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On generalized Iwasawa main conjectures and $p$ -adic Stark conjectures for Artin motives

This paper introduces new $p$ -adic Stark regulators and Iwasawa-Greenberg main conjectures for Artin motives that strengthen existing frameworks, prove their equivalence to Rubin-Stark conjectures for monomial representations, and establish unconditional results on Selmer groups and the Gross-Kuz'min conjecture for abelian extensions of imaginary quadratic fields.

Original authors: Alexandre Maksoud

Published 2026-02-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Alexandre Maksoud

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 solve a massive, cosmic puzzle where the pieces are numbers, shapes, and symmetries. This paper, written by Alexandre Maksoud, is about building a new, more powerful tool to connect two very different worlds of mathematics: the world of algebra (dealing with structures and symmetries) and the world of analysis (dealing with functions, limits, and continuous change).

Here is a breakdown of the paper's main ideas using simple analogies:

1. The Big Picture: Bridging Two Worlds

In mathematics, there are "motives." Think of a motive as a universal blueprint or a DNA strand that contains the essential information about a number system.

The Algebraic Side: This is like counting the number of specific "rooms" or "paths" in a building. In this paper, it involves counting things like "units" (special numbers) and "class groups" (ways numbers can be grouped).
The Analytic Side: This is like measuring the "sound" or "vibration" of that building. It involves complex functions (called L-functions) that encode deep secrets about the number system.

The goal of the paper is to prove that these two sides are actually talking to each other. Specifically, it tries to show that the "sound" (the value of a function) is directly determined by the "structure" (the count of rooms).

2. The Problem: The "Extra Zero" Mystery

Usually, when you try to match the sound to the structure, the numbers line up perfectly. But sometimes, the "sound" hits a dead silence (a zero) at a specific point where you didn't expect it.

The Analogy: Imagine a radio tuned to a station. Most of the time, you hear music. But sometimes, the radio goes silent. If the silence happens exactly when you expect a song to start, that's normal. But if the radio goes silent extra times, or in a way that doesn't make sense, it's a mystery.
The Paper's Contribution: The author introduces a new "tuning knob" (called a p-stabilization) to fix this. He proposes a new rule (a conjecture) that explains why these extra silences happen and how to calculate exactly how loud the silence should be. This is called the "Extra Zero Conjecture."

3. The Main Tool: The "Regulator"

To connect the sound to the structure, mathematicians need a measuring stick called a regulator.

The Analogy: Think of the regulator as a translator. It takes the "units" (the rooms in our building) and translates them into a language the "L-function" (the radio) can understand.
The Innovation: The author creates a new, more precise translator called the p-adic Stark regulator. This translator is designed to work even when the building is very strange or "non-critical" (a technical term meaning it doesn't fit the standard rules).

4. The "Main Conjecture": The Master Key

The paper proposes a Main Conjecture.

The Analogy: Imagine you have a giant, complex lock (the Selmer group, which represents the structure of the number system). You also have a key (the p-adic L-function, which represents the analytic side). The Main Conjecture claims that this specific key fits this specific lock perfectly.
What the Paper Does: The author doesn't just claim the key fits; he builds a factory to make the key. He shows that if you use his new "translator" (the regulator), the key you make will unlock the door. He proves that under certain conditions, this key is the only key that works.

5. Special Cases: When the Puzzle is Easier

The paper looks at specific types of puzzles where the rules are simpler:

Monomial Representations: These are like puzzles where the pieces are all the same shape, just rotated. The author shows that for these, his new theory is essentially the same as other famous theories (Rubin-Stark), but it unifies them under one roof.
Imaginary Quadratic Fields: This is a specific type of number system (like a grid of numbers on a plane). The author proves a new formula for these systems, showing how the "sound" of the system relates to its "units" (special numbers). This is a p-adic Beilinson-Stark formula, which is a fancy way of saying "a new recipe for calculating these values."

6. The "Gross-Kuz'min" Discovery

Along the way, the author solves a long-standing side mystery called the Gross-Kuz'min conjecture.

The Analogy: Imagine you have a bag of marbles (representing ideal classes in a number field). You want to know if the bag is finite or if it keeps growing forever.
The Result: The author proves that for a specific type of number field (imaginary quadratic fields), the bag is finite. He does this by showing that the "silences" (L-invariants) in the system are not zero, which forces the bag to stay small. This is a significant result because it was previously only known for simpler number systems.

Summary of the "Claims"

The paper does not claim to have solved every math problem in the universe. Instead, it claims:

New Definitions: It defines new "regulators" and "L-invariants" that act as better translators between algebra and analysis.
New Conjectures: It formulates a precise guess (Conjecture A) about how these translators work for a wide class of number systems (Artin motives).
Proofs in Specific Cases: It proves that this new guess is true for specific, well-understood types of number systems (like those related to imaginary quadratic fields).
Unconditional Results: It proves that certain mathematical structures (Selmer groups) are "finite" (torsion) without needing to assume any other unproven guesses, provided the new "regulators" work.

In short, the paper builds a new bridge between two islands of mathematics, provides a better map for crossing it, and proves that the bridge holds up in several specific, important locations.

Technical Summary: Generalized Main Conjectures and p-adic Stark Conjectures for Artin Motives

1. Problem Statement and Context
The paper addresses the formulation of Iwasawa main conjectures and $p$ -adic Stark conjectures for Artin motives, specifically those arising from Artin representations $\rho$ of the absolute Galois group $G_{\mathbb{Q}}$ . While Iwasawa theory has been well-developed for motives that are critical and ordinary at $p$ (e.g., via the work of Mazur, Wiles, and Perrin-Riou), Artin motives are typically non-critical and their arithmetic at $s=0$ is governed by "refined Stark conjectures" (Stark, Rubin, Gross).

Existing approaches to these non-critical cases include:

Perrin-Riou and Benois: Who formulated conjectures for $p$ -adic $L$ -functions and Selmer groups using extended logarithm maps and Selmer complexes, particularly handling "trivial zeros" and $L$ -invariants.
Büyükboduk, Sakamoto, and others (BKS17, BS22): Who developed a "higher rank" Iwasawa theory based on Rubin-Stark elements, avoiding explicit $p$ -adic $L$ -functions and $L$ -invariants in favor of algebraic constructions.

The central problem is to unify these perspectives. The author seeks to construct a framework that explicitly incorporates $p$ -adic $L$ -functions and $L$ -invariants (as in Perrin-Riou/Benois) while generalizing the algebraic results on Rubin-Stark elements (as in BKS17/BS22) for general Artin motives, including those that are non-critical and non-ordinary.

2. Methodology and Framework
The author introduces a family of $p$ -stabilized Artin representations. Given an Artin representation $\rho$ and a prime $p$ , a $p$ -stabilization is a choice of a $G_{\mathbb{Q}_p}$ -stable subspace $W_p^+ \subset W_p$ of dimension $d_+ = \dim H^0(\mathbb{R}, W)$ . This choice is treated as an additional parameter, even when no canonical choice exists for non-critical motives.

Key methodological tools include:

Modified Coleman Maps: The author constructs extended Coleman maps to relate the Iwasawa cohomology of global units to the Selmer groups.
Selmer Groups: The study focuses on the Greenberg-style Selmer group $X_\infty(\rho, \rho^+)$ , defined via a specific local condition at $p$ determined by the $p$ -stabilization $\rho^+$ .
Regulators and $L$ -invariants: New $p$ -adic Stark regulators and an $L$ -invariant $L(\rho, \rho^+)$ are defined. These generalize Gross's regulator and Benois' $L$ -invariant, recovering known invariants in special cases (e.g., crystalline motives).
Duality and Descent: The proofs utilize Poitou-Tate duality, the structure of global unit groups in $\mathbb{Z}_p$ -towers (Kuz'min, Belliard), and descent arguments for Selmer complexes (Nekovář, Benois).

3. Main Conjectures
The paper formulates Conjecture A, which consists of three parts:

Existence ( $EX_{\rho, \rho^+}$ ): There exists a $p$ -adic measure $\theta_{\rho, \rho^+}$ in the fraction field of the Iwasawa algebra $\Lambda$ that interpolates the algebraic parts of Artin $L$ -values at $s=0$ , corrected by complex and $p$ -adic Stark regulators.
Main Conjecture ( $IMC_{\rho, \rho^+}$ ): The measure $\theta_{\rho, \rho^+}$ generates the characteristic ideal of the Selmer group $X_\infty(\rho, \rho^+)$ .
Extra Zero Conjecture ( $EZC_{\rho, \rho^+}$ ): If the stabilization is admissible, $\theta_{\rho, \rho^+}$ vanishes at the trivial character with a specific multiplicity related to the dimension of the local invariants. The leading term of this vanishing is expressed via the new $L$ -invariant $L(\rho, \rho^+)$ and a modified Euler factor.

4. Key Results and Theorems
The paper establishes several unconditional and conditional results:

Theorem A (Structure of Selmer Groups): Under the assumption that $\rho$ is unramified at $p$ , the author proves that $X_\infty(\rho, \rho^+)$ is a torsion $\Lambda$ -module if and only if the $p$ -stabilization is $\eta$ -admissible for some (or all but finitely many) characters $\eta$ . Furthermore, a generator of the characteristic ideal can be explicitly constructed using modified Coleman maps and families of "special elements" related to Artin $L$ -values.
Theorem B (Tamagawa Number Conjecture): Assuming the existence of an admissible $p$ -stabilization with a non-vanishing $L$ -invariant and the validity of Conjecture A, the paper proves the $p$ -part of the Bloch-Kato Tamagawa number conjecture for Artin motives at $s=0$ . This relates the leading term of the $L$ -function to the size of the Tate-Shafarevich group and ideal class groups.
Theorem C (Equivalence for Monomial Representations): For monomial representations (induced from a character $\chi$ $χ$ of a number field $k$ $k$ ), the paper proves that Conjecture A is essentially equivalent to the "higher rank Iwasawa main conjecture" and the "Iwasawa-theoretic Mazur-Rubin-Sano conjecture" formulated in terms of Rubin-Stark elements. Specifically:
- The existence of the $p$ -adic $L$ -function ($EX$) follows from the Rubin-Stark conjecture over $\mathbb{Z}_p$ .
- The Main Conjecture ($IMC$) is equivalent to the higher-rank Iwasawa main conjecture.
- The Extra Zero Conjecture ($EZC$) is implied by the Mazur-Rubin-Sano conjecture.
Corollary A (Imaginary Quadratic Fields): In the case where $k$ is an imaginary quadratic field and $p$ is inert, the author derives a new $p$ -adic Beilinson formula for finite-order characters. This formula expresses the derivative of a $p$ -adic $L$ -function in terms of a determinant of $p$ -adic logarithms of units and a Gross-Stark unit analogue.
Theorem D (Gross-Kuz'min Conjecture): The paper proves the Gross-Kuz'min conjecture (finiteness of the coinvariants of the $p$ -split class group) unconditionally for abelian extensions of imaginary quadratic fields. This is achieved by showing the non-vanishing of specific $L$ -invariants, utilizing Roy's strong six exponential theorem.

5. Significance and Claims
The paper claims to provide a unifying approach to the cyclotomic Iwasawa theory of Artin motives. Its significance lies in:

Bridging Approaches: It connects the analytic approach (Perrin-Riou/Benois, involving $p$ -adic $L$ -functions and $L$ -invariants) with the algebraic approach (Rubin-Stark elements, as in BKS17/BS22).
Generalization: It generalizes previous results on Deligne-Ribet and Katz $p$ -adic $L$ -functions, as well as recent constructions for weight one modular forms and their adjoints.
Explicit Formulas: It provides explicit interpolation formulas and $p$ -adic Beilinson formulas in settings where they were previously unknown or only conjectural (e.g., inert primes in imaginary quadratic fields).
Unconditional Results: It establishes the torsionness of Selmer groups and the validity of the Gross-Kuz'min conjecture for a broad class of fields without relying on unproven hypotheses beyond standard conjectures (like Rubin-Stark) for the conditional results.

The author notes that for non-critical motives, the choice of $p$ -stabilization is not canonical, and the conjectures are formulated relative to this choice. The work suggests that the "extra zero" phenomena in this setting are governed by the new $L$ -invariants, which recover known invariants (like Gross's regulator) in specific cases.