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On the Artin formalism for triple product $p$ -adic $L$ -functions

This paper investigates the factorization of triple-product $p$ -adic $L$ -functions in cases where Artin formalism predicts a split that the defining properties do not support, reframing the problem through the lens of the Equivariant Tamagawa Number Conjecture as a comparison between diagonal cycles, Beilinson–Kato elements, and Heegner cycles.

Original authors: Kâzım Büyükboduk, Ryotaro Sakamoto

Published 2026-07-28

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CC BY 4.0

Original authors: Kâzım Büyükboduk, Ryotaro Sakamoto

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 universe of numbers as a vast, cosmic library where every book contains a secret recipe for how the universe behaves. In this library, there are special books called "L-functions." You can think of these not as stories you read, but as complex musical scores. When you play a specific note on a specific instrument (a mathematical operation), the score tells you a hidden number that reveals deep truths about shapes, symmetry, and the very fabric of reality. For decades, mathematicians have been trying to write down these scores for different combinations of instruments, hoping to hear a melody that connects everything together.

One of the most famous rules in this library is the "Artin formalism." It's like a magic rule that says: if you have a massive, complicated orchestra playing a symphony, and you know how the individual sections (like the strings or the brass) work, you should be able to predict exactly how the whole orchestra sounds by simply multiplying the scores of the sections together. It's the mathematical equivalent of saying, "If I know how a violin sounds and how a cello sounds, I can predict the sound of a duet." Usually, this works perfectly. But sometimes, the orchestra gets stuck in a weird silence. The music stops, the notes vanish, and the magic rule seems to break. The score for the whole group is zero, but the scores for the individual parts aren't zero. This is a "regular" silence, and mathematicians know how to handle it. But there is a much stranger, "irregular" silence where the music doesn't just stop; it seems to turn into a derivative, a measure of how fast the music would have changed if it hadn't stopped. This paper dives into that confusing, silent corner of the library to see if the magic rule can be fixed, even when the music seems to have disappeared.

The Mystery of the Vanishing Symphony

In this paper, the authors, Kazim Büyükboduk and Ryotaro Sakamoto, are investigating a very specific, tricky problem in the world of "p-adic L-functions." To understand their quest, imagine you are a detective trying to solve a case where a suspect (a mathematical object) has vanished from a crime scene (a specific range of numbers).

Usually, when mathematicians look at a triple product of three different mathematical "families" (let's call them Family F, Family G, and Family G's mirror image, Gc), they expect to find a formula that breaks the big, complicated product down into smaller, simpler pieces. This is the "Artin formalism" mentioned earlier. It's like taking a giant, complex cake and slicing it into two perfect, smaller cakes that you can easily taste and understand.

However, the authors are looking at a scenario where the "interpolation range"—the part of the cake you are supposed to be able to taste—is completely empty. It's as if you are trying to slice a cake that isn't there. In this specific situation, the standard recipe for the big cake says the result is zero. But the authors suspect that the zero isn't a mistake; it's a clue. They believe that if you look closely at this "zero," you won't just find nothing; you'll find the rate of change of the music, a concept known as a "derivative."

The Main Discovery: A Hidden Connection

The paper's main finding is a bold conjecture (a strong mathematical guess that they have proven the algebraic version of) about how to fix the broken recipe. They propose that when the big, triple-product "cake" vanishes, it doesn't just disappear. Instead, it splits into two very specific ingredients:

A Degree-6 L-function: Think of this as a standard, well-behaved musical score for a smaller orchestra. It's a known quantity that mathematicians are comfortable with.
A "Big Beilinson-Kato Element": This is the star of the show. The authors suggest that the missing piece of the puzzle is a special mathematical object that acts like a "logarithm" of a derivative. In everyday terms, if the music stopped, this object tells you how fast the music was trying to change right before it stopped. It's like measuring the speed of a car that has just hit the brakes and stopped; the car is still (zero), but the speedometer (the derivative) tells you it was moving.

The authors prove that the "algebraic" version of this recipe works. They show that the module of "leading terms" (the most important part of the algebraic structure) for the big, vanished product is exactly equal to the product of the smaller, degree-6 score and this special "derivative" object.

What They Don't Claim (and What They Rule Out)

It is crucial to understand what this paper doesn't say. The authors are very careful not to claim they have solved the entire mystery in every possible scenario.

They do not claim to have proven the full analytic version of their conjecture (the version that deals with the actual numbers and functions directly) in all cases. They admit that the full proof is "out of reach" for now and relies on other deep, ongoing work by different mathematicians (specifically, a "Generalized Gross-Kudla formula" that is still being developed).
They explicitly rule out the idea that the vanishing of the L-function is just a boring, trivial zero with no hidden meaning. They argue against the idea that the Artin formalism simply fails here; instead, they show it transforms into something more subtle involving derivatives.
They do not claim that this works for every possible combination of numbers. Their results are specific to a scenario where the "root number" (a sign that determines if the music is happy or sad) is negative, and where the families of numbers involved meet certain technical criteria (like being "ordinary" and having specific properties at the prime number 2).

How Sure Are They?

The authors are extremely confident about the algebraic part of their story. They have rigorously proved that the algebraic structures (the "Selmer complexes" and "modules of leading terms") behave exactly as their conjecture predicts. They have built a solid mathematical bridge showing that the algebraic side of the equation splits perfectly into the two ingredients they described.

However, regarding the analytic side (the actual p-adic L-functions that interpolate the values), they are more cautious. They state that their conjecture is a "natural variant" of previous work and is supported by a "line of attack" involving comparing different types of cycles (geometric shapes in higher dimensions). They suggest that if a certain formula (the Generalized Gross-Kudla formula) is true, then their conjecture is true. But they stop short of saying the conjecture is a "solved problem" in the analytic sense. They have laid the groundwork and proven the algebraic foundation, but the final, full verification of the analytic formula relies on future work by others.

The "BDP-Principle" and the Magic Logarithm

To make sense of this, the authors introduce a guiding principle they call the "BDP-principle" (named after mathematicians Bertolini, Darmon, and Prasanna). This principle suggests that when a p-adic L-function vanishes in a specific way, it should be thought of as a "p-adic avatar" (a shadow or a digital version) of a family of derivatives.

They use a clever tool called a "large logarithm map" to translate the mysterious "Beilinson-Kato element" into something that looks like a derivative. Imagine you have a secret code (the Beilinson-Kato element) that looks like gibberish. The "logarithm map" is a decoder ring that translates that gibberish into a clear message: "This is the rate of change of the music."

Why This Matters

Why should a curious teenager care about vanishing musical scores and algebraic cakes? Because this work is about finding order in chaos. When mathematics hits a wall where things seem to disappear (vanish), it often means we are on the verge of discovering a deeper layer of reality. By showing that the "zero" is actually a product of a known score and a derivative, the authors are revealing that the universe of numbers is more connected than it appears. They are showing that even when the music stops, the silence itself has a structure, a rhythm, and a story to tell.

They have successfully mapped the algebraic terrain, proving that the pieces fit together in a specific, elegant way. While the final, full picture of the analytic world is still being painted, this paper provides the blueprint and the solid foundation upon which the rest of the building will be constructed. It's a reminder that in mathematics, sometimes the most interesting discoveries are found not in the loud, clear notes, but in the quiet, vanishing moments in between.

Technical Summary: On the Artin formalism for triple product p-adic L-functions

Problem Statement
The primary objective of this work is to investigate the "irregular" factorisation problem for triple product $p$ -adic $L$ -functions, specifically in scenarios where the defining interpolation properties of the $L$ -functions do not naturally support such a factorisation, despite the Artin formalism suggesting one should exist. The authors focus on the restriction of the $g$ -unbalanced triple product $p$ -adic $L$ -function, denoted $L^g_p(f \otimes g \otimes g^c)$ , to a specific 2-dimensional weight space $W_2$ .

In this setting, where $f$ and $g$ are cuspidal primitive Hida families and $g^c$ is the conjugate family, the interpolation range for the restriction of $L^g_p(f \otimes g \otimes g^c)$ to $W_2$ is empty. Consequently, the standard Artin formalism—which predicts a factorisation of the complex $L$ -function $L(f \otimes g \otimes g^c, s)$ into a product of a degree-6 $L$ -function $L(f \otimes \text{ad}_0(g), s)$ and a degree-2 $L$ -function $L(f, s)$ —cannot be directly applied to the $p$ -adic $L$ -functions via interpolation. The problem is further complicated by the fact that the relevant Selmer groups are not all torsion, distinguishing this "irregular" case from "regular" factorisation problems studied in earlier works (e.g., Dasgupta [Das16]).

Methodology
The authors employ a strategy guided by the Equivariant Tamagawa Number Conjecture (ETNC) philosophy. Rather than relying on the construction of the specific $p$ -adic $L$ -functions (which may not exist in full generality for the degree-6 component), the paper recasts the factorisation problem as a comparison of algebraic objects: Selmer complexes and modules of leading terms.

Selmer Complexes and Greenberg Conditions: The authors define Selmer complexes associated with the Galois representations $T^\dagger_3$ (degree 8), $T^\dagger_2$ (degree 6), and $M^\dagger_2$ (degree 6, trace-zero part) using Greenberg local conditions. They establish that these complexes are perfect under specific hypotheses regarding Tamagawa factors and residual representations.
Modules of Leading Terms: A central technical innovation is the introduction of "modules of leading terms" (denoted $\delta(T, \Delta)$ ). These are cyclic submodules constructed within the extended Selmer modules (specifically the first cohomology of the Selmer complex). The construction utilizes exterior bi-duals and is inspired by the theory of Kolyvagin systems. The module $\delta(T, \Delta)$ serves as the algebraic counterpart to the $p$ -adic $L$ -function.
Comparison of Cycles and Logarithms: The factorisation is achieved by comparing the leading terms of different Selmer complexes. The authors utilize exact triangles relating the Selmer complexes of the triple product, the trace-zero product, and the individual forms. Crucially, they employ logarithm maps (specifically the large logarithm map $\text{Log}_{\omega_f}$ ) to relate the leading terms of the degree-8 and degree-6 objects to the Beilinson–Kato element associated with $f$ .

Key Contributions and Results

Conjecture 2.1 (Analytic Formulation): The authors formulate a precise conjecture for the factorisation of the $p$ -adic $L$ -function $L^g_p(f \otimes g \otimes g^c)|_{W_2}$ . They predict that this function factors as the product of a degree-6 $p$ -adic $L$ -function ( $L^{ad}_p(f \otimes \text{ad}_0 g)$ ) and the image of the big Beilinson–Kato element $BK^\dagger_f$ under a large logarithm map $\text{Log}_{\omega_f}$ .
$L^g_p(f \otimes g \otimes g^c)^2|_{W_2} \sim L^{ad}_p(f \otimes \text{ad}_0 g) \cdot \text{Log}_{\omega_f}(BK^\dagger_f)$
The appearance of the logarithm of the Beilinson–Kato element is interpreted as encoding the derivatives of $p$ -adic $L$ -functions, consistent with the "BDP-principle" (Bertolini–Darmon–Prasanna).
Theorem 2.2 / Theorem 6.7 (Algebraic Factorisation): The main proven result is the algebraic analogue of Conjecture 2.1. Under mild hypotheses (including regularity of the Hida family rings and specific conditions on Tamagawa factors and residual representations), the authors establish an equality of ideals in the ring of algebraic $p$ -adic $L$ -functions:
$\delta(T^\dagger_2, \Delta_g) = \text{Exp}^*_{F^-T^\dagger_f}(\delta(M^\dagger_2, \text{tr}^*\Delta_g)) \cdot \varpi^*_{2,1}\text{Log}_{\omega_f}(\delta(T^\dagger_f, \Delta_\emptyset))$
Here, $\delta(T^\dagger_2, \Delta_g)$ corresponds to the leading term of the triple product, $\delta(M^\dagger_2, \text{tr}^*\Delta_g)$ corresponds to the degree-6 component, and $\text{Log}_{\omega_f}(\delta(T^\dagger_f, \Delta_\emptyset))$ corresponds to the logarithm of the Beilinson–Kato element.
Handling Irregularity: The paper demonstrates that the factorisation involves a non-trivial interaction between the Selmer groups. Unlike regular cases where factorisation might be a simple product of characteristic ideals, the irregular nature here necessitates the use of modules of leading terms and logarithmic maps to account for the non-torsion nature of certain Selmer groups.

Significance and Claims
The authors claim that this work provides the first systematic treatment of the irregular factorisation problem for triple product $p$ -adic $L$ -functions where the interpolation range is empty. By shifting the focus from analytic $L$ -functions to algebraic Selmer complexes and modules of leading terms, they bypass the need for the existence of the degree-6 $p$ -adic $L$ -function $L^{ad}_p(f \otimes \text{ad}_0 g)$ to prove the factorisation relation.

The significance lies in:

Generalising Artin Formalism: Extending the Artin formalism to the $p$ -adic setting in cases where standard interpolation fails, revealing a structure involving derivatives of $L$ -functions (via the Beilinson–Kato element).
Algebraic Framework: Establishing a robust algebraic framework (using modules of leading terms) to study factorisation problems that are currently out of reach analytically.
Connection to Arithmetic Geometry: The authors note in Appendix A that a full proof of the analytic Conjecture 2.1 would rely on comparing Gross–Kudla formulae for diagonal cycles with Gross–Zagier formulae for Heegner cycles, linking their algebraic results to deep conjectures in arithmetic geometry.

The paper concludes that while the analytic conjecture remains open in full generality, the algebraic factorisation of the leading terms is established, providing strong evidence for the validity of the predicted factorisation of the $p$ -adic $L$ -functions.

Technical Summary: On the Artin formalism for triple product p-adic L-functions

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