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Cuspidal subgroups associated with non-rational Eisenstein maximal ideals

This paper generalizes Ribet's conjecture on Eisenstein congruences to certain non-square-free levels $N$ by proving that such congruences exist even for non-rational Eisenstein maximal ideals, extending previous results that were limited to rational ideals.

Original authors: Debargha Banerjee, Narasimha Kumar, Dipramit Majumdar

Published 2026-06-26

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CC BY 4.0

Original authors: Debargha Banerjee, Narasimha Kumar, Dipramit Majumdar

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

The Big Picture: Finding Hidden Connections in Numbers

Imagine you are a detective trying to solve a mystery in the world of numbers. Specifically, you are looking at Modular Forms. You can think of these as two different types of musical instruments playing on a complex stage:

Cusp Forms: These are like soloists. They are complex, intricate, and "vanish" at the edges of the stage (the cusps). They are hard to predict and very mysterious.
Eisenstein Series: These are like a choir. They are structured, predictable, and follow a strict, rhythmic pattern. They are the "easy" part of the music.

For a long time, mathematicians knew that sometimes, if you listen closely enough (specifically, if you look at the numbers modulo a certain prime), the soloist (Cusp Form) starts to sound exactly like the choir (Eisenstein Series). This is called a congruence.

The famous mathematician Ramanujan discovered this first. He found that for a specific level of complexity (Level 1, Weight 12), the soloist and the choir sound identical if you ignore the number 691.

The Problem: The "Rational" vs. "Non-Rational" Mystery

Previous research (by mathematicians like Mazur, Ribet, and Yoo) successfully explained when this happens for the "Rational" choir members. These are the choir members whose notes are simple whole numbers or fractions.

However, there is a whole other section of the choir: the Non-Rational members. Their notes involve complex numbers (like square roots of negative numbers or roots of unity). For a long time, no one knew exactly when these complex choir members would start sounding like the soloists.

The Paper's Goal: The authors (Banerjee, Kumar, and Majumdar) wanted to solve this mystery. They wanted to create a rulebook that predicts exactly which "primes" (the special numbers that act as the filter for the congruence) will cause a Non-Rational choir member to mimic a soloist.

The Solution: Building a New Map

To solve this, the authors had to build a new map. Here is how they did it, using an analogy:

1. The "Cuspidal Subgroup" (The Crowd Control)
Imagine the stage (the modular curve) has a specific area where the audience sits, called the "Cusps." The Cuspidal Subgroup is like a specific group of audience members who have special tickets. The size of this group (how many people are in it) tells us how likely a congruence is to happen.

Old Method: Previous mathematicians tried to count this crowd by looking at a "Shimura subgroup" (a specific VIP section) and guessing the rest. This was messy and didn't work well for the complex, Non-Rational choir.
New Method: The authors developed a way to count the entire crowd directly, without relying on the VIP section guesswork. They calculated the exact size of this group for the Non-Rational choir members.

2. The "Eisenstein Series" (The Choir)
The authors constructed specific "Non-Rational" choir members (Eisenstein series) that are tailored to the complex rules of the stage. They used a technique called "refinement" (like tuning an instrument) to create these specific versions.

3. The "Congruence" (The Match)
Once they knew the exact size of the crowd (the Cuspidal Subgroup), they could predict the "magic numbers" (primes).

The Rule: If a prime number divides the size of this crowd, then a congruence exists. The soloist and the Non-Rational choir member will sound the same modulo that prime.

The Main Findings

The paper makes three major claims:

We can predict the magic numbers: They proved that for a wide range of complex levels (specifically levels that are squares of primes, like $N=121$ or $N=725$ ), we can calculate the exact size of the "Cuspidal Subgroup" associated with these Non-Rational choir members.
The connection is real: They confirmed a conjecture by Ribet. If you find a Non-Rational maximal ideal (a specific mathematical condition), there is always a non-zero crowd (Cuspidal Subgroup) associated with it. This means the congruence is guaranteed to exist for the right prime.
We can see it in action: The authors didn't just do the math on paper; they used computers (SAGE and LMFDB) to find real examples.
- Example: They looked at Level 121 ( $11^2$ ). They found a Non-Rational choir member and a soloist that sound identical when you divide by 5.
- Example: They looked at Level 725 ( $5^2 \times 29$ ) and found matches modulo 7.

Why This Matters (In the Context of the Paper)

The paper is a "proof of concept" for a difficult area of number theory.

Before this, we had a map for the simple (Rational) choir.
Now, the authors have drawn the first reliable map for the complex (Non-Rational) choir.
They showed that even though these numbers are complex, the rules governing their "congruences" (their hidden connections to the soloists) are just as predictable as the simple ones, provided you know how to count the crowd correctly.

Summary Analogy

Think of the universe of numbers as a giant library.

Cusp Forms are the rare, handwritten manuscripts.
Eisenstein Series are the mass-produced books.
Congruences are the moments where a handwritten manuscript looks exactly like a mass-produced book if you squint at it through a specific colored filter (the prime number).

For a long time, we knew how to find these matches for the standard mass-produced books. This paper says: "We have now figured out how to find these matches for the weird, complex mass-produced books too. We built a new counting machine to measure the library's structure, and we used it to find specific examples where the rare manuscripts and the complex books match up."

Technical Summary: Cuspidal Subgroups Associated with Non-Rational Eisenstein Maximal Ideals

Problem Statement
The paper addresses the generalization of Ramanujan-like congruences—congruences between cusp forms and Eisenstein series—for congruence subgroups $\Gamma_0(N)$ where $N$ is an arbitrary positive integer. While Ramanujan's original discovery concerned level $N=1$ and weight 12, and subsequent work by Ogg, Mazur, and Ribet established deep connections for square-free levels, the behavior of these congruences for general $N$ (specifically those with square factors) and for non-rational Eisenstein maximal ideals remained less understood.

A central challenge is determining the primes $\ell$ that can serve as the residual characteristic of an Eisenstein maximal ideal $\mathfrak{m}$ and understanding the structure of the cuspidal subgroup $C_{\Gamma_0(N)}$ associated with such ideals. Previous results, such as Ribet's conjecture (proved by Yoo for square-free $N$ and rational ideals), linked the non-triviality of the cuspidal subgroup to the existence of these congruences. However, extending these results to non-rational ideals (where the associated Galois representation is reducible but not split into the trivial character and the cyclotomic character) and to levels with square factors ( $N$ not square-free) required new methods, particularly because existing techniques often relied on the Shimura subgroup, which limits the ability to compute the order of the entire cuspidal group.

Methodology
The authors employ a strategy rooted in the seminal work of Mazur on Eisenstein ideals, adapted for non-rational settings and general levels. The methodology proceeds through several key stages:

Classification of Ideals: The paper first classifies Eisenstein maximal ideals of the Hecke algebra $T(N)$ into rational and non-rational types based on the reducibility of the associated Galois representation $\rho_{\mathfrak{m}}$ . For non-rational ideals with residual characteristic $\ell$ coprime to $6N_1$ (where $N_1$ is the square-full part of $N$ ), the authors associate a unique non-trivial Dirichlet character $\epsilon_{\mathfrak{m}}$ .
Construction of Eisenstein Series: Instead of relying on the Shimura subgroup as in previous works (e.g., Ren), the authors construct specific non-rational Eisenstein series $E_{\phi, M, L}$ of weight 2. These series are derived from a base Eisenstein series $E_\phi$ (associated with a Dirichlet character $\phi$ ) via "ordinary" and "critical" refinements at primes dividing the level $N$ . This construction allows for precise control over the Fourier coefficients and the associated Hecke eigenvalues.
Boundary Map and Cuspidal Divisors: The core technical innovation lies in computing the boundary of these Eisenstein series. The authors calculate the divisor $\delta_{\Gamma_0(N)}(E_{\phi, M, L})$ supported on the cusps of the modular curve $X_0(N)$ . By analyzing the ramification indices of degeneracy maps and utilizing properties of Gauss sums and Bernoulli numbers, they derive an explicit formula for the constant $\beta_{\Gamma_0(N), \phi, M, L}$ that scales the divisor.
Base Change and Module Structure: To handle the non-rational nature of the ideals, the authors perform a base change to a ring $R = \mathbb{Z}[1/6N, \zeta_f, \phi]$ , where $\zeta_f$ is a root of unity related to the conductor of the character. They prove that the cuspidal subgroup associated with the Eisenstein series, when tensored with $R$ , is a cyclic $R$ -module isomorphic to $R/(\beta_{\Gamma_0(N), \phi, M, L})$ . This computation is performed directly, bypassing the dependency on the Shimura subgroup that limited previous approaches.
Connection to Congruences: Using the computed order of the cuspidal subgroup, the authors relate the support of the Eisenstein ideal to the primes dividing the order of the cuspidal subgroup. They establish that if a prime $\ell$ divides the order of the cuspidal subgroup, it must divide the index of the Eisenstein ideal, thereby confirming the existence of congruences between the Eisenstein series and cusp forms.

Key Contributions and Results

Generalization of Ribet's Conjecture: The paper proves a generalization of Ribet's conjecture for non-square-free levels $N$ and non-rational Eisenstein maximal ideals. Specifically, Theorem 1.3 states that for an Eisenstein maximal ideal $\mathfrak{m}$ with residual characteristic $\ell$ coprime to $6N$ , the $\ell$ -torsion of the cuspidal subgroup $C_{\Gamma_0(N)}[\mathfrak{m}]$ is non-zero.
Explicit Order of Cuspidal Subgroups: Theorem 1.2 provides an explicit description of the order of the cuspidal subgroup associated with a non-rational Eisenstein series $E_{\phi, M, L}$ . The authors show that the $R$ -module $C_{\Gamma_0(N)}(E_{\phi, M, L}) \otimes R$ is cyclic and generated by a specific constant $\beta_{\Gamma_0(N), \phi, M, L}$ derived from $L$ -values and Gauss sums.
Classification of Residual Characteristics: Proposition 6.1 identifies the set of possible residual characteristics $\ell$ for non-rational Eisenstein ideals. These primes must divide $6N$ , or belong to sets $S_1(N)$ (primes dividing $q^2-1$ for $q|N$ ) or $S_2(N)$ (primes dividing specific values involving generalized Bernoulli numbers).
Explicit Examples: The authors provide concrete examples of Eisenstein congruences for non-rational series at square-full levels (e.g., $N=121, 725, 234$ ). Using computational tools (SAGE and LMFDB), they demonstrate congruences between constructed Eisenstein series and newforms modulo specific primes, verifying the theoretical predictions.

Significance and Claims
The paper claims to be the first to provide explicit examples of Eisenstein congruences for non-rational Eisenstein series at square-full levels. Its primary significance lies in overcoming the limitations of previous methods that depended on the Shimura subgroup, which could only compute the order of certain components of the cuspidal group. By modifying Stevens' arguments and working directly with the boundary of Eisenstein series over a carefully chosen ring $R$ , the authors achieve a complete determination of the cuspidal subgroup's order for these cases.

The work confirms the Faltings-Jordan conjecture for non-rational Eisenstein maximal ideals of weight 2, establishing a correspondence between these ideals and characteristic-zero Eisenstein series. Furthermore, it extends the understanding of Ramanujan-like congruences beyond the square-free case, offering a robust framework to predict which primes can produce such congruences for general levels. The authors emphasize that their results are a "routine generalization" of Mazur's techniques in the context of the base-changed ring, yet the explicit computation of the cuspidal subgroup order independent of the Shimura subgroup represents a novel contribution to the field.