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Towards a generalized Maeda conjecture for modular forms with quadratic nebentypus

This paper extends the framework for lower-bounding the number of Galois orbits of newforms to the case of quadratic nebentypus by determining local inertial type counts and establishing Galois equivariance of Atkin-Li pseudo-eigenvalues, ultimately deriving a lower bound for non-CM orbits and revealing a strict inequality between local and global equivalences in small weights.

Original authors: Debargha Banerjee, Dhrubajyoti Das, Srijan Das, Tathagata Mandal, Sudipa Mondal

Published 2026-05-27

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CC BY 4.0

Original authors: Debargha Banerjee, Dhrubajyoti Das, Srijan Das, Tathagata Mandal, Sudipa Mondal

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 librarian trying to organize a massive, ever-growing collection of books. These aren't ordinary books; they are "modular forms," which are complex mathematical objects that behave like intricate, repeating patterns.

In this library, the books are grouped into "shelves" based on their weight (how heavy the pattern is) and their "nebentypus" (a specific type of label or signature attached to them). The main question the authors are asking is: As the books get heavier and heavier, how many distinct groups (or "orbits") of these books will we end up with?

This is a modern version of a famous guess called Maeda's Conjecture. The guess suggests that for certain simple libraries, all the heavy books eventually merge into just one giant group. But for more complex libraries (with specific labels), the books split into many groups. The authors want to predict exactly how many groups there will be.

Here is how they solve the puzzle, explained through simple analogies:

1. The Problem: Sorting the Books

The authors are looking at a specific type of library where the books have a "quadratic nebentypus." Think of this as a special label that can only be one of two things (like a red or blue sticker).

The Goal: Count the number of distinct groups of books as they get heavier.
The Challenge: You can't just count the books one by one because the library is infinite. You need a rule to predict the number of groups without looking at every single book.

2. The Strategy: Two ID Cards

To sort these books, the authors realized that every book has two "ID cards" that act as fingerprints. If two books have different ID cards, they must belong to different groups.

ID Card #1: The Local Inertial Type (The "Neighborhood ID")
Imagine every book lives in a specific neighborhood (a prime number). The "Local Inertial Type" describes the book's behavior right in that neighborhood.

The authors figured out how many different "neighborhood behaviors" are possible for these specific quadratic labels.
They created a catalog (Tables in the paper) that lists exactly how many unique neighborhood IDs exist for any given weight.

ID Card #2: The Atkin-Li Pseudo-eigenvalue (The "Signature")
This is a second piece of information, like a secret signature on the book.

In simpler libraries, this signature was a simple "Yes" or "No" (like a sign on a door).
In this complex library, the signature is a bit more fluid. It can change slightly depending on who is looking at it (mathematical "Galois conjugation"), but it changes in a predictable way.
The authors invented a new rule: If two signatures can be transformed into each other by this predictable change, they count as the same signature. This is like saying "Red" and "Dark Red" are the same color for the purpose of sorting.

3. The Twist: Symmetric vs. Asymmetric Groups

Here is where it gets tricky. Sometimes, a book's neighborhood ID allows for two possible signatures (let's call them Signature A and Signature B).

Symmetric Case: The library's rules say Signature A and Signature B are actually the same group. (They merge).
Asymmetric Case: The rules say Signature A and Signature B are different groups. (They stay separate).

The authors had to count how many neighborhood IDs fall into the "Symmetric" bucket and how many fall into the "Asymmetric" bucket. This count is crucial because it determines the final number of groups.

4. The Big Prediction (The Lower Bound)

The authors combined these two ID cards to create a formula. They calculated the total number of unique combinations of "Neighborhood ID" + "Signature."

The Main Result:
They proved that for very heavy books (large weights), the number of distinct groups will be at least the number of these unique combinations they calculated.

Think of it like saying: "No matter how the books are arranged, you will never have fewer than X groups."
They proved that for sufficiently heavy books, every single one of these theoretical combinations actually exists in the library.

5. The Surprise: The "Strict Inequality"

When the authors checked the library with small, light books (low weights), they found something interesting.

The Theory: "We predict at least 2 groups."
The Reality: "We actually found 3 groups."

Why? Because sometimes, the library's rules (the global Galois action) are stricter than the local rules. Two signatures that looked like they should be different based on the neighborhood ID actually turned out to be the same group when viewed from a distance. However, in some cases, the library's "Hecke field" (the collection of numbers describing the book) wasn't big enough to realize all the theoretical possibilities, leaving some "local" differences visible as "global" differences.

Summary

The paper is a mathematical census. The authors built a system to count the minimum number of distinct families of modular forms with quadratic labels.

They identified two key features (Local Type and Signature).
They counted how many unique combinations of these features are possible.
They proved that for heavy enough books, the library will contain at least that many distinct families.
They showed that for light books, the actual number can sometimes be higher than their minimum prediction, revealing that the local rules don't always tell the whole story.

This work extends a famous mathematical guess (Maeda's Conjecture) to a more complex type of library, providing a solid lower bound for how many distinct families of these mathematical objects exist.

Technical Summary: Towards a Generalized Maeda Conjecture for Modular Forms with Quadratic Nebentypus

Problem Statement
The paper addresses the asymptotic behavior of the number of Galois orbits of newforms in the space $S_k(\Gamma_0(N), \Psi)$ as the weight $k$ increases. While Maeda's conjecture predicts a single Galois orbit for level 1, higher levels generally yield multiple orbits. Previous work by Dieulefait, Pacetti, and Tsaknias (DPT21) established a lower bound for the number of non-CM Galois orbits, $N_{CM}(N, k)$ , for trivial nebentypus using local inertial types and Atkin-Lehner signs as global invariants. This paper extends that framework to the case of non-trivial quadratic nebentypus $\Psi$ . A primary challenge in this setting is that the classical Atkin-Lehner theory is not directly available, and the Galois action does not preserve the space $S_k(\Gamma_0(N), \Psi)$ unless $\Psi$ is Galois invariant (which holds for quadratic characters).

Methodology
The authors develop a local-global framework to count admissible pairs of invariants that define Galois orbits. The methodology proceeds in three main stages:

Local Analysis of Inertial Types:
The paper analyzes the Galois orbits of local inertial types for $GL_2(\mathbb{Q}_p)$ under the constraint of a quadratic central character. Using the local Langlands correspondence, the authors classify representations into principal series, Steinberg (special), and supercuspidal types. They explicitly compute the number of distinct Galois orbits of local inertial types, denoted $LT(p^n, \Psi)$ , for odd primes $p$ , distinguishing between tamely ramified and unramified local nebentypus cases. This involves detailed group-theoretic analysis of unit groups in quadratic extensions and the action of the absolute Galois group on primitive characters.
Global Invariants and Atkin-Li Pseudo-eigenvalues:
To refine the count, the authors introduce the Atkin-Li pseudo-eigenvalue as a second invariant. They establish the Galois equivariance of these pseudo-eigenvalues (Theorem 4.5), showing that under Galois conjugation $\sigma$ , the eigenvalue transforms as $\lambda_p(f^\sigma) = \Psi_p(\chi_p(\sigma))\sigma(\lambda_p(f))$ . Since $\Psi$ is quadratic, this leads to an equivalence relation $\sim$ on the unit circle $S^1$ .
Crucially, the authors address the issue that high ramification twists can obscure the original eigenvalue. They define the minimal Atkin-Li pseudo-eigenvalue (via minimal quadratic twists) to ensure the invariant is well-defined and Galois equivariant.
Theorem 4.13 determines when a local type uniquely fixes the pseudo-eigenvalue and when it allows two values differing by a sign. This leads to a classification of local types into symmetric orbits (where $\lambda \sim -\lambda$ ) and asymmetric orbits (where $\lambda \not\sim -\lambda$ ).
Existence and Realization:
The paper proves that every compatible pair of a local inertial type orbit and a minimal Atkin-Li pseudo-eigenvalue equivalence class is realized by a non-CM newform for sufficiently large weights. This relies on limit multiplicity results by Weinstein (for principal series) and Kim-Shin-Templier (for discrete series). A restriction is placed on the level $N$ : it must be a prime power or satisfy $v_p(N) \ge 3$ (odd) for all $p|N$ , ensuring that prescribed local representations can be fixed simultaneously via trace formula methods.

Key Contributions and Results

Extension of the Lower Bound Framework: The paper generalizes the DPT21 lower bound to quadratic nebentypus. It defines $LO(p^n, \Psi)$ as the number of compatible pairs of local-type Galois orbits and minimal Atkin-Li pseudo-eigenvalue equivalence classes.
Explicit Counting Formulas: The authors provide explicit formulas for $LT(p^n, \Psi)$ and $LO(p^n, \Psi)$ in Tables 4–7, covering cases for tamely ramified and unramified local nebentypus, and distinguishing between $p=3$ and $p \neq 3$ .
The Main Theorem (Theorem 1.1): For $N$ satisfying the specified conditions and $\Psi$ a quadratic character, there exists a $k_0$ such that for all $k \ge k_0$ :
$\prod_{p|N} LO(p^{v_p(N)}, \Psi) \le N_{CM}(N, k, \Psi)$
This establishes a concrete lower bound for the number of non-CM Galois orbits.
Symmetric vs. Asymmetric Orbits: A novel contribution is the distinction between symmetric and asymmetric orbits regarding the pseudo-eigenvalue sign. The authors show that while local analysis determines the number of types, the global Galois action determines whether the two possible signs ( $\lambda, -\lambda$ ) collapse into one invariant or remain distinct. This introduces a parameter $|S_{asym}|$ into the final counting formula, which cannot always be computed explicitly without global data.
Computational Evidence: Using data from the LMFDB, the authors demonstrate that for small weights, the strict inequality $LO < N_{CM}$ can occur. This indicates that local invariants that are distinct globally may be identified by the Galois action in a way that the local coefficient fields of small-weight forms cannot yet realize.

Significance and Claims
The paper claims to provide the first systematic lower bound for the number of Galois orbits in the quadratic nebentypus setting, filling a gap left by previous work on trivial nebentypus. The authors assert that their framework successfully adapts the "local-global" counting strategy to a setting where the central character imposes strict constraints and the classical Atkin-Lehner signs are replaced by pseudo-eigenvalues.

The paper remains modest regarding the exact asymptotic count. While it conjectures (based on the generalized Maeda philosophy) that the lower bound $LO(N)$ might eventually equal $N_{CM}(N, k, \Psi)$ , the computational evidence presented suggests that for quadratic nebentypus, the inequality can be strict even for large weights. This implies that certain local equivalences (specifically regarding the sign of the pseudo-eigenvalue) are not always realized globally by Galois conjugation over the coefficient fields of the modular forms, a phenomenon not present in the trivial nebentypus case. The work sets the stage for understanding the asymptotic behavior of modular forms with non-trivial characters, though the explicit determination of the asymmetric orbit contribution remains a challenge for general levels.