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The non-abelian Leopoldt conjecture and equalities of $\mathcal{L}$ -invariants

Assuming the non-abelian Leopoldt conjecture, this paper establishes that Hansen's $p$ -adic eigenvariety is étale at certain classical points and uses this to prove the equality of Fontaine–Mazur and automorphic $\mathcal{L}$ -invariants for regular algebraic cuspidal representations, thereby verifying Gehrmann's conjecture on their independence from cohomological degree and demonstrating functoriality for symmetric power lifts.

Original authors: Daniel Barrera Salazar, Andrew Graham, Chris Williams

Published 2026-03-20

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CC BY 4.0

Original authors: Daniel Barrera Salazar, Andrew Graham, Chris Williams

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 detective trying to solve a mystery about numbers. In the world of advanced mathematics, there are two different "languages" used to describe the same hidden objects: Automorphic forms (which are like complex musical patterns or symmetries) and Galois representations (which are like secret codes or maps of how numbers interact).

For a long time, mathematicians have suspected that these two languages describe the exact same reality, but they haven't always been able to prove it perfectly, especially when looking at specific "clues" called L-invariants.

This paper, written by Daniel Barrera Salazar, Andrew Graham, and Chris Williams, is a major breakthrough in connecting these two languages. Here is the story of what they did, explained simply.

1. The Map and the Terrain: Eigenvarieties

Imagine you have a giant, multi-dimensional map called an Eigenvariety. This map doesn't show cities and roads; it shows families of mathematical objects (like our musical patterns).

The Problem: Sometimes, this map is smooth and easy to walk on. Other times, it's jagged, broken, or has "dead ends."
The Discovery: The authors proved that under a specific, famous hypothesis (called the Non-Abelian Leopoldt Conjecture), this map is actually very smooth and well-behaved at certain important points.
The Analogy: Think of the map as a hiking trail. Usually, you might worry about falling off a cliff or hitting a wall. The authors proved that if you assume a certain rule about the mountain's shape (the conjecture), then at the specific spots where we are looking, the trail is perfectly flat and safe to walk on. This "flatness" (mathematically called being étale) is crucial because it allows us to make precise measurements.

2. The Two Clues: L-Invariants

Now, let's talk about the L-invariants.

The Automorphic L-invariant: This is a clue found by listening to the "music" (the automorphic forms). It tells us how the music behaves when it gets very loud or very quiet near a specific point (the prime number $p$ ).
The Fontaine-Mazur L-invariant: This is a clue found by decoding the "secret map" (the Galois representation). It tells us how the numbers in the map are stretching or twisting.

For decades, mathematicians have guessed that these two clues are actually the same number. If they are the same, it means our music and our map are describing the exact same thing.

3. The Big Breakthrough

The authors used their new "smooth map" discovery to finally prove that these two clues are indeed equal.

The Setup: They looked at a specific type of musical pattern (a representation of $GL_n$ ) that has a special, slightly chaotic feature at the number $p$ (called being "Steinberg").
The Method: They used a clever trick involving "infinitesimal deformations." Imagine you have a sculpture. If you nudge it slightly, how does it change? The authors nudged their mathematical objects in a specific direction.
The Result: Because the map was smooth (thanks to their first discovery), they could measure the "nudge" perfectly. They found that the change in the musical pattern matched the change in the secret map exactly.

In short: They proved that the "Automorphic L-invariant" equals the "Fontaine-Mazur L-invariant."

4. Why Does This Matter? (The "So What?")

You might ask, "Why do we care if two numbers are equal?"

The "Exceptional Zero" Mystery: In the world of elliptic curves (which are shapes used in cryptography and number theory), there is a famous mystery called the "Exceptional Zero Conjecture." It asks: "If a certain number (an L-value) is zero, does it mean something special is happening?"
- The authors' result helps solve this mystery for much more complex shapes than just simple curves. It confirms that the "zero" isn't a mistake; it's a deep signal that the music and the map are perfectly synchronized.
Symmetry and Patterns: They also showed that these L-invariants don't change depending on which "layer" of the music you look at. It's like saying a song sounds the same whether you listen to the bass, the drums, or the melody. This confirms a prediction made by a mathematician named Gehrmann.

5. The Special Case: Symmetric Powers

Finally, the authors looked at a specific, very famous family of musical patterns called Symmetric Powers (which are built by taking a simple melody and repeating it in complex ways).

They showed that for these specific patterns, all the conditions for their proof are met (except for the one big assumption about the map's shape).
This means that for these patterns, the "music" and the "map" are definitely speaking the same language, and their L-invariants are identical.

Summary

Think of this paper as building a perfect bridge between two islands:

Island A (Automorphic): The world of symmetries and patterns.
Island B (Galois): The world of number codes and secrets.

For a long time, we had a shaky rope bridge between them. The authors, assuming a specific rule about the landscape (the Non-Abelian Leopoldt Conjecture), built a solid, smooth highway. They drove across it and proved that the "distance markers" (L-invariants) on both sides are exactly the same.

This confirms that our understanding of these mathematical objects is consistent and unified, bringing us closer to solving some of the deepest mysteries in number theory.

1. Problem Statement and Context

The paper addresses two interconnected problems in arithmetic geometry and the Langlands program:

The Geometry of Eigenvarieties: Understanding the local structure of $p$ -adic eigenvarieties associated with reductive groups $G$ that do not admit discrete series representations (specifically when the "defect" $\ell(G) > 0$ ). In such cases, the cuspidal eigenvariety is not flat over the weight space, and classical points are often isolated within families of "non-classical" points.
Equality of L-Invariants: Establishing an equality between Fontaine–Mazur L-invariants (constructed from Galois representations) and automorphic L-invariants (constructed from $p$ -arithmetic cohomology) for $p$ -ordinary regular algebraic cuspidal automorphic representations (RACARs) $\pi$ of $\text{GL}_n(\mathbb{A})$ where the local component $\pi_p$ is Steinberg.

The authors aim to generalize previous results (known for $n=2, 3$ and groups with $\ell=0$ ) to arbitrary $n$ and groups with positive defect, under the assumption of the Non-Abelian Leopoldt Conjecture (NALC).

2. Methodology

The authors employ a combination of $p$ -adic Hodge theory, overconvergent cohomology, and spectral sequence techniques. The methodology is divided into two main pillars:

A. Geometric Structure of Eigenvarieties (The "Étaleness" Result)

Overconvergent Cohomology: The authors utilize Hansen's construction of eigenvarieties via overconvergent cohomology groups $H^*_c(K, D_\Omega)$ .
The Non-Abelian Leopoldt Conjecture (NALC): They assume the NALC, which predicts that the dimension of an irreducible component of the eigenvariety through a classical point is $\dim(\mathcal{W}) - \ell(G)$ .
Hansen–Thorne Techniques: They adapt arguments from Hansen and Thorne (specifically regarding Tor spectral sequences) to prove a freeness result for overconvergent cohomology.
Koszul Complexes: To handle the case where $\ell(G) > 0$ (where cohomology appears in multiple degrees), they utilize the Koszul complex associated with locally analytic principal series representations. This allows them to relate cohomology in different degrees and prove that the eigenvariety is étale over a specific Zariski-closed subspace of the weight space at non-critical classical points of "cohomological multiplicity one."

B. Comparison of L-Invariants (The "BCGS Formula")

The authors prove two distinct "Benois–Colmez–Greenberg–Stevens" (BCGS) formulas and equate them:

Automorphic BCGS Formula: They study the infinitesimal variation of Hecke eigenvalues along a tangent vector $v$ in the weight space. Using the Koszul complex and a Tor spectral sequence for locally analytic principal series, they show that the derivative of the character associated with the family, denoted $\partial \chi_v[c]$ , lies in the automorphic L-invariant subspace $L_{c}^{r, \varepsilon}(\pi)$ .
Galois BCGS Formula: They utilize their previous work on local-global compatibility for torsion classes to show that the Galois representation $\rho_\pi$ associated with a Hida family through $\pi$ is trianguline. By analyzing the parameters of this triangulation, they deduce that the same infinitesimal character $\partial \chi_v[c]$ lies in the Fontaine–Mazur L-invariant subspace $L_{c}^{FM}(\pi)$ .

3. Key Contributions and Theorems

Theorem A: Structure of Overconvergent Cohomology

Under the NALC and assuming cohomological multiplicity one, the authors prove:

There exists a rigid Zariski-closed subspace $\Sigma \subset \mathcal{W}$ of dimension $\dim \mathcal{W} - \ell(G)$ containing the classical weight.
The eigenvariety is étale over $\Sigma$ at the classical point.
The overconvergent cohomology groups $H^{q+i}_c(K, D_\Sigma)_{\leq h, \varepsilon}$ are free of rank $\binom{\ell(G)}{i}$ over the local ring $\mathcal{O}(\Sigma)_\lambda$ .
Significance: This generalizes étaleness results to the "defect" case ( $\ell > 0$ ), providing a robust geometric framework for $p$ -adic families in higher-rank groups.

Theorem B: Equality of L-Invariants

Let $\pi$ be a $p$ -ordinary RACAR of $\text{GL}_n$ with $\pi_p$ Steinberg. Assuming:

NALC holds for families through $\pi$ .
$\pi$ admits non- $c$ -parabolic deformations (a tangent vector $v$ exists such that $v_c \neq v_{c+1}$ ).
$p > 2n$ and the residual Galois representation is irreducible and decomposed generic.
The local Galois representation is non-split at $c$ .

Then, for all degrees $r$ and signs $\varepsilon$ :
$L_{c}^{FM}(\pi) = L_{c}^{r, \varepsilon}(\pi)$
Furthermore, if $v_c \neq v_{c+1}$ , these L-invariants are 1-dimensional and spanned by the infinitesimal character $\partial \chi_v[c]$ .

Corollary C: Independence of Degree and Sign

The NALC implies Gehrmann's Conjecture: The automorphic L-invariant $L_{c}^{r, \varepsilon}(\pi)$ is independent of the cohomological degree $r$ and the sign $\varepsilon$ .

Theorem D: Symmetric Power Functoriality

For $\pi = \text{Sym}^{n-1} f$ (the symmetric power lift of a modular form $f$ ):

The condition of non- $c$ -parabolic deformations is verified unconditionally.
Assuming NALC, the automorphic L-invariants of the symmetric power lift are equal to the L-invariant of the original modular form:
$L_{c}^{r, \varepsilon}(\text{Sym}^{n-1} \pi) = L_{1}^{FM}(\pi)$
This establishes a functoriality result for L-invariants.

4. Technical Innovations

Tor Spectral Sequence for Principal Series: The authors construct a new Tor spectral sequence relating the cohomology of locally analytic principal series representations over different affinoids. This is crucial for handling the "defect" $\ell > 0$ , where standard methods (which work for $\ell=0$ ) fail to control the cohomology in intermediate degrees.
Generalization of Gehrmann–Rosso: They generalize the strategy of Gehrmann and Rosso (originally for $\ell=0$ ) to the general case by replacing the reliance on specific degree arguments with the Tor spectral sequence and Koszul complex analysis.
Local-Global Compatibility at $p$ : They leverage recent results on the local-global compatibility of Galois representations attached to torsion classes to establish the trianguline nature of the Galois representation in families, bridging the gap between automorphic and Galois data.

5. Significance and Impact

Resolution of Exceptional Zero Conjectures: The equality of L-invariants is a prerequisite for proving the Greenberg–Benois exceptional zero conjecture (predicting the vanishing of $p$ -adic L-functions at central critical points). This paper provides the necessary equality for arbitrary $n$ and Steinberg local components, removing previous restrictions to small $n$ or self-dual cases.
Geometric Understanding of Eigenvarieties: Theorem A provides a definitive structural description of eigenvarieties for groups with positive defect, confirming that they behave like complete intersections locally, which is essential for constructing $p$ -adic L-functions in families for groups like $\text{GL}_3$ over imaginary quadratic fields.
Functoriality of L-Invariants: The result on symmetric powers (Theorem D) is a significant step toward understanding how L-invariants behave under Langlands functoriality, linking the arithmetic of modular forms to higher-rank automorphic forms.
Conditional Nature: The results are conditional on the Non-Abelian Leopoldt Conjecture. However, the paper demonstrates that if NALC holds, a vast array of arithmetic consequences (including independence of degree for L-invariants) follow naturally.

In summary, this paper unifies the geometric study of $p$ -adic families with the arithmetic of L-invariants, providing a powerful framework to tackle exceptional zero phenomena in the context of the Langlands program for $\text{GL}_n$ .