough: The "Big" Family

The authors successfully built a "Big Hirzebruch–Zagier Cycle."

The Analogy: Imagine you have a single, static statue. Usually, to see how it changes, you have to build a new statue for every tiny change. The authors found a way to build a master mold (a "Big Cycle") that contains all the possible versions of the statue at once.
This master mold lives in a special mathematical space called Iwasawa cohomology. Think of this as a "super-library" that holds every possible variation of the sculpture in a single, organized volume.
They proved that if you take a specific "snapshot" (a specialization) of this master mold, you get back the original, known sculptures. This means they successfully packaged these complex shapes into a p-adic family (a family that works with a specific type of number system called p-adic numbers, which are like a different way of measuring distance).

4. The Application: Measuring the "Adjoint L-Value"

Why build this family? To measure a specific number.

In number theory, there are "L-functions." These are like complex recipes that, when you plug in a number, give you a result.
One specific result, called the Adjoint L-value (at $s=1$ ), is very important. It tells us deep things about the symmetry of the modular forms.
The Connection: The authors used a famous formula (by Hida) that says: If you take the "dot product" (a way of measuring overlap) between a specific wave (a differential form) and our Hirzebruch–Zagier sculpture, the result is exactly this Adjoint L-value.

5. The Final Result: The p-adic L-Function

By using their "Big Family" of sculptures, the authors constructed a p-adic L-function.

The Analogy: Imagine you have a machine that can calculate the Adjoint L-value for any modular form in a specific family.
Usually, these machines only work for specific, isolated forms. The authors built a machine that works for the entire family at once.
They call this the Adjoint p-adic L-function. It is a single mathematical object that "interpolates" (connects the dots between) the L-values of all the forms in the family.

Summary of the Journey

Identify the shapes: They looked at special paths (cycles) connecting two mathematical gardens.
Build the family: They created a "Big Cycle" that holds all variations of these paths in a continuous p-adic family.
Measure the value: They used this family to construct a new tool (the p-adic L-function) that calculates a specific, important number (the Adjoint L-value) for any form in the family.

What they did NOT do:
The paper is purely theoretical mathematics. They did not apply this to physics, medicine, or engineering. They did not claim this solves a specific real-world problem like weather prediction or cryptography. The "application" mentioned in the abstract is strictly within the realm of number theory: constructing a geometric object to understand the values of L-functions.

In short, they built a universal mold for a complex geometric shape, which allowed them to create a universal calculator for a specific number that mathematicians have been trying to understand for decades.

Technical Summary: Hirzebruch–Zagier Cycles in p-adic Families and Adjoint L-Values

Problem Statement
The paper addresses a classical gap in the literature regarding the $p$ -adic variation of special cycles on Hilbert modular varieties. While the $p$ -adic variation of cohomology classes (such as Heegner points) has been extensively studied for various Shimura varieties, the specific case of Hilbert modular varieties presents technical obstructions. Specifically, these varieties do not satisfy Milne's axiom SV5, which typically ensures the center of the group contains no $R$ -split torus that is not already $\mathbb{Q}$ -split. This obstruction prevents the direct application of existing methods (e.g., from [Loe21], [LRZ]) used to construct $p$ -adic families of cycles.

The authors aim to construct $p$ -adic families of generalized Hirzebruch–Zagier cycles arising from Hilbert modular varieties associated with a quadratic extension $E/F$ of totally real number fields. A primary application of this construction is the geometric definition of a multivariable $p$ -adic adjoint $L$ -function, twisted by the Hecke character of $E/F$ , attached to Hida families of Hilbert modular forms over $F$ .

Methodology
The authors employ a combination of Iwasawa theory, the theory of base change for automorphic representations, and explicit geometric constructions of cycles.

Overcoming the SV5 Obstruction: The authors adapt the framework of Loeffler [Loe21] and Loeffler–Rossi–Zerbes [LRZ]. They demonstrate that the hypothesis of Milne's axiom SV5 can be removed in the context of Hilbert modular varieties. This is achieved by carefully analyzing the intersection of the level subgroups in the $p$ -adic tower with the center of the group $G = \text{Res}_{E/\mathbb{Q}}\text{GL}_2$ . They construct a specific tower of level subgroups $(V_r)_r$ indexed by multi-indices $r \in \mathbb{Z}_{\geq 0}^{|\Upsilon_{F,p}|}$ , where $\Upsilon_{F,p}$ is the set of primes of $F$ above $p$ .
Construction of the Big Cycle: Using a spherical variety argument involving an element $u \in G(\mathbb{Z}_p)$ and the diagonal embedding $\iota: H \hookrightarrow G$ (where $H = \text{Res}_{F/\mathbb{Q}}\text{GL}_2$ ), the authors define Hirzebruch–Zagier cycles $Z^k_r$ at finite levels. By utilizing the ordinary projector $e^{n.ord}$ associated with a specific Hecke operator $T_\eta$ , they prove the existence of a "Big Hirzebruch–Zagier class" $Z_\infty$ in the middle-degree ordinary Iwasawa cohomology group $H^{2d}_{\text{Iw}}(Y_G(V_\infty), \mathcal{O})$ .
Moment Maps and Specialization: The authors define moment maps $\text{mom}^k_r$ that relate the universal Iwasawa class $Z_\infty$ to the classical cycles at finite levels. They establish control theorems showing that the ordinary part of the Iwasawa cohomology is a finite module over the Iwasawa algebra $\Lambda_{G/H}$ .
Base Change and $L$ -functions: Leveraging the theory of base change (Getz–Goresky [GG12]), the authors relate the pairing of the Hirzebruch–Zagier cycles with differential forms to the residue at $s=1$ of the Asai $L$ -function. This residue, in turn, is linked to the adjoint $L$ -function $L(\text{Ad}(f) \otimes \eta_{E/F}, 1)$ of a Hilbert modular form $f$ on $F$ .
Geometric Construction of the $p$ -adic $L$ -function: By evaluating the Big Hirzebruch–Zagier class against a specific idempotent associated with a Hida family $f$ (and its base change $f_E$ ), the authors define the $p$ -adic $L$ -function $L_p(\text{Ad}(f) \otimes \eta_{E/F})$ as an element of the coefficient field of the family.

Key Contributions and Results

Theorem 1.1 (Existence of the Big Class): The authors prove the existence of a Big Hirzebruch–Zagier class $Z_\infty$ in the ordinary Iwasawa cohomology. They show that the specialization of this class at a weight $k$ yields the classical Hirzebruch–Zagier cycle (twisted by Hecke operators) up to an explicit Euler factor. This result fills the gap in the literature regarding $p$ -adic families of cycles for Hilbert modular varieties, specifically handling the case where the center is not $\mathbb{Q}$ -split.
Theorem 1.2 (Construction of the Adjoint $p$ -adic $L$ -function): The paper constructs a multivariable $p$ -adic $L$ -function $L_p(\text{Ad}(f) \otimes \eta_{E/F})$ for a Hida family $f$ of Hilbert modular forms over $F$ . This function interpolates the algebraic parts of the special values $L(\text{Ad}(f) \otimes \eta_{E/F}, 1)$ as $f$ varies in the family.
Explicit Interpolation Formulae: The authors provide precise interpolation formulae (Theorem 7.8) for the $p$ -adic $L$ -function at classical weights. These formulae involve Euler factors at $p$ , Gauss sums, and the residue of partial Dedekind zeta functions, depending on the local behavior of the form (unramified, Steinberg, or ramified principal series).
Control Theorems: The paper establishes a control theorem (Theorem 4.34) for the Iwasawa cohomology of the Hilbert modular varieties, ensuring that the specialization map is an isomorphism under certain conditions (torsion-freeness and concentration in a single degree).

Significance
The paper claims significance in two main areas:

Geometric Construction: Unlike previous constructions of adjoint $p$ -adic $L$ -functions for Hilbert modular forms (e.g., [Hid90], [Wu01], [Ros16]) which often rely on interpreting the $L$ -value as a congruence number or using analytic formulas without a geometric cycle family, this work provides a geometric construction. It realizes the $p$ -adic $L$ -function as the image of a universal cohomology class (the Big Hirzebruch–Zagier cycle) under a specific idempotent. This connects the $L$ -values directly to the geometry of the Hilbert modular varieties via the Kudla program framework.
Extension of the Kudla Program: The work generalizes the Hirzebruch–Zagier theorem and the Gross–Kohnen–Zagier theorem to the setting of $p$ -adic families for Hilbert modular varieties. It demonstrates that the deep conjectures linking special cycles to $L$ -values hold in the context of $p$ -adic variation, even in the absence of the SV5 axiom.

The authors note that while the Gross–Kohnen–Zagier theorem's $p$ -adic variation via Big Heegner points was treated by Longo and Nicole [LN19, LN20], this paper extends such techniques to the higher-dimensional setting of Hilbert modular surfaces and varieties, utilizing the relative quadratic base change for automorphic representations as a central tool. The construction is limited to the ordinary case, though the authors suggest that techniques from [Roc22] could potentially extend these results to non-ordinary situations in future work.